Finite Group DatabasebetaFinite Group DatabasebetaOrder 23040.
156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5760 |
| commutator, cosocle | C5×C9×C128 | C22 |
| Frattini | C3×C128 | C2×D15 |
| Fitting | C5×C9×C256 | C2 |
| socle | C2×C3×C5 | D384 |
| hypercenter | C256 | D45 |
| hypocenter | C5×C9 | D256 |
| LCS | G ▷ C5×C9×C128 ▷ C5×C9×C64 ▷ C5×C9×C32 ▷ C5×C9×C16 ▷ C5×C8×C9 ▷ C4×C5×C9 ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ C128 ◁ C256 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D5760 | C2 | semidirect |
| 10. C5×C9×C256 | C2 | semidirect |
| 14. C9×C256 | D5 | semidirect |
| 20. C5×C256 | D9 | semidirect |
| 45. C256 | D45 | semidirect |
| 74. C5×C9 | D256 | semidirect |
| 96. C9 | D1280 | semidirect |
| 101. C5 | D2304 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5760 | 2 |
| D3840 | 3 |
| D2304 | 5 |
| C5×C9×C256 | 2 |