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 | #123×C317 | C2 |
| hypercenter | #123 | D317 |
| hypocenter | C317 | #560 |
| 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. #23778776 | C2 | semidirect |
| 2. #23778775 | C2 | semidirect |
| 3. #23778762 | C2 | semidirect |
| 4. #23778755 | C2 | semidirect |
| 5. #23778754 | C2 | semidirect |
| 6. #23778753 | C2 | semidirect |
| 7. #23778750 | C2 | semidirect |
| 8. #23778748 | C2 | semidirect |
| 9. #123×C317 | C2 | semidirect |
| 13. C2×D1268 | C22 | semidirect |
| 16. C2×C22⋊D317 | C22 | semidirect |
| 17. C2×C4×D317 | C22 | semidirect |
| 18. C4⋊Dic317 | C22 | semidirect |
| 19. #19099553 | C22 | semidirect |
| 24. #19099551 | C22 | semidirect |
| 25. C4×Dic317 | C22 | semidirect |
| 27. C22×C4×C317 | C22 | semidirect |
| 28. C4⋊C4×C317 | C22 | semidirect |
| 30. C22⋊C4×C317 | C22 | semidirect |
| 74. C317 | #560 | semidirect |
| 79. #123 | D317 | semidirect |
| 94. C22×C4 | C2×D317 | semidirect |
| 97. C4⋊C4 | C2×D317 | semidirect |
| 104. C22⋊C4 | C2×D317 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23778776 | 2 |
| #23778775 | 2 |
| #23778762 | 2 |
| #23778755 | 2 |
| #23778754 | 2 |
| #23778753 | 2 |
| #23778750 | 2 |
| #23778748 | 2 |
| #123×C317 | 2 |
| #560 | 317 |