CHICAGO -- The White Sox have 34 players who have worn their uniform and eventually been elected to the Hall of Fame, with that number increasing by two when Minnie Miñoso and Jim Kaat were elected via the 16-member Golden Days Era Committee on Dec. 5, 2021. But who are the franchise’s best players not in the Hall? Here’s a look at the top five, with a few others in consideration.

To be eligible for this distinction, the player must be retired and currently eligible for the Hall of Fame ballot. And it’s not about greatest tenure with the White Sox, but there must be some connection.

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1) Dick Allen (1963-77)
Miñoso was No. 1 on this list, and the super-talented Allen just missed joining him in leaving this list with 11 votes from the Golden Days Era Committee in 2021. He needed 12 for Hall of Fame selection.

Allen hit 351 home runs, stole 133 bases and had a slash line of .292/.378/.534 over 15 seasons. The first baseman’s legacy with the White Sox was not exactly a long one, playing in 348 games over three years after coming to the White Sox on Dec. 2, 1971, in a trade sending Tommy John and Steve Huntz to the Dodgers. But Allen's 1972 season, which is now one of five MVP performances in franchise history after Abreu, was dominant enough on its own. His spectacular performance, propelling the White Sox to an 87-67 record, helped bring the franchise back to life.

The right-handed-hitting slugger led the AL with 37 home runs and 113 RBIs, and he finished a mere 10 points behind Rod Carew’s .318 mark from winning the Triple Crown. Tales of Allen’s high-level on-field ability came from those who played with him, with his White Sox tenure resulting in 85 home runs, a .307 average, a .988 OPS and three All-Star appearances. In ’72, Allen also was tops in the AL in offensive bWAR, slugging percentage, on-base percentage and OPS, and he tied for the AL lead in walks and extra-base hits. Allen somehow never topped 18.9% in Hall of Fame support and would also need to be a Veterans Committee selection at this point.

“He just did legendary things on the field,” said Kaat of his former teammate and good friend. Kaat pitched from 1973-75 with the White Sox and won 20 games twice. “I hope I'm around for the next five-year cycle when Dick gets in, because he certainly deserves it.”

2) Mark Buehrle (2000-15)
Buehrle stayed on the Hall of Fame ballot for another year by garnering 5.8 percent of the vote and 23 votes in his second appearance in 2022. He earned 11 percent of the vote and 44 votes during his Hall of Fame ballot debut in '21. The veteran southpaw presents an interesting case with 214 victories, 33 complete games, 10 shutouts, four Gold Gloves and five All-Star appearances. An argument could be made for Buehrle as one of the most popular players to put on a White Sox uniform -- both among teammates and fans.

After all, there aren’t many White Sox players who have a résumé that includes a perfect game and a perfect rain-delay tarp dive early in his career. Buehrle joined the White Sox rotation in 2001, and for the next 14 seasons, he won at least 10 games, pitched at least 200 innings and made at least 30 starts. He just missed a 15th straight season in 2015 with the Blue Jays, throwing 198 2/3 innings. He had a career average fastball velocity of 85.6 mph, per Fangraphs, but Buehrle simply could pitch.

His 161 victories rank sixth in franchise history, and he has the unique accomplishment of starting Game 2 of the 2005 World Series and allowing four runs over seven innings and 100 pitches, then earning the save in Game 3 by recording the final out in the 14th inning. Buehrle, a 38th-round pick in the 1998 Draft, had his No. 56 jersey retired in 2017. He made nine Opening Day starts for the White Sox, throwing a no-hitter against Texas in '07 and then that perfect game against the Rays in '09, leading to a then-record of 45 straight hitters retired.

3) Billy Pierce (1949-61)
Although he didn’t begin his career with the White Sox, arriving in a trade with Detroit on Nov. 10, 1948, Pierce finished with the most victories by a White Sox southpaw as part of his 211 career wins. His White Sox tenure featured 186 victories, 35 shutouts and a 3.19 ERA over 13 seasons, 456 games and 391 starts. Pierce also pitched in three games during the 1959 World Series without allowing a run.

Pierce never topped 200 strikeouts in a season, but his total of 1,796 (among 1,999 for his career) rank No. 1 in White Sox history. He ranks in the top 10 for the White Sox in 11 statistical categories, including starts (third), innings pitched (fourth), shutouts (third) and quality starts (third). The seven-time All-Star remained a part of the White Sox organization long after he retired (Pierce died in 2015), and he is remembered as a classy individual who was a popular presence with fans and members of the club.

As a side note, Pierce had 60 RBIs with 13 doubles, two triples and one stolen base over his time with the White Sox.

4) Paul Konerko (1999-2014)
Konerko received 10 votes, or 2.5%, in his first and only year on the ballot in 2020. But even without that Hall of Fame recognition, Konerko’s numbers over 16 seasons with the White Sox and 18 overall make him a true standout presence.

Although he had a couple of chances to leave via free agency, Konerko stayed with the White Sox and served as the team’s last captain, as named by then-manager Ozzie Guillen. Konerko played 1,843 games at first base for the White Sox and finished second in franchise history with 432 home runs (439 for his career). He topped all White Sox hitters with 4,010 total bases and 13 seasons of at least 20 home runs. Konerko also was a consummate clubhouse leader and served as the voice of the franchise.

He caught the last out of the 2005 World Series sweep over Houston and then presented said baseball to White Sox chairman Jerry Reinsdorf at the victory celebration. Konerko’s No. 14 jersey was retired in 2015, after his ballpark sculpture was unveiled in ’14.

5) Robin Ventura (1989-98)
Ventura is arguably the greatest third baseman in franchise history with 171 homers, 741 RBIs and five Gold Gloves during his decade with the White Sox. He was the team’s top pick in the 1988 Draft and also managed the club for five seasons with a 375-435 career record and a second-place finish in the AL Central in 2012.

Others worth considering
• Sammy Sosa: Sosa, who hit 609 home runs in his career, was more of a five-tool talent during his three years with the White Sox from 1989-91 before being traded across town to the Cubs.

• Tommy John: Eighty-two of his 288 victories came over seven seasons with the White Sox. The surgery bearing his name should be enough to get John serious Hall of Fame consideration, as it changed the game.

• Albert Belle: Belle is better known as part of the dominant Cleveland teams of the '90s. But he hit 79 of his 381 homers and had 268 of his 1,239 RBIs in two years with the White Sox.

• Magglio Ordonez: Konerko often referred to Ordonez as his most talented teammate overall during his White Sox tenure. Ordonez’s 187 home runs rank sixth in franchise history, to go with his 240 doubles, .307 average and .889 OPS over 1,001 games.