Finite Group DatabasebetaFinite Group DatabasebetaOrder 27830.
40 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×#10597244 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C23×C11⋊D11 | C5 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C112×C23 | C2×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C23×AGL(1,𝔽11) | — | 11 | 2 | no | maximal |
| 5 | C23×C11⋊C5 | — | 11 | 2 | no | — |
| 6 | #11925894 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 7 | #10597244 | C2×C23 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | D11×C23 | — | 11 | 2 | no | — |
| 9 | D11×C23 | — | 55 | 2 | no | — |
| 10 | C11×C23 | — | 5 | 2 | no | — |
| 11 | C11×C23 | AGL(1,𝔽11) | 1 | 2 | yes | complemented |
| 12 | C11⋊D11 | C5×C23 | 1 | 1 | yes | characteristic, complemented, Hall |
| 13 | C2×C5×C23 | — | 121 | 1 | no | Hall |
| 14 | C112 | C2×C5×C23 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C5×C23 | — | 121 | 1 | no | Hall |
| 16 | AGL(1,𝔽11) | — | 11 | 2 | no | — |
| 17 | C11⋊C5 | — | 11 | 2 | no | — |
| 18 | C2×C23 | — | 121 | 1 | no | Hall |
| 19 | C23 | #11925894 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 20 | D11 | — | 11 | 2 | no | — |
| 21 | D11 | — | 55 | 2 | no | — |
| 22 | C11 | — | 5 | 2 | no | — |
| 23 | C11 | C23×AGL(1,𝔽11) | 1 | 2 | yes | complemented |
| 24 | C2×C5 | — | 121 | 1 | no | Hall |
| 25 | C5 | — | 121 | 1 | no | Sylow |
| 26 | C2 | — | 121 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #11925894 |
| commutator, cosocle, hypocenter | C112 | C2×C5×C23 |
| Frattini | C1 | G |
| Fitting, socle | C112×C23 | C2×C5 |
| LCS | G ▷ C112 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#10597244 | C2 | semidirect |
| 2. C23×C11⋊D11 | C5 | semidirect |
| 3. C112×C23 | C2×C5 | semidirect |
| 6. #11925894 | C23 | direct (N × H) |
| 7. #10597244 | C2×C23 | semidirect |
| 11. C11×C23 | AGL(1,𝔽11) | semidirect |
| 12. C11⋊D11 | C5×C23 | semidirect |
| 14. C112 | C2×C5×C23 | semidirect |
| 19. C23 | #11925894 | direct (N × H) |
| 23. C11 | C23×AGL(1,𝔽11) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#10597244 | 2 |
| C23×C11⋊D11 | 5 |
| C23×AGL(1,𝔽11) | 11 |
| #11925894 | 23 |