Finite Group DatabasebetaFinite Group DatabasebetaOrder 180.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C9×D5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | ΓL(1,𝔽16) | — | 3 | 1 | no | maximal |
| 3 | C5×C9 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | Dic9 | — | 5 | 1 | no | maximal, Hall |
| 5 | C3×D5 | D3 | 1 | 1 | yes | characteristic |
| 6 | AGL(1,𝔽5) | — | 9 | 1 | no | Hall |
| 7 | C2×C9 | — | 5 | 1 | no | — |
| 8 | C3×C5 | Dic3 | 1 | 1 | yes | characteristic |
| 9 | Dic3 | — | 15 | 1 | no | — |
| 10 | D5 | D9 | 1 | 1 | yes | characteristic |
| 11 | C9 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C2×C3 | — | 5 | 1 | no | — |
| 13 | C5 | Dic9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C4 | — | 45 | 1 | no | Sylow |
| 15 | C3 | ΓL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 16 | C2 | — | 5 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C5×C9 | C4 |
| Frattini | C3 | ΓL(1,𝔽16) |
| socle | C3×C5 | Dic3 |
| prime radical | C3×D5 | D3 |
| cosocle | C9×D5 | C2 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C5×C9 | C4 | semidirect |
| 11. C9 | AGL(1,𝔽5) | semidirect |
| 13. C5 | Dic9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×D5 | 2 |
| ΓL(1,𝔽16) | 3 |
| Dic9 | 5 |