Finite Group DatabasebetaFinite Group DatabasebetaOrder 189.
22 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32×C7 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×ΓL(1,𝔽8) | C3 | 1 | 3 | yes | maximal, complemented |
| 3 | H(3,𝔽3) | — | 7 | 1 | no | maximal, Sylow |
| 4 | C3×C7 | — | 3 | 1 | no | — |
| 5 | C3×C7 | C32 | 1 | 1 | yes | characteristic |
| 6 | ΓL(1,𝔽8) | — | 3 | 3 | no | — |
| 7 | C32 | — | 7 | 3 | no | — |
| 8 | C32 | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 9 | C7 | H(3,𝔽3) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C3 | — | 3 | 1 | no | — |
| 11 | C3 | — | 21 | 3 | no | — |
| 12 | C3 | C3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C3×C7 | C32 |
| Fitting | C32×C7 | C3 |
| hypercenter | C32 | ΓL(1,𝔽8) |
| hypocenter | C7 | H(3,𝔽3) |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×C7 | C3 | semidirect |
| 2. C3×ΓL(1,𝔽8) | C3 | semidirect |
| 8. C32 | ΓL(1,𝔽8) | semidirect |
| 9. C7 | H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C7 | 3 |
| C3×ΓL(1,𝔽8) | 3 |
| H(3,𝔽3) | 7 |