Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
232 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 63 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 | (C3⋊D3)⋊D5 |
| commutator, hypocenter | C32×C5 | C22×C4 |
| Frattini | C2 | C2×(C3⋊D3)⋊D5 |
| Fitting | C32×C4×C5 | C22 |
| socle, cosocle | C2×C32×C5 | C23 |
| prime radical | C2×(C3⋊D3)⋊D5 | 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×C5×C3⋊D3 | C2 | semidirect |
| 3. C3×C4×D15 | C2 | semidirect |
| 4. #67155 | C2 | semidirect |
| 5. #67154 | C2 | semidirect |
| 9. (C3⋊D3)⋊D5 | C4 | direct (N × H) |
| 10. C32×C4×C5 | C22 | semidirect |
| 11. C5×C3⋊Dic3 | C22 | semidirect |
| 12. C3×Dic15 | C22 | semidirect |
| 15. C4×D15 | D3 | semidirect |
| 23. C5×C3⋊D3 | C2×C4 | semidirect |
| 24. C3×D15 | C2×C4 | semidirect |
| 27. C4×C3⋊D3 | D5 | semidirect |
| 37. C3×C4×C5 | C2×D3 | semidirect |
| 38. Dic15 | C2×D3 | semidirect |
| 43. C32×C5 | C22×C4 | semidirect |
| 52. C32×C4 | C2×D5 | semidirect |
| 53. C3⋊Dic3 | C2×D5 | semidirect |
| 57. D15 | C4×D3 | semidirect |
| 73. C4×C5 | D32 | semidirect |
| 76. C3⋊D3 | C4×D5 | semidirect |
| 80. C3×C5 | C2×C4×D3 | semidirect |
| 88. C3×C4 | D3×D5 | semidirect |
| 95. C32 | C2×C4×D5 | semidirect |
| 107. C5 | C4×D32 | semidirect |
| 113. C4 | (C3⋊D3)⋊D5 | direct (N × H) |
| 115. C3 | C4×D3×D5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×(C3⋊D3)⋊D5 | 2 |
| C4×C5×C3⋊D3 | 2 |
| C3×C4×D15 | 2 |
| #67155 | 2 |
| #67154 | 2 |
| C4×D3×D5 | 3 |
| C4×D32 | 5 |