| 1 | C2×C7×ΓL(1,𝔽8)×C53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×Dic371 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C7×ΓL(1,𝔽8)×C53 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C2×C72×C53 | C2×C3 | 1 | 1 | yes | characteristic |
| 5 | ΓL(1,𝔽8)⋊Dic53 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 6 | C3×C7×Dic53 | — | 7 | 1 | no | maximal |
| 7 | C72×C53 | C3×C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | C2×ΓL(1,𝔽8)×C53 | C2×C7 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×C7×C53 | — | 7 | 1 | no | — |
| 10 | Dic371 | C3×C7 | 1 | 1 | yes | characteristic, complemented |
| 11 | C7×Dic53 | — | 7 | 1 | no | — |
| 12 | C3×C7×C53 | — | 7 | 1 | no | — |
| 13 | ΓL(1,𝔽8)×C53 | C4×C7 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×C7×C53 | — | 6 | 1 | no | — |
| 15 | C2×C7×C53 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 16 | C2×C7×C53 | C2×C3×C7 | 1 | 1 | yes | characteristic |
| 17 | C3×Dic53 | — | 7 | 1 | no | Hall |
| 18 | C7×ΓL(1,𝔽8)⋊C4 | — | 53 | 1 | no | maximal, Hall |
| 19 | C7×C53 | — | 6 | 1 | no | — |
| 20 | C7×C53 | ΓL(1,𝔽8)⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 21 | C7×C53 | C3×C4×C7 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2×C3×C53 | — | 7 | 1 | no | — |
| 23 | C2×C7×ΓL(1,𝔽8) | D53 | 1 | 1 | yes | characteristic |
| 24 | Dic53 | — | 7 | 1 | no | Hall |
| 25 | C7×Dic7 | — | 53 | 1 | no | Hall |
| 26 | C3×C53 | — | 7 | 1 | no | Hall |
| 27 | C7×ΓL(1,𝔽8) | Dic53 | 1 | 1 | yes | characteristic, complemented, Hall |
| 28 | C2×C53 | C7×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 29 | C2×C72 | C3×D53 | 1 | 1 | yes | characteristic |
| 30 | C3×C4×C7 | — | 371 | 1 | no | — |
| 31 | ΓL(1,𝔽8)⋊C4 | — | 53 | 1 | no | — |
| 32 | C53 | C7×ΓL(1,𝔽8)⋊C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 33 | C72 | C3×Dic53 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 34 | C2×C3×C7 | — | 7 | 1 | no | — |
| 35 | C2×ΓL(1,𝔽8) | C7×D53 | 1 | 1 | yes | characteristic |
| 36 | C4×C7 | — | 371 | 1 | no | — |
| 37 | Dic7 | — | 53 | 1 | no | — |
| 38 | C3×C7 | — | 7 | 1 | no | — |
| 39 | ΓL(1,𝔽8) | C7×Dic53 | 1 | 1 | yes | characteristic, complemented |
| 40 | C2×C7 | — | 6 | 1 | no | — |
| 41 | C2×C7 | C3×C7×D53 | 1 | 1 | yes | characteristic |
| 42 | C2×C7 | ΓL(1,𝔽8)⋊D53 | 1 | 1 | yes | characteristic |
| 43 | C3×C4 | — | 371 | 1 | no | Hall |
| 44 | C7 | — | 6 | 1 | no | — |
| 45 | C7 | C3×C7×Dic53 | 1 | 1 | yes | characteristic, complemented |
| 46 | C7 | ΓL(1,𝔽8)⋊Dic53 | 1 | 1 | yes | characteristic, direct product factor |
| 47 | C2×C3 | — | 7 | 1 | no | — |
| 48 | C4 | — | 371 | 1 | no | Sylow |
| 49 | C3 | — | 7 | 1 | no | Sylow |
| 50 | C2 | C7×ΓL(1,𝔽8)⋊D53 | 1 | 1 | yes | characteristic |
Finite Group Databasebeta