Finite Group DatabasebetaFinite Group DatabasebetaOrder 32226.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C41×C131 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D5371 | — | 3 | 1 | no | maximal, Hall |
| 3 | C41×C131 | D3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | D393 | — | 41 | 1 | no | maximal, Hall |
| 5 | C3×C131 | D41 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | D131 | — | 123 | 1 | no | Hall |
| 7 | D123 | — | 131 | 1 | no | maximal, Hall |
| 8 | C131 | D123 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C3×C41 | D131 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | D41 | — | 393 | 1 | no | Hall |
| 11 | C41 | D393 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | D3 | — | 5371 | 1 | no | Hall |
| 13 | C3 | D5371 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 16113 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C41×C131 | C2 |
| LCS | G ▷ C3×C41×C131 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C41×C131 | C2 | semidirect |
| 3. C41×C131 | D3 | semidirect |
| 5. C3×C131 | D41 | semidirect |
| 8. C131 | D123 | semidirect |
| 9. C3×C41 | D131 | semidirect |
| 11. C41 | D393 | semidirect |
| 13. C3 | D5371 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C41×C131 | 2 |
| D5371 | 3 |
| D393 | 41 |
| D123 | 131 |