Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
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×D239 |
| commutator, socle, cosocle | C22×C239 | C24 |
| Fitting | #120×C239 | C2 |
| hypercenter | #120 | D239 |
| hypocenter | C239 | #560 |
| LCS | G ▷ C22×C239 ▷ C239 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21439956 | C2 | semidirect |
| 2. #21439953 | C2 | semidirect |
| 3. #21439942 | C2 | semidirect |
| 4. #21439941 | C2 | semidirect |
| 5. #21439933 | C2 | semidirect |
| 6. #21439932 | C2 | semidirect |
| 7. #21439931 | C2 | semidirect |
| 8. #21439929 | C2 | semidirect |
| 9. #120×C239 | C2 | semidirect |
| 10. C2×D956 | C22 | semidirect |
| 14. C2×C22⋊D239 | C22 | semidirect |
| 15. C22×Dic239 | C22 | semidirect |
| 16. C2×C4×D239 | C22 | semidirect |
| 17. C4⋊Dic239 | C22 | semidirect |
| 18. Dic239⋊C4 | C22 | semidirect |
| 19. Dic239⋊C4 | C22 | semidirect |
| 23. C22⋊Dic239 | C22 | semidirect |
| 24. (C2×D239)⋊C4 | C22 | semidirect |
| 25. C4×Dic239 | C22 | semidirect |
| 74. C239 | #560 | semidirect |
| 84. #120 | D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439956 | 2 |
| #21439953 | 2 |
| #21439942 | 2 |
| #21439941 | 2 |
| #21439933 | 2 |
| #21439932 | 2 |
| #21439931 | 2 |
| #21439929 | 2 |
| #120×C239 | 2 |
| #560 | 239 |