Finite Group DatabasebetaFinite Group DatabasebetaOrder 23456.
76 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#19154462 | C2 | 1 | 6 | yes | maximal, direct product factor |
| 2 | C22×C4×C733 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #19154462 | C22 | 1 | 4 | yes | direct product factor |
| 4 | C23×C733 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×C4×C733 | C4 | 1 | 3 | yes | — |
| 6 | C2×C4×C733 | C22 | 1 | 3 | yes | — |
| 7 | C22×C733 | C8 | 1 | 4 | yes | complemented |
| 8 | C22×C733 | C2×C4 | 1 | 3 | yes | — |
| 9 | C4×C733 | C2×C4 | 1 | 3 | yes | — |
| 10 | C4×C733 | C23 | 1 | 1 | yes | characteristic |
| 11 | C2×C733 | C2×C8 | 1 | 6 | yes | complemented |
| 12 | C2×C733 | C22×C4 | 1 | 1 | yes | characteristic |
| 13 | C733 | C22×C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C22×C8 | — | 733 | 1 | no | maximal, Sylow |
| 15 | C22×C4 | D733 | 1 | 1 | yes | characteristic |
| 16 | C2×C8 | — | 733 | 6 | no | — |
| 17 | C23 | Dic733 | 1 | 1 | yes | characteristic |
| 18 | C2×C4 | Dic733 | 1 | 3 | yes | — |
| 19 | C2×C4 | C2×D733 | 1 | 3 | yes | — |
| 20 | C8 | — | 733 | 4 | no | — |
| 21 | C22 | #19154462 | 1 | 4 | yes | direct product factor |
| 22 | C22 | C2×Dic733 | 1 | 3 | yes | — |
| 23 | C4 | C2×Dic733 | 1 | 3 | yes | — |
| 24 | C4 | C22×D733 | 1 | 1 | yes | characteristic |
| 25 | C2 | C2×#19154462 | 1 | 6 | yes | direct product factor |
| 26 | C2 | C22×Dic733 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | D733 |
| commutator, hypocenter | C733 | C22×C8 |
| Frattini | C4 | C22×D733 |
| Fitting | C22×C4×C733 | C2 |
| socle, prime radical | C23×C733 | C4 |
| cosocle | C4×C733 | C23 |
| LCS | G ▷ C733 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19154462 | C2 | direct (N × H) |
| 3. #19154462 | C22 | direct (N × H) |
| 7. C22×C733 | C8 | semidirect |
| 11. C2×C733 | C2×C8 | semidirect |
| 13. C733 | C22×C8 | semidirect |
| 21. C22 | #19154462 | direct (N × H) |
| 25. C2 | C2×#19154462 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19154462 | 2 |
| C22×C4×C733 | 2 |
| C22×C8 | 733 |