Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
134 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×D59 |
| commutator, cosocle | C32×C59 | C23 |
| Frattini | C32 | C22×D59 |
| Fitting | C59×SD128 | C2 |
| socle | C2×C59 | C2×D32 |
| hypercenter | SD128 | D59 |
| hypocenter | C59 | #13742 |
| LCS | G ▷ C32×C59 ▷ C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ SD128 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21426406 | C2 | semidirect |
| 2. D32×D59 | C2 | semidirect |
| 3. #21422310 | C2 | semidirect |
| 4. D3776 | C2 | semidirect |
| 5. #21422308 | C2 | semidirect |
| 6. #21422306 | C2 | semidirect |
| 7. C59×SD128 | C2 | semidirect |
| 12. D1888 | C22 | semidirect |
| 15. C59⋊C64 | C22 | semidirect |
| 16. C59×Q64 | C22 | semidirect |
| 18. C59×C64 | C22 | semidirect |
| 64. SD128 | D59 | semidirect |
| 75. Q64 | C2×D59 | semidirect |
| 83. C64 | C2×D59 | semidirect |
| 84. C59 | #13742 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21426406 | 2 |
| D32×D59 | 2 |
| #21422310 | 2 |
| D3776 | 2 |
| #21422308 | 2 |
| #21422306 | 2 |
| C59×SD128 | 2 |
| #13742 | 59 |