Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
26 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×#622 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×C3⋊Dic3 | C4 | 1 | 1 | yes | characteristic |
| 3 | #3771 | — | 5 | 1 | no | maximal, Hall |
| 4 | C2×C32×C5 | C8 | 1 | 1 | yes | characteristic |
| 5 | #672 | — | 9 | 1 | no | maximal, Hall |
| 6 | #622 | D5 | 1 | 1 | yes | characteristic |
| 7 | C5×Dic3 | — | 12 | 1 | no | — |
| 8 | C32×C5 | C16 | 1 | 1 | yes | characteristic, complemented, Hall |
| 9 | C5×C8 | — | 9 | 1 | no | — |
| 10 | C3⋊Dic3 | Dic5 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C5 | — | 4 | 1 | no | — |
| 12 | C4×C5 | — | 9 | 1 | no | — |
| 13 | C2×C32 | #168 | 1 | 1 | yes | characteristic |
| 14 | C16 | — | 45 | 1 | no | Sylow |
| 15 | C3×C5 | — | 4 | 1 | no | — |
| 16 | Dic3 | — | 12 | 1 | no | — |
| 17 | C2×C5 | AGL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 18 | C32 | #672 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | C8 | — | 9 | 1 | no | — |
| 20 | C2×C3 | — | 4 | 1 | no | — |
| 21 | C5 | #3771 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 22 | C4 | — | 9 | 1 | no | — |
| 23 | C3 | — | 4 | 1 | no | — |
| 24 | C2 | #67194 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #67194 |
| commutator, hypocenter | C32×C5 | C16 |
| Fitting, socle, prime radical | C2×C32×C5 | C8 |
| cosocle | C5×#622 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C32×C5 | C16 | semidirect |
| 18. C32 | #672 | semidirect |
| 21. C5 | #3771 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#622 | 2 |
| #3771 | 5 |
| #672 | 9 |