Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×D239 |
| commutator, socle, cosocle | C22×C239 | C24 |
| Fitting | #123×C239 | C2 |
| prime radical | #21439932 | C2 |
| hypercenter | #123 | D239 |
| hypocenter | C239 | #542 |
| LCS | G ▷ C22×C239 ▷ C239 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21439959 | C2 | semidirect |
| 2. #21439955 | C2 | semidirect |
| 3. #21439953 | C2 | semidirect |
| 4. #21439947 | C2 | semidirect |
| 5. #21439946 | C2 | semidirect |
| 6. #21439946 | C2 | semidirect |
| 7. #21439942 | C2 | semidirect |
| 8. #21439939 | C2 | semidirect |
| 9. #21439938 | C2 | semidirect |
| 10. #21439937 | C2 | semidirect |
| 12. #21439925 | C2 | semidirect |
| 13. C4×C22⋊D239 | C2 | semidirect |
| 14. #21439916 | C2 | semidirect |
| 15. #123×C239 | C2 | semidirect |
| 103. C239 | #542 | semidirect |
| 112. #123 | D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439959 | 2 |
| #21439955 | 2 |
| #21439953 | 2 |
| #21439947 | 2 |
| #21439946 | 2 |
| #21439942 | 2 |
| #21439939 | 2 |
| #21439938 | 2 |
| #21439937 | 2 |
| #21439932 | 2 |
| #21439925 | 2 |
| C4×C22⋊D239 | 2 |
| #21439916 | 2 |
| #123×C239 | 2 |
| #542 | 239 |