Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
168 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×D5 |
| commutator, Fitting, socle, cosocle | C2×C32×C5 | C23 |
| prime radical | C2×C5×C3⋊D3 | C22 |
| hypocenter | C32×C5 | C2×Q8 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #67153 | C2 | semidirect |
| 5. C3×#1204 | C2 | semidirect |
| 17. #1204 | D3 | semidirect |
| 22. C5×C3⋊D3 | Q8 | semidirect |
| 40. C32×C5 | C2×Q8 | semidirect |
| 69. C3⋊D3 | Dic10 | semidirect |
| 72. C3×C5 | D3×Q8 | semidirect |
| 85. C32 | C2×Dic10 | semidirect |
| 95. C5 | #3797 | semidirect |
| 102. C3 | D3×Dic10 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #67155 | 2 |
| #67153 | 2 |
| C5×#624 | 2 |
| C3⋊D3×Dic5 | 2 |
| C3×#1204 | 2 |
| D3×Dic10 | 3 |
| #3797 | 5 |