Finite Group DatabasebetaFinite Group DatabasebetaOrder 360.
28 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3⋊Dic15 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C32×C5 | C4 | 1 | 1 | yes | characteristic |
| 3 | #622 | — | 5 | 1 | no | maximal, Hall |
| 4 | Dic15 | — | 6 | 2 | no | — |
| 5 | C32×C5 | C8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | #170 | — | 9 | 1 | no | maximal, Hall |
| 7 | C3⋊Dic3 | — | 5 | 1 | no | — |
| 8 | C2×C3×C5 | — | 2 | 2 | no | — |
| 9 | Dic5 | — | 9 | 1 | no | — |
| 10 | C2×C32 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 11 | C3×C5 | — | 2 | 2 | no | — |
| 12 | Dic3 | — | 30 | 2 | no | — |
| 13 | C2×C5 | #157 | 1 | 1 | yes | characteristic |
| 14 | C32 | #170 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C8 | — | 45 | 1 | no | Sylow |
| 16 | C2×C3 | — | 2 | 2 | no | — |
| 17 | C5 | #622 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 18 | C4 | — | 45 | 1 | no | — |
| 19 | C3 | — | 2 | 2 | no | — |
| 20 | C2 | #4380 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #4380 |
| commutator, hypocenter | C32×C5 | C8 |
| Fitting, socle, prime radical | C2×C32×C5 | C4 |
| cosocle | C3⋊Dic15 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C32×C5 | C8 | semidirect |
| 14. C32 | #170 | semidirect |
| 17. C5 | #622 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊Dic15 | 2 |
| #622 | 5 |
| #170 | 9 |