Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
162 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D5×D9 |
| commutator, cosocle | C2×C5×C9 | C23 |
| Frattini | C2×C3 | C2×D3×D5 |
| Fitting | C22×C5×C9 | C22 |
| socle | C2×C3×C5 | C22×D3 |
| hypercenter | C22 | D5×D9 |
| hypocenter | C5×C9 | C2×D4 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C9×D5 | C2 | semidirect |
| 2. C2×D5×D9 | C2 | semidirect |
| 3. #67099 | C2 | semidirect |
| 4. C5×#611 | C2 | semidirect |
| 5. #67083 | C2 | semidirect |
| 6. #67082 | C2 | semidirect |
| 7. D5×Dic9 | C2 | semidirect |
| 9. C22×C5×C9 | C22 | semidirect |
| 10. C2×D45 | C22 | semidirect |
| 11. C2×C5×D9 | C22 | semidirect |
| 14. C2×C9×D5 | C22 | semidirect |
| 16. Dic45 | C22 | semidirect |
| 17. C5×Dic9 | C22 | semidirect |
| 30. C9×D5 | D4 | semidirect |
| 38. #611 | D5 | direct (N × H) |
| 50. C5×C9 | C2×D4 | semidirect |
| 52. C22×D5 | D9 | semidirect |
| 61. C22×C9 | C2×D5 | semidirect |
| 64. C2×D9 | C2×D5 | semidirect |
| 66. Dic9 | C2×D5 | semidirect |
| 81. C22×C5 | C2×D9 | semidirect |
| 86. C2×D5 | C2×D9 | semidirect |
| 111. D5 | #611 | direct (N × H) |
| 112. C9 | D4×D5 | semidirect |
| 126. C5 | C2×#611 | semidirect |
| 133. C22 | D5×D9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C9×D5 | 2 |
| C2×D5×D9 | 2 |
| #67099 | 2 |
| C5×#611 | 2 |
| #67083 | 2 |
| #67082 | 2 |
| D5×Dic9 | 2 |
| D5×#67 | 3 |
| C2×#611 | 5 |