Finite Group DatabasebetaFinite Group DatabasebetaOrder 504.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#157 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×C3⋊D3 | C4 | 1 | 1 | yes | characteristic |
| 3 | AGL(1,𝔽9) | — | 7 | 1 | no | maximal, Hall |
| 4 | C32×C7 | C8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C7⋊C8 | — | 9 | 1 | no | maximal, Hall |
| 6 | D3×C7 | — | 12 | 1 | no | — |
| 7 | #157 | D7 | 1 | 1 | yes | characteristic |
| 8 | C4×C7 | — | 9 | 1 | no | — |
| 9 | C3×C7 | — | 4 | 1 | no | — |
| 10 | C3⋊D3 | Dic7 | 1 | 1 | yes | characteristic |
| 11 | C2×C7 | — | 9 | 1 | no | — |
| 12 | C32 | C7⋊C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C8 | — | 63 | 1 | no | Sylow |
| 14 | C7 | AGL(1,𝔽9) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | D3 | — | 12 | 1 | no | — |
| 16 | C4 | — | 9 | 1 | no | — |
| 17 | C3 | — | 4 | 1 | no | — |
| 18 | C2 | — | 9 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C7 | C8 |
| prime radical | C7×C3⋊D3 | C4 |
| cosocle | C7×#157 | C2 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C32×C7 | C8 | semidirect |
| 12. C32 | C7⋊C8 | semidirect |
| 14. C7 | AGL(1,𝔽9) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#157 | 2 |
| AGL(1,𝔽9) | 7 |
| C7⋊C8 | 9 |