Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 312 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 | C4×D4×C239 | C2 |
| hypercenter | C4×D4 | D239 |
| hypocenter | C239 | #550 |
| 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. C2×#16838548 | C2 | semidirect |
| 2. #21439957 | C2 | semidirect |
| 3. #21439956 | C2 | semidirect |
| 4. #21439934 | C2 | semidirect |
| 5. #21439933 | C2 | semidirect |
| 6. C4×Dic478 | C2 | semidirect |
| 7. #21439922 | C2 | semidirect |
| 8. C4×D4×C239 | C2 | semidirect |
| 27. Dic478 | D4 | semidirect |
| 53. C4×C239 | #41 | semidirect |
| 61. C239 | #550 | semidirect |
| 70. C4×D4 | D239 | semidirect |
| 117. C4 | #16838548 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16838548 | 2 |
| #21439957 | 2 |
| #21439956 | 2 |
| #21439934 | 2 |
| #21439933 | 2 |
| C4×Dic478 | 2 |
| #21439922 | 2 |
| C4×D4×C239 | 2 |
| #550 | 239 |