| 1 | C2×C3×ΓL(1,𝔽8)×C71 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×Dic3×C71 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C4×ΓL(1,𝔽8)×C71 | — | 3 | 1 | no | maximal |
| 4 | C3×ΓL(1,𝔽8)×C71 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C2×ΓL(1,𝔽8)×C71 | — | 2 | 1 | no | — |
| 6 | C2×ΓL(1,𝔽8)×C71 | D3 | 1 | 1 | yes | characteristic |
| 7 | C2×C3×C7×C71 | C2×C3 | 1 | 1 | yes | characteristic |
| 8 | C3×Dic3×C71 | — | 7 | 1 | no | maximal, Hall |
| 9 | C4×C7×C71 | — | 3 | 1 | no | Hall |
| 10 | C3×C7×C71 | C3×C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | ΓL(1,𝔽8)×C71 | — | 2 | 1 | no | — |
| 12 | ΓL(1,𝔽8)×C71 | Dic3 | 1 | 1 | yes | characteristic, direct product factor |
| 13 | C2×C32×C71 | — | 7 | 1 | no | — |
| 14 | C2×C7×C71 | C3×D3 | 1 | 1 | yes | characteristic |
| 15 | C3×C4×C71 | — | 21 | 1 | no | — |
| 16 | Dic3×C71 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 17 | C32×C71 | — | 7 | 1 | no | Hall |
| 18 | C7×C71 | C3×Dic3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 19 | C2×C3×C71 | — | 7 | 1 | no | — |
| 20 | C2×C3×C71 | — | 14 | 1 | no | — |
| 21 | C2×C3×C71 | C2×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 22 | C4×C71 | — | 3 | 1 | no | Hall |
| 23 | Dic3×ΓL(1,𝔽8) | C71 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 24 | C3×C71 | — | 7 | 1 | no | — |
| 25 | C3×C71 | — | 14 | 1 | no | — |
| 26 | C3×C71 | C4×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 27 | C2×C71 | D3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 28 | C2×C3×ΓL(1,𝔽8) | C2×C71 | 1 | 1 | yes | characteristic |
| 29 | C7×Dic3 | C3×C71 | 1 | 1 | yes | characteristic, complemented |
| 30 | C4×ΓL(1,𝔽8) | — | 3 | 1 | no | — |
| 31 | C71 | Dic3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 32 | C3×ΓL(1,𝔽8) | C4×C71 | 1 | 1 | yes | characteristic, complemented, Hall |
| 33 | C2×C3×C7 | C2×C3×C71 | 1 | 1 | yes | characteristic |
| 34 | C2×ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 35 | C2×ΓL(1,𝔽8) | D3×C71 | 1 | 1 | yes | characteristic |
| 36 | C3×Dic3 | — | 7 | 1 | no | Hall |
| 37 | C4×C7 | — | 3 | 1 | no | Hall |
| 38 | C3×C7 | C3×C4×C71 | 1 | 1 | yes | characteristic, complemented |
| 39 | ΓL(1,𝔽8) | — | 2 | 1 | no | — |
| 40 | ΓL(1,𝔽8) | Dic3×C71 | 1 | 1 | yes | characteristic, direct product factor |
| 41 | C2×C32 | — | 7 | 1 | no | — |
| 42 | C2×C7 | C3×D3×C71 | 1 | 1 | yes | characteristic |
| 43 | C3×C4 | — | 21 | 1 | no | — |
| 44 | Dic3 | ΓL(1,𝔽8)×C71 | 1 | 1 | yes | characteristic, direct product factor |
| 45 | C32 | — | 7 | 1 | no | Sylow |
| 46 | C7 | C3×Dic3×C71 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 47 | C2×C3 | — | 7 | 1 | no | — |
| 48 | C2×C3 | — | 14 | 1 | no | — |
| 49 | C2×C3 | C2×ΓL(1,𝔽8)×C71 | 1 | 1 | yes | characteristic |
| 50 | C4 | — | 3 | 1 | no | Sylow |
| 51 | C3 | — | 7 | 1 | no | — |
| 52 | C3 | — | 14 | 1 | no | — |
| 53 | C3 | C4×ΓL(1,𝔽8)×C71 | 1 | 1 | yes | characteristic, complemented |
| 54 | C2 | D3×ΓL(1,𝔽8)×C71 | 1 | 1 | yes | characteristic |
Finite Group Databasebeta