Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
232 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C3⋊D3)⋊D5 |
| commutator, hypocenter | C32×C5 | C22×C4 |
| Frattini | C2 | C2×(C3⋊D3)⋊D5 |
| Fitting, socle | C22×C32×C5 | C22 |
| prime radical | C22×C3×D15 | C2 |
| cosocle | C2×C32×C5 | C23 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C5×C3⋊Dic3 | C2 | semidirect |
| 3. C2×C3×Dic15 | C2 | semidirect |
| 4. #67154 | C2 | direct (N × H) |
| 8. C2×C3×D15 | C4 | semidirect |
| 10. C5×C3⋊Dic3 | C22 | semidirect |
| 11. C3×Dic15 | C22 | semidirect |
| 15. C2×Dic15 | D3 | semidirect |
| 22. C2×C32×C5 | C2×C4 | semidirect |
| 24. C3×D15 | C2×C4 | semidirect |
| 27. C2×C3⋊Dic3 | D5 | semidirect |
| 33. C2×D15 | Dic3 | semidirect |
| 37. Dic15 | C2×D3 | semidirect |
| 44. C32×C5 | C22×C4 | semidirect |
| 52. C3⋊Dic3 | C2×D5 | semidirect |
| 56. C2×C3×C5 | C4×D3 | semidirect |
| 57. C2×C3×C5 | C2×Dic3 | semidirect |
| 60. D15 | C2×Dic3 | semidirect |
| 76. C2×C32 | C4×D5 | semidirect |
| 81. C3×C5 | C2×C4×D3 | semidirect |
| 82. C3×C5 | C22×Dic3 | semidirect |
| 95. C2×C5 | D3×Dic3 | semidirect |
| 98. C32 | C2×C4×D5 | semidirect |
| 106. C2×C3 | D5×Dic3 | semidirect |
| 107. C2×C3 | #1200 | semidirect |
| 111. C5 | C2×D3×Dic3 | semidirect |
| 118. C3 | C2×D5×Dic3 | semidirect |
| 119. C3 | C2×#1200 | semidirect |
| 121. C2 | #67154 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×D15 | 2 |
| C2×C5×C3⋊Dic3 | 2 |
| C2×C3×Dic15 | 2 |
| #67154 | 2 |
| C2×#1200 | 3 |
| C2×D5×Dic3 | 3 |
| C2×D3×Dic3 | 5 |