Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
172 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×D15 |
| commutator, socle, cosocle | C2×C32×C5 | C23 |
| Fitting | C32×C4×C5 | C22 |
| hypercenter | C4 | D3×D15 |
| hypocenter | C32×C5 | #41 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #67180 | C2 | semidirect |
| 2. C3×C4×D15 | C2 | semidirect |
| 3. C3×C5×D12 | C2 | semidirect |
| 4. #67150 | C2 | semidirect |
| 5. D3×Dic15 | C2 | semidirect |
| 17. C4×D15 | D3 | semidirect |
| 21. C5×D12 | D3 | semidirect |
| 30. C3×D12 | D5 | semidirect |
| 48. C32×C5 | #41 | semidirect |
| 71. D12 | D15 | semidirect |
| 87. C3×C5 | #235 | semidirect |
| 88. C3×C5 | #237 | semidirect |
| 104. C32 | #711 | semidirect |
| 117. C5 | #3795 | semidirect |
| 124. C3 | #6772 | semidirect |
| 125. C3 | #6819 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #67180 | 2 |
| C3×C4×D15 | 2 |
| C3×C5×D12 | 2 |
| #67150 | 2 |
| D3×Dic15 | 2 |
| #6819 | 3 |
| #6772 | 3 |
| #3795 | 5 |