Finite Group DatabasebetaFinite Group DatabasebetaOrder 22242.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C11×C337 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D3707 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C11×C337 | C2×C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | C3×D337 | — | 11 | 1 | no | maximal, Hall |
| 5 | C3×C337 | D11 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | D337 | — | 11 | 1 | no | Hall |
| 7 | C337 | C3×D11 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | C3×D11 | — | 337 | 1 | no | maximal, Hall |
| 9 | C3×C11 | D337 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | D11 | — | 337 | 1 | no | Hall |
| 11 | C11 | C3×D337 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C3 | — | 3707 | 1 | no | Hall |
| 13 | C3 | D3707 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C2 | — | 3707 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3707 |
| commutator, cosocle, hypocenter | C11×C337 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C3×C11×C337 | C2 |
| LCS | G ▷ C11×C337 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C11×C337 | C2 | semidirect |
| 2. D3707 | C3 | direct (N × H) |
| 3. C11×C337 | C2×C3 | semidirect |
| 5. C3×C337 | D11 | semidirect |
| 7. C337 | C3×D11 | semidirect |
| 9. C3×C11 | D337 | semidirect |
| 11. C11 | C3×D337 | semidirect |
| 13. C3 | D3707 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C11×C337 | 2 |
| D3707 | 3 |
| C3×D337 | 11 |
| C3×D11 | 337 |