Finite Group DatabasebetaFinite Group DatabasebetaOrder 22592.
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×D353 |
| commutator, socle, cosocle | C22×C353 | C24 |
| Fitting | C22⋊Q8×C353 | C2 |
| hypercenter | C22⋊Q8 | D353 |
| hypocenter | C353 | #562 |
| LCS | G ▷ C22×C353 ▷ C353 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26014982 | C2 | semidirect |
| 2. #26014981 | C2 | semidirect |
| 3. #26014978 | C2 | semidirect |
| 4. #26014977 | C2 | semidirect |
| 5. #26014977 | C2 | semidirect |
| 6. #26014975 | C2 | semidirect |
| 7. #26014974 | C2 | semidirect |
| 8. #26014972 | C2 | semidirect |
| 9. #26014970 | C2 | semidirect |
| 10. #26014968 | C2 | semidirect |
| 11. #26014967 | C2 | semidirect |
| 12. #26014966 | C2 | semidirect |
| 13. #26014965 | C2 | semidirect |
| 14. #26014965 | C2 | semidirect |
| 15. C22⋊Q8×C353 | C2 | semidirect |
| 109. C353 | #562 | semidirect |
| 121. C22⋊Q8 | D353 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26014982 | 2 |
| #26014981 | 2 |
| #26014978 | 2 |
| #26014977 | 2 |
| #26014975 | 2 |
| #26014974 | 2 |
| #26014972 | 2 |
| #26014970 | 2 |
| #26014968 | 2 |
| #26014967 | 2 |
| #26014966 | 2 |
| #26014965 | 2 |
| C22⋊Q8×C353 | 2 |
| #562 | 353 |