Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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 | #546 |
| 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. C2×#16838547 | C2 | semidirect |
| 2. C22×C22⋊D239 | C2 | semidirect |
| 3. #21439946 | C2 | semidirect |
| 4. #21439944 | C2 | semidirect |
| 5. #21439942 | C2 | semidirect |
| 6. #21439937 | C2 | semidirect |
| 7. #21439932 | C2 | semidirect |
| 8. #21439932 | C2 | semidirect |
| 9. #21439931 | C2 | semidirect |
| 10. #21439928 | C2 | semidirect |
| 11. D4×Dic239 | C2 | semidirect |
| 12. (C22⋊D239)⋊C4 | C2 | semidirect |
| 13. C2×C22⋊Dic239 | C2 | semidirect |
| 14. C22⋊C4×D239 | C2 | semidirect |
| 15. #120×C239 | C2 | semidirect |
| 18. C2×Dic478 | C22 | semidirect |
| 24. C2×C22⋊D239 | C22 | semidirect |
| 25. C2×C22⋊D239 | C22 | semidirect |
| 27. C22×Dic239 | C22 | semidirect |
| 28. C22×Dic239 | C22 | semidirect |
| 29. C22×Dic239 | C22 | semidirect |
| 30. C2×C4×D239 | C22 | semidirect |
| 31. C2×C4×D239 | C22 | semidirect |
| 32. C4⋊Dic239 | C22 | semidirect |
| 33. C4⋊Dic239 | C22 | semidirect |
| 34. Dic239⋊C4 | C22 | semidirect |
| 35. Dic239⋊C4 | C22 | semidirect |
| 38. C22⋊Dic239 | C22 | semidirect |
| 39. C22⋊Dic239 | C22 | semidirect |
| 40. C22⋊Dic239 | C22 | semidirect |
| 41. (C2×D239)⋊C4 | C22 | semidirect |
| 42. (C2×D239)⋊C4 | C22 | semidirect |
| 43. (C2×D239)⋊C4 | C22 | semidirect |
| 44. (C2×D239)⋊C4 | C22 | semidirect |
| 45. C4×Dic239 | C22 | semidirect |
| 48. C2×D4×C239 | C22 | semidirect |
| 49. C2×D4×C239 | C22 | semidirect |
| 51. C22⋊C4×C239 | C22 | semidirect |
| 52. C22⋊C4×C239 | C22 | semidirect |
| 64. C22⋊D239 | D4 | semidirect |
| 102. C22×C239 | #41 | semidirect |
| 129. C239 | #546 | semidirect |
| 141. #120 | D239 | semidirect |
| 158. C2×D4 | C2×D239 | semidirect |
| 159. C2×D4 | C2×D239 | semidirect |
| 180. C22⋊C4 | C2×D239 | semidirect |
| 181. C22⋊C4 | C2×D239 | semidirect |
| 239. C22 | #16838547 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16838547 | 2 |
| C22×C22⋊D239 | 2 |
| #21439946 | 2 |
| #21439944 | 2 |
| #21439942 | 2 |
| #21439937 | 2 |
| #21439932 | 2 |
| #21439931 | 2 |
| #21439928 | 2 |
| D4×Dic239 | 2 |
| (C22⋊D239)⋊C4 | 2 |
| C2×C22⋊Dic239 | 2 |
| C22⋊C4×D239 | 2 |
| #120×C239 | 2 |
| #546 | 239 |