Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×C128 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×C64 | C4 | 1 | 1 | yes | characteristic |
| 3 | C256 | — | 7 | 1 | no | maximal, Sylow |
| 4 | C7×C32 | C8 | 1 | 1 | yes | characteristic |
| 5 | C128 | D7 | 1 | 1 | yes | characteristic |
| 6 | C7×C16 | C16 | 1 | 1 | yes | characteristic |
| 7 | C64 | Dic7 | 1 | 1 | yes | characteristic |
| 8 | C7×C8 | C32 | 1 | 1 | yes | characteristic |
| 9 | C32 | C7⋊C8 | 1 | 1 | yes | characteristic |
| 10 | C4×C7 | C64 | 1 | 1 | yes | characteristic |
| 11 | C16 | C7⋊C16 | 1 | 1 | yes | characteristic |
| 12 | C2×C7 | C128 | 1 | 1 | yes | characteristic |
| 13 | C8 | C7⋊C32 | 1 | 1 | yes | characteristic |
| 14 | C7 | C256 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C4 | C7⋊C64 | 1 | 1 | yes | characteristic |
| 16 | C2 | C7⋊C128 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C128 | D7 |
| commutator, hypocenter | C7 | C256 |
| Fitting, cosocle | C7×C128 | C2 |
| socle, prime radical | C2×C7 | C128 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C128 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. C7 | C256 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C128 | 2 |
| C256 | 7 |