Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #124×C239 | C2 |
| prime radical | #21439931 | C2 |
| hypercenter | #124 | D239 |
| hypocenter | C239 | #542 |
| LCS | G ▷ C22×C239 ▷ C239 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21439960 | C2 | semidirect |
| 2. #21439953 | C2 | semidirect |
| 3. #21439946 | C2 | semidirect |
| 4. #21439944 | C2 | semidirect |
| 5. #21439937 | C2 | semidirect |
| 6. #21439936 | C2 | semidirect |
| 8. Q8×Dic239 | C2 | semidirect |
| 9. D4×Dic239 | C2 | semidirect |
| 10. #21439917 | C2 | semidirect |
| 11. #124×C239 | C2 | semidirect |
| 73. C239 | #542 | semidirect |
| 78. #124 | D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439960 | 2 |
| #21439953 | 2 |
| #21439946 | 2 |
| #21439944 | 2 |
| #21439937 | 2 |
| #21439936 | 2 |
| #21439931 | 2 |
| Q8×Dic239 | 2 |
| D4×Dic239 | 2 |
| #21439917 | 2 |
| #124×C239 | 2 |
| #542 | 239 |