Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
238 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 238 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 |
| hypercenter | C4×D4 | D239 |
| hypocenter | C239 | #553 |
| 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. #21439954 | C2 | semidirect |
| 2. #21439953 | C2 | semidirect |
| 3. #21439950 | C2 | semidirect |
| 4. #21439945 | C2 | semidirect |
| 5. #21439943 | C2 | semidirect |
| 6. #21439934 | C2 | semidirect |
| 7. #21439933 | C2 | semidirect |
| 8. #21439932 | C2 | semidirect |
| 9. #21439932 | C2 | semidirect |
| 10. C4×C22⋊D239 | C2 | semidirect |
| 11. (C22⋊D239)⋊C4 | C2 | semidirect |
| 12. (C22⋊D239)⋊C4 | C2 | semidirect |
| 13. #21439917 | C2 | semidirect |
| 14. C2×Dic239⋊C4 | C2 | semidirect |
| 15. C4×D4×C239 | C2 | semidirect |
| 109. C239 | #553 | semidirect |
| 123. C4×D4 | D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439954 | 2 |
| #21439953 | 2 |
| #21439950 | 2 |
| #21439945 | 2 |
| #21439943 | 2 |
| #21439934 | 2 |
| #21439933 | 2 |
| #21439932 | 2 |
| C4×C22⋊D239 | 2 |
| (C22⋊D239)⋊C4 | 2 |
| #21439917 | 2 |
| C2×Dic239⋊C4 | 2 |
| C4×D4×C239 | 2 |
| #553 | 239 |