Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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 | C4×D4×C239 | C2 |
| prime radical | #21439928 | C2 |
| hypercenter | C4×D4 | D239 |
| hypocenter | C239 | #551 |
| LCS | G ▷ C22×C239 ▷ C239 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21439955 | C2 | semidirect |
| 2. #21439954 | C2 | semidirect |
| 3. #21439946 | C2 | semidirect |
| 4. #21439943 | C2 | semidirect |
| 5. #21439942 | C2 | semidirect |
| 6. #21439941 | C2 | semidirect |
| 7. #21439939 | C2 | semidirect |
| 8. #21439938 | C2 | semidirect |
| 10. C4×C22⋊D239 | C2 | semidirect |
| 11. #21439917 | C2 | semidirect |
| 12. #21439916 | C2 | semidirect |
| 13. C2×(C2×D239)⋊C4 | C2 | semidirect |
| 14. C22⋊C4×D239 | C2 | semidirect |
| 15. C4×D4×C239 | C2 | semidirect |
| 17. C2×Dic478 | C22 | semidirect |
| 23. C4⋊Dic239 | C22 | semidirect |
| 24. Dic239⋊C4 | C22 | semidirect |
| 25. Dic239⋊C4 | C22 | semidirect |
| 26. Dic239⋊C4 | C22 | semidirect |
| 27. Dic239⋊C4 | C22 | semidirect |
| 28. Dic239⋊C4 | C22 | semidirect |
| 29. Dic239⋊C4 | C22 | semidirect |
| 32. C22⋊Dic239 | C22 | semidirect |
| 33. C22⋊Dic239 | C22 | semidirect |
| 42. C4×Dic239 | C22 | semidirect |
| 43. C4×Dic239 | C22 | semidirect |
| 46. C22×C4×C239 | C22 | semidirect |
| 47. C4⋊C4×C239 | C22 | semidirect |
| 49. C22⋊C4×C239 | C22 | semidirect |
| 50. C42×C239 | C22 | semidirect |
| 110. C239 | #551 | semidirect |
| 122. C4×D4 | D239 | semidirect |
| 136. C22×C4 | C2×D239 | semidirect |
| 144. C4⋊C4 | C2×D239 | semidirect |
| 158. C22⋊C4 | C2×D239 | semidirect |
| 161. C42 | C2×D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439955 | 2 |
| #21439954 | 2 |
| #21439946 | 2 |
| #21439943 | 2 |
| #21439942 | 2 |
| #21439941 | 2 |
| #21439939 | 2 |
| #21439938 | 2 |
| #21439928 | 2 |
| C4×C22⋊D239 | 2 |
| #21439917 | 2 |
| #21439916 | 2 |
| C2×(C2×D239)⋊C4 | 2 |
| C22⋊C4×D239 | 2 |
| C4×D4×C239 | 2 |
| #551 | 239 |