Finite Group DatabasebetaFinite Group DatabasebetaOrder 10404.
26 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C172⋊C3 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | #14579585 | — | 4 | 1 | no | maximal, Hall |
| 3 | C2×C172⋊C3 | — | 3 | 1 | no | — |
| 4 | C22×C172 | C9 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C172⋊C3 | A4 | 1 | 1 | yes | characteristic |
| 6 | C2×C172 | — | 3 | 1 | no | — |
| 7 | C172 | C22⋊C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | C22×C17 | — | 9 | 2 | no | — |
| 9 | C22⋊C9 | — | 289 | 1 | no | maximal, Hall |
| 10 | C2×C17 | — | 9 | 6 | no | — |
| 11 | C17 | — | 9 | 2 | no | — |
| 12 | C22×C3 | — | 289 | 1 | no | — |
| 13 | C9 | — | 1156 | 1 | no | Sylow |
| 14 | C2×C3 | — | 867 | 1 | no | — |
| 15 | C22 | #14579585 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | C3 | — | 289 | 1 | no | — |
| 17 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C172 | C9 |
| prime radical, cosocle | C22×C172⋊C3 | C3 |
| LCS | G ▷ C22×C172 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C22×C172 | C9 | semidirect |
| 7. C172 | C22⋊C9 | semidirect |
| 15. C22 | #14579585 | semidirect |