Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
176 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×D32 |
| commutator, cosocle | C2×C32 | C23×C5 |
| Frattini | C2 | C2×C5×D32 |
| Fitting | C22×C32×C5 | C22 |
| socle | C2×C32×C5 | C23 |
| hypercenter | C22×C5 | D32 |
| hypocenter | C32 | C5×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#638 | C2 | semidirect |
| 2. C3×C5×#67 | C2 | semidirect |
| 3. C2×C3×C5×Dic3 | C2 | semidirect |
| 4. C5×#627 | C2 | semidirect |
| 5. C5×#626 | C2 | semidirect |
| 6. C5×#624 | C2 | semidirect |
| 7. C5×D3×Dic3 | C2 | semidirect |
| 17. #3804 | C5 | direct (N × H) |
| 24. C5×#67 | D3 | semidirect |
| 26. C2×C5×Dic3 | D3 | semidirect |
| 38. #638 | C2×C5 | semidirect |
| 39. C3×#67 | C2×C5 | semidirect |
| 40. C2×C3×Dic3 | C2×C5 | semidirect |
| 41. #627 | C2×C5 | semidirect |
| 42. #626 | C2×C5 | semidirect |
| 43. #624 | C2×C5 | semidirect |
| 44. D3×Dic3 | C2×C5 | semidirect |
| 64. C32×C5 | #41 | semidirect |
| 96. #67 | C5×D3 | semidirect |
| 98. C2×Dic3 | C5×D3 | semidirect |
| 118. C3×C5 | #235 | semidirect |
| 119. C3×C5 | #237 | semidirect |
| 141. C32 | C5×#41 | semidirect |
| 160. C5 | #3804 | direct (N × H) |
| 169. C3 | C5×#235 | semidirect |
| 170. C3 | C5×#237 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#638 | 2 |
| C3×C5×#67 | 2 |
| C2×C3×C5×Dic3 | 2 |
| C5×#627 | 2 |
| C5×#626 | 2 |
| C5×#624 | 2 |
| C5×D3×Dic3 | 2 |
| C5×#237 | 3 |
| C5×#235 | 3 |
| #3804 | 5 |