Finite Group DatabasebetaFinite Group DatabasebetaOrder 22592.
238 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #126×C353 | C2 |
| hypercenter | #126 | D353 |
| hypocenter | C353 | #553 |
| LCS | G ▷ C22×C353 ▷ C353 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26014987 | C2 | semidirect |
| 2. #26014982 | C2 | semidirect |
| 3. #26014977 | C2 | semidirect |
| 4. #26014976 | C2 | semidirect |
| 5. #26014967 | C2 | semidirect |
| 6. #26014965 | C2 | semidirect |
| 7. #26014965 | C2 | semidirect |
| 8. #26014965 | C2 | semidirect |
| 9. C4×D1412 | C2 | semidirect |
| 10. #26014954 | C2 | semidirect |
| 11. #26014952 | C2 | semidirect |
| 12. #26014952 | C2 | semidirect |
| 13. #26014949 | C2 | semidirect |
| 14. C4⋊C4×D353 | C2 | semidirect |
| 15. #126×C353 | C2 | semidirect |
| 109. C353 | #553 | semidirect |
| 112. #126 | D353 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26014987 | 2 |
| #26014982 | 2 |
| #26014977 | 2 |
| #26014976 | 2 |
| #26014967 | 2 |
| #26014965 | 2 |
| C4×D1412 | 2 |
| #26014954 | 2 |
| #26014952 | 2 |
| #26014949 | 2 |
| C4⋊C4×D353 | 2 |
| #126×C353 | 2 |
| #553 | 353 |