Finite Group DatabasebetaFinite Group DatabasebetaOrder 17728.
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 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 | C22⋊Q8×C277 | C2 |
| hypercenter | C22⋊Q8 | D277 |
| hypocenter | C277 | #562 |
| LCS | G ▷ C22×C277 ▷ C277 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22579870 | C2 | semidirect |
| 2. #22579869 | C2 | semidirect |
| 3. #22579866 | C2 | semidirect |
| 4. #22579865 | C2 | semidirect |
| 5. #22579865 | C2 | semidirect |
| 6. #22579863 | C2 | semidirect |
| 7. #22579862 | C2 | semidirect |
| 8. #22579860 | C2 | semidirect |
| 9. #22579858 | C2 | semidirect |
| 10. #22579856 | C2 | semidirect |
| 11. #22579855 | C2 | semidirect |
| 12. #22579854 | C2 | semidirect |
| 13. #22579853 | C2 | semidirect |
| 14. #22579853 | C2 | semidirect |
| 15. C22⋊Q8×C277 | C2 | semidirect |
| 109. C277 | #562 | semidirect |
| 121. C22⋊Q8 | D277 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22579870 | 2 |
| #22579869 | 2 |
| #22579866 | 2 |
| #22579865 | 2 |
| #22579863 | 2 |
| #22579862 | 2 |
| #22579860 | 2 |
| #22579858 | 2 |
| #22579856 | 2 |
| #22579855 | 2 |
| #22579854 | 2 |
| #22579853 | 2 |
| C22⋊Q8×C277 | 2 |
| #562 | 277 |