Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 93 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 | C22×D277 | C23 |
| hypercenter | #125 | D277 |
| hypocenter | C277 | #541 |
| 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 |
| 6. #22579871 | C2 | semidirect |
| 7. #22579870 | C2 | semidirect |
| 8. #22579870 | C2 | semidirect |
| 9. #22579870 | C2 | semidirect |
| 12. C4×Dic554 | C2 | semidirect |
| 13. #22579845 | C2 | semidirect |
| 15. #125×C277 | C2 | semidirect |
| 91. C277 | #541 | semidirect |
| 102. #125 | D277 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22579879 | 2 |
| #22579875 | 2 |
| #22579874 | 2 |
| #22579871 | 2 |
| #22579870 | 2 |
| #22579858 | 2 |
| C4×Dic554 | 2 |
| #22579845 | 2 |
| #22579838 | 2 |
| #125×C277 | 2 |
| #541 | 277 |