Finite Group DatabasebetaFinite Group DatabasebetaOrder 22407.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C11×C97 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C11×C97⋊C3 | — | 7 | 1 | no | maximal, Hall |
| 3 | #14523271 | C11 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | C11×C97 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C7×C97 | C3×C11 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C97⋊C3 | — | 7 | 1 | no | Hall |
| 7 | C11×ΓL(1,𝔽8) | — | 97 | 1 | no | maximal, Hall |
| 8 | C97 | C11×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C7×C11 | C97⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | C3×C11 | — | 679 | 1 | no | Hall |
| 11 | ΓL(1,𝔽8) | — | 97 | 1 | no | Hall |
| 12 | C11 | #14523271 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 13 | C7 | C11×C97⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C3 | — | 679 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C11 | #14523271 |
| commutator, cosocle, hypocenter | C7×C97 | C3×C11 |
| Frattini | C1 | G |
| Fitting, socle | C7×C11×C97 | C3 |
| LCS | G ▷ C7×C97 | |
| UCS | C1 ◁ C11 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C11×C97 | C3 | semidirect |
| 3. #14523271 | C11 | direct (N × H) |
| 4. C11×C97 | ΓL(1,𝔽8) | semidirect |
| 5. C7×C97 | C3×C11 | semidirect |
| 8. C97 | C11×ΓL(1,𝔽8) | semidirect |
| 9. C7×C11 | C97⋊C3 | semidirect |
| 12. C11 | #14523271 | direct (N × H) |
| 13. C7 | C11×C97⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C11×C97 | 3 |
| C11×C97⋊C3 | 7 |
| #14523271 | 11 |
| C11×ΓL(1,𝔽8) | 97 |