Finite Group DatabasebetaFinite Group DatabasebetaOrder 8228.
84 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C112×D17 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C112×C17 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | C11×#595 | C11 | 1 | 12 | yes | maximal, direct product factor |
| 4 | C4×C112 | — | 17 | 1 | no | maximal, Hall |
| 5 | C11×D17 | C2×C11 | 1 | 12 | yes | — |
| 6 | C2×C112 | — | 17 | 1 | no | — |
| 7 | C11×C17 | C4×C11 | 1 | 12 | yes | complemented |
| 8 | C112 | #595 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | #595 | C112 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 10 | C4×C11 | — | 17 | 12 | no | — |
| 11 | D17 | C2×C112 | 1 | 1 | yes | characteristic |
| 12 | C2×C11 | — | 17 | 12 | no | — |
| 13 | C17 | C4×C112 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C11 | C11×#595 | 1 | 12 | yes | direct product factor |
| 15 | C4 | — | 17 | 1 | no | Sylow |
| 16 | C2 | — | 17 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C112 | #595 |
| commutator, hypocenter | C17 | C4×C112 |
| Frattini | C1 | G |
| Fitting, socle | C112×C17 | C4 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C112 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C112×C17 | C4 | semidirect |
| 3. C11×#595 | C11 | direct (N × H) |
| 7. C11×C17 | C4×C11 | semidirect |
| 8. C112 | #595 | direct (N × H) |
| 9. #595 | C112 | direct (N × H) |
| 13. C17 | C4×C112 | semidirect |
| 14. C11 | C11×#595 | direct (N × H) |