Finite Group DatabasebetaFinite Group DatabasebetaOrder 15104.
142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #16837237 |
| commutator, cosocle | C32×C59 | C23 |
| Frattini | C32 | C22×D59 |
| Fitting | C2×C59×D32 | C2 |
| socle | C22×C59 | D32 |
| hypercenter | C2×D32 | D59 |
| hypocenter | C59 | C2×D64 |
| LCS | G ▷ C32×C59 ▷ C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×D32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D1888 | C2 | semidirect |
| 2. #21422308 | C2 | direct (N × H) |
| 3. C2×C59⋊C64 | C2 | semidirect |
| 4. C2×C59×D32 | C2 | semidirect |
| 7. D1888 | C22 | semidirect |
| 8. C59⋊C64 | C22 | semidirect |
| 11. C59×D32 | C22 | semidirect |
| 49. C2×D32 | D59 | semidirect |
| 53. C2×C59 | D64 | semidirect |
| 61. D32 | C2×D59 | semidirect |
| 64. C59 | C2×D64 | semidirect |
| 101. C2 | #21422308 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D1888 | 2 |
| #21422308 | 2 |
| C2×C59⋊C64 | 2 |
| C2×C59×D32 | 2 |
| C2×D64 | 59 |