Finite Group DatabasebetaFinite Group DatabasebetaOrder 23276.
60 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #26042787 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #19154120 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | #14523747 | C11 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 4 | C23⋊D23 | C2×C11 | 1 | 1 | yes | characteristic |
| 5 | C232 | C4×C11 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | AGL(1,𝔽23) | — | 46 | 12 | no | — |
| 7 | C23⋊C11 | — | 46 | 12 | no | — |
| 8 | D23 | — | 46 | 12 | no | — |
| 9 | C4×C11 | — | 529 | 1 | no | maximal, Hall |
| 10 | C23 | — | 2 | 12 | no | — |
| 11 | C2×C11 | — | 529 | 1 | no | — |
| 12 | C11 | — | 529 | 1 | no | Sylow |
| 13 | C4 | — | 529 | 1 | no | Sylow |
| 14 | C2 | — | 529 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C232 | C4×C11 |
| prime radical | #26042787 | C2 |
| cosocle | C23⋊D23 | C2×C11 |
| LCS | G ▷ C232 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #19154120 | C4 | semidirect |
| 3. #14523747 | C11 | semidirect |
| 5. C232 | C4×C11 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26042787 | 2 |
| #14523747 | 11 |
| C4×C11 | 529 |