Finite Group DatabasebetaFinite Group DatabasebetaOrder 23210.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #26042440 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D11×C211 | C5 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C11×C211 | C2×C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C2×C211⋊C5 | — | 11 | 1 | no | maximal, Hall |
| 5 | C211⋊C5 | — | 11 | 1 | no | Hall |
| 6 | C2×C211 | — | 55 | 1 | no | Hall |
| 7 | C211 | AGL(1,𝔽11) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | AGL(1,𝔽11) | — | 211 | 1 | no | maximal, Hall |
| 9 | C11⋊C5 | — | 211 | 1 | no | Hall |
| 10 | D11 | C211⋊C5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 11 | C11 | C2×C211⋊C5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C5 | — | 2321 | 1 | no | Hall |
| 13 | C5 | — | 2321 | 1 | no | Sylow |
| 14 | C2 | — | 55 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C11×C211 | C2×C5 |
| LCS | G ▷ C11×C211 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26042440 | C2 | semidirect |
| 2. D11×C211 | C5 | semidirect |
| 3. C11×C211 | C2×C5 | semidirect |
| 7. C211 | AGL(1,𝔽11) | semidirect |
| 10. D11 | C211⋊C5 | semidirect |
| 11. C11 | C2×C211⋊C5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26042440 | 2 |
| D11×C211 | 5 |
| C2×C211⋊C5 | 11 |
| AGL(1,𝔽11) | 211 |