Finite Group DatabasebetaFinite Group DatabasebetaOrder 20032.
234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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×D313 |
| commutator, socle, cosocle | C22×C313 | C24 |
| Fitting | #123×C313 | C2 |
| prime radical | #23756469 | C2 |
| hypercenter | #123 | D313 |
| hypocenter | C313 | #529 |
| LCS | G ▷ C22×C313 ▷ C313 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23756492 | C2 | semidirect |
| 2. #23756490 | C2 | semidirect |
| 3. #23756487 | C2 | semidirect |
| 4. #23756483 | C2 | semidirect |
| 5. #23756480 | C2 | semidirect |
| 6. #23756474 | C2 | semidirect |
| 8. #23756462 | C2 | semidirect |
| 9. D4×Dic313 | C2 | semidirect |
| 10. #23756457 | C2 | semidirect |
| 11. #23756457 | C2 | semidirect |
| 12. #23756455 | C2 | semidirect |
| 13. #23756453 | C2 | semidirect |
| 14. C2×C4×Dic313 | C2 | semidirect |
| 15. #123×C313 | C2 | semidirect |
| 103. C313 | #529 | semidirect |
| 110. #123 | D313 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23756492 | 2 |
| #23756490 | 2 |
| #23756487 | 2 |
| #23756483 | 2 |
| #23756480 | 2 |
| #23756474 | 2 |
| #23756469 | 2 |
| #23756462 | 2 |
| D4×Dic313 | 2 |
| #23756457 | 2 |
| #23756455 | 2 |
| #23756453 | 2 |
| C2×C4×Dic313 | 2 |
| #123×C313 | 2 |
| #529 | 313 |