Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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 | #117×C277 | C2 |
| hypercenter | #117 | D277 |
| hypocenter | C277 | #555 |
| LCS | G ▷ C22×C277 ▷ C277 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22579879 | C2 | semidirect |
| 2. #22579868 | C2 | semidirect |
| 3. #22579866 | C2 | semidirect |
| 4. #22579855 | C2 | semidirect |
| 5. #22579854 | C2 | semidirect |
| 6. #22579853 | C2 | semidirect |
| 7. #22579842 | C2 | semidirect |
| 8. C4×C22⋊D277 | C2 | semidirect |
| 9. #22579838 | C2 | semidirect |
| 10. #22579837 | C2 | semidirect |
| 11. #117×C277 | C2 | semidirect |
| 78. C277 | #555 | semidirect |
| 90. #117 | D277 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22579879 | 2 |
| #22579868 | 2 |
| #22579866 | 2 |
| #22579855 | 2 |
| #22579854 | 2 |
| #22579853 | 2 |
| #22579842 | 2 |
| C4×C22⋊D277 | 2 |
| #22579838 | 2 |
| #22579837 | 2 |
| #117×C277 | 2 |
| #555 | 277 |