Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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 | #126×C277 | C2 |
| prime radical | C2×C22⋊D277 | C22 |
| hypercenter | #126 | D277 |
| hypocenter | C277 | #563 |
| LCS | G ▷ C22×C277 ▷ C277 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22579880 | C2 | semidirect |
| 2. #22579879 | C2 | semidirect |
| 3. #22579875 | C2 | semidirect |
| 4. #22579874 | C2 | semidirect |
| 5. #22579873 | C2 | semidirect |
| 6. #22579871 | C2 | semidirect |
| 7. #22579870 | C2 | semidirect |
| 11. #22579860 | C2 | semidirect |
| 12. #22579858 | C2 | semidirect |
| 13. #22579857 | C2 | semidirect |
| 14. #22579857 | C2 | semidirect |
| 15. #126×C277 | C2 | semidirect |
| 97. C277 | #563 | semidirect |
| 104. #126 | D277 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22579880 | 2 |
| #22579879 | 2 |
| #22579875 | 2 |
| #22579874 | 2 |
| #22579873 | 2 |
| #22579871 | 2 |
| #22579870 | 2 |
| #22579866 | 2 |
| #22579864 | 2 |
| #22579862 | 2 |
| #22579860 | 2 |
| #22579858 | 2 |
| #22579857 | 2 |
| #126×C277 | 2 |
| #563 | 277 |