Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
176 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 176 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 | C22×C32×C5 | C22 |
| hypercenter | C22 | D3×D15 |
| hypocenter | C32×C5 | #41 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #67184 | C2 | semidirect |
| 2. C3×#1220 | C2 | semidirect |
| 3. C2×C3×C5×Dic3 | C2 | semidirect |
| 4. #67153 | C2 | semidirect |
| 5. #67151 | C2 | semidirect |
| 6. #67149 | C2 | semidirect |
| 7. Dic3×D15 | C2 | semidirect |
| 22. #1220 | D3 | semidirect |
| 27. C2×C5×Dic3 | D3 | semidirect |
| 40. C2×C3×Dic3 | D5 | semidirect |
| 64. C32×C5 | #41 | semidirect |
| 98. C2×Dic3 | D15 | semidirect |
| 118. C3×C5 | #235 | semidirect |
| 119. C3×C5 | #237 | semidirect |
| 141. C32 | #709 | semidirect |
| 160. C5 | #3804 | semidirect |
| 169. C3 | #6779 | semidirect |
| 170. C3 | #6817 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #67184 | 2 |
| C3×#1220 | 2 |
| C2×C3×C5×Dic3 | 2 |
| #67153 | 2 |
| #67151 | 2 |
| #67149 | 2 |
| Dic3×D15 | 2 |
| #6817 | 3 |
| #6779 | 3 |
| #3804 | 5 |