Finite Group DatabasebetaFinite Group DatabasebetaOrder 22592.
236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 236 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 | #123×C353 | C2 |
| prime radical | #26014988 | C2 |
| hypercenter | #123 | D353 |
| hypocenter | C353 | #559 |
| LCS | G ▷ C22×C353 ▷ C353 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #26014978 | C2 | semidirect |
| 3. #26014978 | C2 | semidirect |
| 4. #26014977 | C2 | semidirect |
| 5. #26014973 | C2 | semidirect |
| 6. #26014957 | C2 | semidirect |
| 7. C4×C22⋊D353 | C2 | semidirect |
| 8. #26014952 | C2 | semidirect |
| 9. #123×C353 | C2 | semidirect |
| 67. C353 | #559 | semidirect |
| 74. #123 | D353 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26014988 | 2 |
| #26014978 | 2 |
| #26014977 | 2 |
| #26014973 | 2 |
| #26014957 | 2 |
| C4×C22⋊D353 | 2 |
| #26014952 | 2 |
| #123×C353 | 2 |
| #559 | 353 |