Finite Group DatabasebetaFinite Group DatabasebetaOrder 22592.
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×D353 |
| commutator, socle, cosocle | C22×C353 | C24 |
| Fitting | #126×C353 | C2 |
| prime radical | C2×C22⋊D353 | C22 |
| hypercenter | #126 | D353 |
| hypocenter | C353 | #563 |
| 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. #26014992 | C2 | semidirect |
| 2. #26014991 | C2 | semidirect |
| 3. #26014987 | C2 | semidirect |
| 4. #26014986 | C2 | semidirect |
| 5. #26014985 | C2 | semidirect |
| 6. #26014983 | C2 | semidirect |
| 7. #26014982 | C2 | semidirect |
| 11. #26014972 | C2 | semidirect |
| 12. #26014970 | C2 | semidirect |
| 13. #26014969 | C2 | semidirect |
| 14. #26014969 | C2 | semidirect |
| 15. #126×C353 | C2 | semidirect |
| 97. C353 | #563 | semidirect |
| 104. #126 | D353 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26014992 | 2 |
| #26014991 | 2 |
| #26014987 | 2 |
| #26014986 | 2 |
| #26014985 | 2 |
| #26014983 | 2 |
| #26014982 | 2 |
| #26014978 | 2 |
| #26014976 | 2 |
| #26014974 | 2 |
| #26014972 | 2 |
| #26014970 | 2 |
| #26014969 | 2 |
| #126×C353 | 2 |
| #563 | 353 |