Finite Group DatabasebetaFinite Group DatabasebetaOrder 6111.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #14523272 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14523271 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C7×C97⋊C3 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | ΓL(1,𝔽8)×C97 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×C97⋊C3 | — | 7 | 1 | no | maximal, Hall |
| 6 | C7×C97 | C32 | 1 | 1 | yes | characteristic, complemented, Hall |
| 7 | C3×C97 | — | 7 | 1 | no | — |
| 8 | C97⋊C3 | — | 7 | 1 | no | — |
| 9 | C97⋊C3 | — | 7 | 1 | no | — |
| 10 | C97⋊C3 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 11 | C97 | C3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C3×ΓL(1,𝔽8) | — | 97 | 1 | no | maximal, Hall |
| 13 | C3×C7 | — | 97 | 1 | no | — |
| 14 | ΓL(1,𝔽8) | — | 97 | 1 | no | — |
| 15 | ΓL(1,𝔽8) | — | 97 | 1 | no | — |
| 16 | ΓL(1,𝔽8) | C97⋊C3 | 1 | 1 | yes | characteristic, direct product factor |
| 17 | C32 | — | 679 | 1 | no | Sylow |
| 18 | C7 | C3×C97⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | C3 | — | 7 | 1 | no | — |
| 20 | C3 | — | 97 | 1 | no | — |
| 21 | C3 | — | 679 | 1 | no | — |
| 22 | C3 | — | 679 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C97 | C32 |
| LCS | G ▷ C7×C97 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14523272 | C3 | semidirect |
| 2. #14523271 | C3 | semidirect |
| 3. C7×C97⋊C3 | C3 | semidirect |
| 4. ΓL(1,𝔽8)×C97 | C3 | semidirect |
| 6. C7×C97 | C32 | semidirect |
| 10. C97⋊C3 | ΓL(1,𝔽8) | direct (N × H) |
| 11. C97 | C3×ΓL(1,𝔽8) | semidirect |
| 16. ΓL(1,𝔽8) | C97⋊C3 | direct (N × H) |
| 18. C7 | C3×C97⋊C3 | semidirect |