Finite Group DatabasebetaFinite Group DatabasebetaOrder 20288.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 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×D317 |
| commutator, socle, cosocle | C22×C317 | C24 |
| Fitting | #120×C317 | C2 |
| hypercenter | #120 | D317 |
| hypocenter | C317 | #560 |
| LCS | G ▷ C22×C317 ▷ C317 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23778776 | C2 | semidirect |
| 2. #23778773 | C2 | semidirect |
| 3. #23778762 | C2 | semidirect |
| 4. #23778761 | C2 | semidirect |
| 5. #23778753 | C2 | semidirect |
| 6. #23778752 | C2 | semidirect |
| 7. #23778751 | C2 | semidirect |
| 8. #23778749 | C2 | semidirect |
| 9. #120×C317 | C2 | semidirect |
| 10. C2×D1268 | C22 | semidirect |
| 14. C2×C22⋊D317 | C22 | semidirect |
| 15. C22×Dic317 | C22 | semidirect |
| 16. C2×C4×D317 | C22 | semidirect |
| 17. C4⋊Dic317 | C22 | semidirect |
| 18. #19099553 | C22 | semidirect |
| 19. #19099553 | C22 | semidirect |
| 23. C22⋊Dic317 | C22 | semidirect |
| 24. #19099551 | C22 | semidirect |
| 25. C4×Dic317 | C22 | semidirect |
| 74. C317 | #560 | semidirect |
| 84. #120 | D317 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23778776 | 2 |
| #23778773 | 2 |
| #23778762 | 2 |
| #23778761 | 2 |
| #23778753 | 2 |
| #23778752 | 2 |
| #23778751 | 2 |
| #23778749 | 2 |
| #120×C317 | 2 |
| #560 | 317 |