Finite Group DatabasebetaFinite Group DatabasebetaOrder 23457.
8 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C1117 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C1117⋊C3 | — | 7 | 1 | no | maximal, Hall |
| 3 | C1117 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | ΓL(1,𝔽8) | — | 1117 | 1 | no | maximal, Hall |
| 5 | C7 | C1117⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C3 | — | 7819 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C1117 | C3 |
| LCS | G ▷ C7×C1117 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C1117 | C3 | semidirect |
| 3. C1117 | ΓL(1,𝔽8) | semidirect |
| 5. C7 | C1117⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C1117 | 3 |
| C1117⋊C3 | 7 |
| ΓL(1,𝔽8) | 1117 |