Finite Group DatabasebetaFinite Group DatabasebetaOrder 15296.
238 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 99 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 | C2×C4⋊C4×C239 | C2 |
| hypercenter | C2×C4⋊C4 | D239 |
| hypocenter | C239 | #553 |
| LCS | G ▷ C22×C239 ▷ C239 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21439955 | C2 | semidirect |
| 2. #21439951 | C2 | semidirect |
| 3. #21439945 | C2 | semidirect |
| 4. #21439935 | C2 | semidirect |
| 5. #21439934 | C2 | semidirect |
| 6. #21439934 | C2 | semidirect |
| 7. #21439922 | C2 | semidirect |
| 8. C4×C22⋊D239 | C2 | semidirect |
| 9. C4×C22⋊D239 | C2 | semidirect |
| 10. #21439918 | C2 | semidirect |
| 11. C2×C4⋊C4×C239 | C2 | semidirect |
| 78. C239 | #553 | semidirect |
| 90. C2×C4⋊C4 | D239 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21439955 | 2 |
| #21439951 | 2 |
| #21439945 | 2 |
| #21439935 | 2 |
| #21439934 | 2 |
| #21439922 | 2 |
| C4×C22⋊D239 | 2 |
| #21439918 | 2 |
| C2×C4⋊C4×C239 | 2 |
| #553 | 239 |