Finite Group DatabasebetaFinite Group DatabasebetaOrder 28096.
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×D439 |
| commutator, socle, cosocle | C22×C439 | C24 |
| Fitting | #123×C439 | C2 |
| prime radical | #28380715 | C2 |
| hypercenter | #123 | D439 |
| hypocenter | C439 | #529 |
| LCS | G ▷ C22×C439 ▷ C439 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #28380738 | C2 | semidirect |
| 2. #28380736 | C2 | semidirect |
| 3. #28380733 | C2 | semidirect |
| 4. #28380729 | C2 | semidirect |
| 5. #28380726 | C2 | semidirect |
| 6. #28380720 | C2 | semidirect |
| 8. #28380708 | C2 | semidirect |
| 9. D4×Dic439 | C2 | semidirect |
| 10. (C22⋊D439)⋊C4 | C2 | semidirect |
| 11. (C22⋊D439)⋊C4 | C2 | semidirect |
| 12. #28380701 | C2 | semidirect |
| 13. #28380699 | C2 | semidirect |
| 14. C2×C4×Dic439 | C2 | semidirect |
| 15. #123×C439 | C2 | semidirect |
| 103. C439 | #529 | semidirect |
| 110. #123 | D439 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28380738 | 2 |
| #28380736 | 2 |
| #28380733 | 2 |
| #28380729 | 2 |
| #28380726 | 2 |
| #28380720 | 2 |
| #28380715 | 2 |
| #28380708 | 2 |
| D4×Dic439 | 2 |
| (C22⋊D439)⋊C4 | 2 |
| #28380701 | 2 |
| #28380699 | 2 |
| C2×C4×Dic439 | 2 |
| #123×C439 | 2 |
| #529 | 439 |