Finite Group DatabasebetaFinite Group DatabasebetaOrder 20288.
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×D317 |
| commutator, socle, cosocle | C22×C317 | C24 |
| Fitting | #123×C317 | C2 |
| prime radical | #23778752 | C2 |
| hypercenter | #123 | D317 |
| hypocenter | C317 | #529 |
| LCS | G ▷ C22×C317 ▷ C317 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23778775 | C2 | semidirect |
| 2. #23778773 | C2 | semidirect |
| 3. #23778770 | C2 | semidirect |
| 4. #23778766 | C2 | semidirect |
| 5. #23778763 | C2 | semidirect |
| 6. #23778757 | C2 | semidirect |
| 8. #23778745 | C2 | semidirect |
| 9. D4×Dic317 | C2 | semidirect |
| 10. #23778740 | C2 | semidirect |
| 11. #23778740 | C2 | semidirect |
| 12. #23778738 | C2 | semidirect |
| 13. #23778736 | C2 | semidirect |
| 14. C2×C4×Dic317 | C2 | semidirect |
| 15. #123×C317 | C2 | semidirect |
| 103. C317 | #529 | semidirect |
| 110. #123 | D317 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23778775 | 2 |
| #23778773 | 2 |
| #23778770 | 2 |
| #23778766 | 2 |
| #23778763 | 2 |
| #23778757 | 2 |
| #23778752 | 2 |
| #23778745 | 2 |
| D4×Dic317 | 2 |
| #23778740 | 2 |
| #23778738 | 2 |
| #23778736 | 2 |
| C2×C4×Dic317 | 2 |
| #123×C317 | 2 |
| #529 | 317 |