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The Indians are effectively out of playoff contention, but most of the Cleveland minor-league clubs are either in line to make the postseason or in serious contention for a spot.
Each league does the playoffs a little bit differently, so I'll be listing the playoff format along with the number of games left.
Columbus (AAA)
Current record: 61-49 (2.5 games back division, 2.5 game lead wild card)
33 games left
There are four teams that make the playoffs in the International League: the three division winners and the best second-place club. The Clippers are in a pretty good division to make the playoffs, assuming of course that their roster doesn't get plundered by the big-league club.
Playoff chances: Good
Akron (AA)
Current record: 57-53 (5.0 games back division, 1.0 game back of second place)
31 games left
The Eastern League has two divisions, and the top two teams in each division will make the playoffs. Note that even if the second-place club in one division has a worse record the third-place club in the other division, the second-place club will qualify for the playoffs. The Ducks are a game behind Altoona and Richmond for second place in the EL Western Division.
Playoff chances: Decent
Lynchburg (A+)
Second-half record: 21-18 (1.0 game division lead)
31 games left
In the Carolina League (and the Midwest League), the schedule is divided into two halves; the first division winners in both halves qualifies for the playoffs. Because in this part of the minors, a lot of prospects get promoted halfway into the season, this gives teams a chance to make the playoffs after getting new players. If one team wins the division in both the first and second halves, then the second-place team in the second half will go to the playoffs.
Lynchburg is the only minor-league affiliate leading its division, with its main competition being the Frederick Keys.
Playoff chances: Good
Lake County (A-)
Second-half record: 18-19 (4.5 games back division, 2.0 games back of second place)
32 games left
The Midwest League also follows the first half/second half format. Eight teams make the playoffs, so if you finish in first or second place in your division in either half of the season, you'll make the playoffs. If the same team qualifies for the playoffs in both halves, then the third-place team in the second half will qualify.
The Captains are one of bunch of clubs fighting for the second place position, so making the playoffs will be a tall order.
Playoff chances: Unlikely
Mahoning Valley (Short-season A)
Current record: 20-22 (7.0 games back division, 2.5 games back wild card)
33 games left
Four teams make the playoffs in the NYPL, made up of the three division winners plus the best second-place club. The Scrappers are in a difficult position, as they are in a group of four teams behind in the wild card leader.
Playoff chances: Unlikely
AZL Indians (Rookie)
Second-half record: 2-5 (3.0 games back division)
20 games left
Despite being a short-season league, the AZL has a first-half and second-half format, with six teams (the first and second-winners winners of the three division) making an abbreviated playoffs (single-elimination).The AZL Indians didn't win their division in the first half of the season, and are 3 games behind the division leader with about three weeks left in the regular season.
Playoff Chances: Unlikely
DSL Indians (Rookie)
Current record: 29-27 (4.5 games back division, 9.0 games back wild card)
15 games left
In the DSL, each of the five division winners plus the best second-place team makes the playoffs. The DSL Indians have a better shot at winning their division than qualifying via the wild card spot, but there's just over two weeks left to play, so there isn't much time to make up that much ground.
Playoff Chances: Very unlikely