Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D277 |
| commutator, socle, cosocle | C22×C277 | C24 |
| Fitting | #125×C277 | C2 |
| prime radical | C2×D1108 | C22 |
| hypercenter | #125 | D277 |
| hypocenter | C277 | #554 |
| LCS | G ▷ C22×C277 ▷ C277 | |
| UCS | C1 ◁ C22 ◁ #125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22579879 | C2 | semidirect |
| 2. #22579871 | C2 | semidirect |
| 3. #22579860 | C2 | semidirect |
| 4. #22579858 | C2 | semidirect |
| 5. #22579858 | C2 | semidirect |
| 6. C4×Dic554 | C2 | semidirect |
| 10. C4⋊C4×D277 | C2 | semidirect |
| 11. #125×C277 | C2 | semidirect |
| 38. D1108 | Q8 | semidirect |
| 70. C277 | #554 | semidirect |
| 74. #125 | D277 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22579879 | 2 |
| #22579871 | 2 |
| #22579860 | 2 |
| #22579858 | 2 |
| C4×Dic554 | 2 |
| C4×D1108 | 2 |
| #22579842 | 2 |
| C4⋊C4×D277 | 2 |
| #125×C277 | 2 |
| #554 | 277 |