| 1 | C4×C5×AGL(1,𝔽7) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | ΓL(1,𝔽8)⋊#168 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | ΓL(1,𝔽8)×#168 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10587838 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 5 | C2×C5×AGL(1,𝔽7) | C4 | 1 | 1 | yes | characteristic |
| 6 | C4×C5×ΓL(1,𝔽8) | C22 | 1 | 1 | yes | characteristic |
| 7 | C5×ΓL(1,𝔽8)⋊C4 | C4 | 1 | 1 | yes | characteristic |
| 8 | #1131⋊C3 | — | 5 | 1 | no | maximal, Hall |
| 9 | C4×C5×D7 | C2×C3 | 1 | 1 | yes | characteristic |
| 10 | C7⋊#168 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C7×#168 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 12 | C3×#681 | — | 7 | 1 | no | maximal, Hall |
| 13 | C2×C5×ΓL(1,𝔽8) | C2×C4 | 1 | 1 | yes | characteristic |
| 14 | C5×AGL(1,𝔽7) | — | 2 | 1 | no | — |
| 15 | C4×AGL(1,𝔽7) | D5 | 1 | 1 | yes | characteristic |
| 16 | C8×ΓL(1,𝔽8) | — | 5 | 1 | no | — |
| 17 | ΓL(1,𝔽8)⋊C8 | — | 5 | 1 | no | — |
| 18 | C2×C5×D7 | C3×C4 | 1 | 1 | yes | characteristic |
| 19 | C4×C5×C7 | C22×C3 | 1 | 1 | yes | characteristic |
| 20 | C5×Dic7 | C3×C4 | 1 | 1 | yes | characteristic |
| 21 | C2×C3×C4×C5 | — | 7 | 1 | no | — |
| 22 | C3×#168 | — | 7 | 1 | no | — |
| 23 | C3×#168 | — | 7 | 1 | no | — |
| 24 | #1131 | — | 5 | 1 | no | Hall |
| 25 | C5×ΓL(1,𝔽8) | M16 | 1 | 1 | yes | characteristic, complemented, Hall |
| 26 | C2×AGL(1,𝔽7) | Dic5 | 1 | 1 | yes | characteristic |
| 27 | C4×ΓL(1,𝔽8) | C2×D5 | 1 | 1 | yes | characteristic |
| 28 | ΓL(1,𝔽8)⋊C4 | Dic5 | 1 | 1 | yes | characteristic |
| 29 | #681 | — | 7 | 1 | no | Hall |
| 30 | C2×C5×C7 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 31 | C5×D7 | — | 2 | 1 | no | — |
| 32 | C22×C3×C5 | — | 7 | 1 | no | — |
| 33 | C3×C4×C5 | — | 7 | 1 | no | — |
| 34 | C3×C4×C5 | — | 7 | 1 | no | — |
| 35 | C4×D7 | C3×D5 | 1 | 1 | yes | characteristic |
| 36 | C7×C8 | — | 5 | 1 | no | — |
| 37 | C7⋊C8 | — | 5 | 1 | no | — |
| 38 | C3×M16 | — | 35 | 1 | no | Hall |
| 39 | C2×ΓL(1,𝔽8) | C2×Dic5 | 1 | 1 | yes | characteristic |
| 40 | AGL(1,𝔽7) | — | 2 | 1 | no | — |
| 41 | C2×C4×C5 | — | 7 | 1 | no | — |
| 42 | #168 | — | 7 | 1 | no | — |
| 43 | #168 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 44 | C5×C7 | C3×M16 | 1 | 1 | yes | characteristic, complemented, Hall |
| 45 | C2×C3×C5 | — | 7 | 1 | no | — |
| 46 | C2×C3×C5 | — | 14 | 1 | no | — |
| 47 | C2×D7 | C3×Dic5 | 1 | 1 | yes | characteristic |
| 48 | C4×C7 | C2×C3×D5 | 1 | 1 | yes | characteristic |
| 49 | Dic7 | C3×Dic5 | 1 | 1 | yes | characteristic |
| 50 | C2×C3×C4 | — | 7 | 1 | no | — |
| 51 | C3×C8 | — | 35 | 1 | no | — |
| 52 | C3×C8 | — | 35 | 1 | no | — |
| 53 | ΓL(1,𝔽8) | #681 | 1 | 1 | yes | characteristic, complemented, Hall |
| 54 | C22×C5 | — | 7 | 1 | no | — |
| 55 | C4×C5 | — | 7 | 1 | no | — |
| 56 | C4×C5 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 57 | M16 | — | 35 | 1 | no | Sylow |
| 58 | C3×C5 | — | 7 | 1 | no | Hall |
| 59 | C2×C7 | C2×C3×Dic5 | 1 | 1 | yes | characteristic |
| 60 | D7 | — | 2 | 1 | no | — |
| 61 | C22×C3 | — | 7 | 1 | no | — |
| 62 | C3×C4 | — | 7 | 1 | no | — |
| 63 | C3×C4 | — | 7 | 1 | no | — |
| 64 | C2×C5 | — | 14 | 1 | no | — |
| 65 | C2×C5 | C4×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 66 | C2×C4 | — | 7 | 1 | no | — |
| 67 | C8 | — | 5 | 1 | no | — |
| 68 | C8 | — | 35 | 1 | no | — |
| 69 | C7 | C3×#681 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 70 | C2×C3 | — | 7 | 1 | no | — |
| 71 | C2×C3 | — | 14 | 1 | no | — |
| 72 | C5 | #1131⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 73 | C22 | — | 7 | 1 | no | — |
| 74 | C4 | — | 7 | 1 | no | — |
| 75 | C4 | D5×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 76 | C3 | — | 7 | 1 | no | Sylow |
| 77 | C2 | — | 14 | 1 | no | — |
| 78 | C2 | Dic5×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
Finite Group Databasebeta