| 1 | C2×C5×ΓL(1,𝔽8)×C71 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×Dic497 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C5×ΓL(1,𝔽8)×C71 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | ΓL(1,𝔽8)⋊Dic71 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | C2×C5×C7×C71 | C2×C3 | 1 | 1 | yes | characteristic |
| 6 | C3×C5×Dic71 | — | 7 | 1 | no | maximal, Hall |
| 7 | C2×ΓL(1,𝔽8)×C71 | C2×C5 | 1 | 1 | yes | characteristic |
| 8 | C5×C7×C71 | C3×C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 9 | C2×C3×C5×C71 | — | 7 | 1 | no | — |
| 10 | Dic497 | C3×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 11 | ΓL(1,𝔽8)×C71 | C4×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 12 | C5×Dic71 | — | 7 | 1 | no | Hall |
| 13 | C3×C5×C71 | — | 7 | 1 | no | Hall |
| 14 | C2×C7×C71 | C2×C3×C5 | 1 | 1 | yes | characteristic |
| 15 | C3×Dic71 | — | 7 | 1 | no | Hall |
| 16 | C2×C5×C71 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 17 | C7×C71 | C3×C4×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 18 | C2×C3×C71 | — | 7 | 1 | no | — |
| 19 | C5×ΓL(1,𝔽8)⋊C4 | — | 71 | 1 | no | maximal, Hall |
| 20 | C5×C71 | ΓL(1,𝔽8)⋊C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 21 | Dic71 | — | 7 | 1 | no | Hall |
| 22 | C3×C71 | — | 7 | 1 | no | Hall |
| 23 | C2×C5×ΓL(1,𝔽8) | D71 | 1 | 1 | yes | characteristic |
| 24 | C2×C71 | C5×AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 25 | C5×Dic7 | — | 71 | 1 | no | Hall |
| 26 | C5×ΓL(1,𝔽8) | Dic71 | 1 | 1 | yes | characteristic, complemented, Hall |
| 27 | ΓL(1,𝔽8)⋊C4 | — | 71 | 1 | no | Hall |
| 28 | C71 | C5×ΓL(1,𝔽8)⋊C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 29 | C2×C5×C7 | C3×D71 | 1 | 1 | yes | characteristic |
| 30 | C3×C4×C5 | — | 497 | 1 | no | Hall |
| 31 | C2×ΓL(1,𝔽8) | C5×D71 | 1 | 1 | yes | characteristic |
| 32 | C5×C7 | C3×Dic71 | 1 | 1 | yes | characteristic, complemented, Hall |
| 33 | C2×C3×C5 | — | 7 | 1 | no | — |
| 34 | Dic7 | — | 71 | 1 | no | Hall |
| 35 | ΓL(1,𝔽8) | C5×Dic71 | 1 | 1 | yes | characteristic, complemented, Hall |
| 36 | C4×C5 | — | 497 | 1 | no | Hall |
| 37 | C3×C5 | — | 7 | 1 | no | Hall |
| 38 | C2×C7 | C3×C5×D71 | 1 | 1 | yes | characteristic |
| 39 | C3×C4 | — | 497 | 1 | no | Hall |
| 40 | C2×C5 | ΓL(1,𝔽8)⋊D71 | 1 | 1 | yes | characteristic |
| 41 | C7 | C3×C5×Dic71 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 42 | C2×C3 | — | 7 | 1 | no | — |
| 43 | C5 | ΓL(1,𝔽8)⋊Dic71 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 44 | C4 | — | 497 | 1 | no | Sylow |
| 45 | C3 | — | 7 | 1 | no | Sylow |
| 46 | C2 | C5×ΓL(1,𝔽8)⋊D71 | 1 | 1 | yes | characteristic |
Finite Group Databasebeta