Finite Group DatabasebetaFinite Group DatabasebetaOrder 23488.
216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×D367 |
| commutator, socle, cosocle | C22×C367 | C24 |
| Fitting | #123×C367 | C2 |
| prime radical | #26044233 | C2 |
| hypercenter | #123 | D367 |
| hypocenter | C367 | #542 |
| LCS | G ▷ C22×C367 ▷ C367 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26044260 | C2 | semidirect |
| 2. #26044256 | C2 | semidirect |
| 3. #26044254 | C2 | semidirect |
| 4. #26044248 | C2 | semidirect |
| 5. #26044247 | C2 | semidirect |
| 6. #26044247 | C2 | semidirect |
| 7. #26044243 | C2 | semidirect |
| 8. #26044240 | C2 | semidirect |
| 9. #26044239 | C2 | semidirect |
| 10. #26044238 | C2 | semidirect |
| 12. #26044226 | C2 | semidirect |
| 13. C4×C22⋊D367 | C2 | semidirect |
| 14. #26044217 | C2 | semidirect |
| 15. #123×C367 | C2 | semidirect |
| 103. C367 | #542 | semidirect |
| 112. #123 | D367 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26044260 | 2 |
| #26044256 | 2 |
| #26044254 | 2 |
| #26044248 | 2 |
| #26044247 | 2 |
| #26044243 | 2 |
| #26044240 | 2 |
| #26044239 | 2 |
| #26044238 | 2 |
| #26044233 | 2 |
| #26044226 | 2 |
| C4×C22⋊D367 | 2 |
| #26044217 | 2 |
| #123×C367 | 2 |
| #542 | 367 |