Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
434 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 434 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 |
|---|---|---|
| 8. C7×#501 | C4 | semidirect |
| 20. #9440 | C7 | direct (N × H) |
| 79. #501 | C4×C7 | semidirect |
| 196. C7 | #9440 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3166 | 2 |
| C7×#2371 | 2 |
| C7×#1534 | 2 |
| C7×#1506 | 2 |
| C7×#1300 | 2 |
| #9440 | 7 |