Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
232 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×D15 |
| commutator, hypocenter | C32×C5 | C22×C4 |
| Frattini | C2 | C2×D3×D15 |
| Fitting | C32×C4×C5 | C22 |
| socle, cosocle | C2×C32×C5 | C23 |
| prime radical | C2×D3×D15 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C3⋊D15 | C2 | semidirect |
| 3. C3×C4×D15 | C2 | semidirect |
| 4. C3×C4×C5×D3 | C2 | semidirect |
| 5. #67149 | C2 | semidirect |
| 6. D3×Dic15 | C2 | semidirect |
| 7. Dic3×D15 | C2 | semidirect |
| 13. D3×D15 | C4 | direct (N × H) |
| 14. C32×C4×C5 | C22 | semidirect |
| 15. C3⋊Dic15 | C22 | semidirect |
| 16. C3×Dic15 | C22 | semidirect |
| 17. C3×C5×Dic3 | C22 | semidirect |
| 28. C4×D15 | D3 | direct (N × H) |
| 29. C4×C5×D3 | D3 | semidirect |
| 35. C3⋊D15 | C2×C4 | semidirect |
| 36. C3×D15 | C2×C4 | semidirect |
| 37. C3×C5×D3 | C2×C4 | semidirect |
| 42. C3×C4×D3 | D5 | semidirect |
| 57. C3×C4×C5 | C2×D3 | semidirect |
| 58. C3×C4×C5 | C2×D3 | semidirect |
| 62. Dic15 | C2×D3 | semidirect |
| 64. C5×Dic3 | C2×D3 | semidirect |
| 67. C32×C5 | C22×C4 | semidirect |
| 79. C32×C4 | C2×D5 | semidirect |
| 82. C3×Dic3 | C2×D5 | semidirect |
| 90. D15 | C4×D3 | direct (N × H) |
| 92. C5×D3 | C4×D3 | semidirect |
| 107. C4×D3 | D15 | direct (N × H) |
| 113. C4×C5 | D32 | semidirect |
| 119. C3×D3 | C4×D5 | semidirect |
| 122. C3×C5 | C2×C4×D3 | semidirect |
| 123. C3×C5 | C2×C4×D3 | semidirect |
| 136. C3×C4 | D3×D5 | semidirect |
| 137. C3×C4 | C2×D15 | semidirect |
| 142. Dic3 | C2×D15 | semidirect |
| 147. C32 | C2×C4×D5 | semidirect |
| 164. D3 | C4×D15 | direct (N × H) |
| 165. C5 | C4×D32 | semidirect |
| 173. C4 | D3×D15 | direct (N × H) |
| 175. C3 | C4×D3×D5 | semidirect |
| 176. C3 | C2×C4×D15 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×D15 | 2 |
| C4×C3⋊D15 | 2 |
| C3×C4×D15 | 2 |
| C3×C4×C5×D3 | 2 |
| #67149 | 2 |
| D3×Dic15 | 2 |
| Dic3×D15 | 2 |
| C2×C4×D15 | 3 |
| C4×D3×D5 | 3 |
| C4×D32 | 5 |