Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
422 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C23×C7 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C7×#487 | C4 | semidirect |
| 18. #9426 | C7 | direct (N × H) |
| 75. #487 | C4×C7 | semidirect |
| 181. C7 | #9426 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3139 | 2 |
| C7×#2381 | 2 |
| C7×#1558 | 2 |
| C7×#1515 | 2 |
| C7×#1288 | 2 |
| #9426 | 7 |