Finite Group DatabasebetaFinite Group DatabasebetaOrder 225.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C52 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C52⋊C3 | C3 | 1 | 3 | yes | maximal, direct product factor |
| 3 | C52 | C32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | C3×C5 | — | 3 | 2 | no | — |
| 5 | C32 | — | 25 | 1 | no | maximal, Sylow |
| 6 | C5 | — | 3 | 2 | no | — |
| 7 | C3 | — | 25 | 3 | no | — |
| 8 | C3 | C52⋊C3 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C52⋊C3 |
| commutator, cosocle, hypocenter | C52 | C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52 | C3 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C52 | C3 | semidirect |
| 2. C52⋊C3 | C3 | direct (N × H) |
| 3. C52 | C32 | semidirect |
| 8. C3 | C52⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C52 | 3 |
| C52⋊C3 | 3 |
| C32 | 25 |