Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
104 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #67156 |
| commutator, Fitting, cosocle | C32×C4×C5 | C22 |
| Frattini, hypercenter | C4 | (C3⋊D3)⋊D5 |
| socle | C2×C32×C5 | D4 |
| hypocenter | C32×C5 | D8 |
| LCS | G ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C4⋊(C3⋊D3) | C2 | semidirect |
| 2. C3×D60 | C2 | semidirect |
| 3. C3×C3⋊#168 | C2 | semidirect |
| 10. D60 | D3 | semidirect |
| 15. C3⋊#168 | D3 | semidirect |
| 21. C4⋊(C3⋊D3) | D5 | semidirect |
| 34. C32×C5 | D8 | semidirect |
| 64. C3×C5 | D24 | semidirect |
| 65. C3×C5 | #213 | semidirect |
| 77. C32 | #686 | semidirect |
| 89. C5 | #3714 | semidirect |
| 94. C3 | #6652 | semidirect |
| 95. C3 | #6654 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C4⋊(C3⋊D3) | 2 |
| C3×D60 | 2 |
| C3×C3⋊#168 | 2 |
| #6654 | 3 |
| #6652 | 3 |
| #3714 | 5 |