Finite Group DatabasebetaFinite Group DatabasebetaOrder 10404.
58 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#14579583 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #19104513 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×C17×D51 | — | 3 | 1 | no | maximal |
| 4 | #14579583 | C22 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C2×C3×C172 | D3 | 1 | 1 | yes | characteristic |
| 6 | C17×D51 | — | 3 | 2 | no | — |
| 7 | C2×C17×D17 | — | 9 | 1 | no | Hall |
| 8 | C3×C172 | C2×D3 | 1 | 1 | yes | characteristic |
| 9 | C2×C172 | D9 | 1 | 1 | yes | characteristic, complemented |
| 10 | C17×D17 | — | 9 | 2 | no | — |
| 11 | C172 | C2×D9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×D51 | — | 3 | 1 | no | — |
| 13 | C2×D3×C17 | — | 51 | 1 | no | — |
| 14 | C2×C3×C17 | — | 3 | 1 | no | — |
| 15 | C2×C3×C17 | — | 3 | 1 | no | — |
| 16 | C2×C3×C17 | — | 6 | 2 | no | — |
| 17 | D51 | — | 3 | 2 | no | — |
| 18 | D3×C17 | — | 51 | 2 | no | — |
| 19 | C22×C17 | — | 153 | 1 | no | — |
| 20 | C2×D17 | — | 9 | 1 | no | — |
| 21 | C3×C17 | — | 3 | 1 | no | — |
| 22 | C3×C17 | — | 3 | 1 | no | — |
| 23 | C3×C17 | — | 6 | 2 | no | — |
| 24 | C2×D9 | — | 289 | 1 | no | maximal, Hall |
| 25 | C2×C17 | — | 3 | 1 | no | — |
| 26 | C2×C17 | — | 3 | 1 | no | — |
| 27 | C2×C17 | — | 6 | 2 | no | — |
| 28 | C2×C17 | — | 153 | 2 | no | — |
| 29 | D17 | — | 9 | 2 | no | — |
| 30 | C2×C9 | — | 289 | 1 | no | — |
| 31 | D9 | — | 289 | 2 | no | — |
| 32 | C17 | — | 3 | 1 | no | — |
| 33 | C17 | — | 3 | 1 | no | — |
| 34 | C17 | — | 6 | 2 | no | — |
| 35 | C2×D3 | — | 51 | 1 | no | — |
| 36 | C9 | — | 289 | 1 | no | Sylow |
| 37 | C2×C3 | #13181892 | 1 | 1 | yes | characteristic |
| 38 | D3 | — | 51 | 2 | no | — |
| 39 | C22 | — | 153 | 1 | no | Sylow |
| 40 | C3 | C2×#13181892 | 1 | 1 | yes | characteristic |
| 41 | C2 | — | 153 | 2 | no | — |
| 42 | C2 | #19104513 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #19104513 |
| commutator, cosocle, hypocenter | #14579583 | C22 |
| Frattini | C3 | C2×#13181892 |
| Fitting, socle | C2×C3×C172 | D3 |
| LCS | G ▷ #14579583 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14579583 | C2 | semidirect |
| 2. #19104513 | C2 | direct (N × H) |
| 4. #14579583 | C22 | semidirect |
| 9. C2×C172 | D9 | semidirect |
| 11. C172 | C2×D9 | semidirect |
| 42. C2 | #19104513 | direct (N × H) |