Finite Group DatabasebetaFinite Group DatabasebetaOrder 171.
6 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C19⋊C3 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C19 | C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 3 | C9 | — | 19 | 1 | no | maximal, Sylow |
| 4 | C3 | — | 19 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C19 | C9 |
| prime radical, cosocle | C19⋊C3 | C3 |
| LCS | G ▷ C19 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C19 | C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C19⋊C3 | 3 |
| C9 | 19 |