Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
156 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1409 |
| commutator | C4×C8 | C2×C4×C7 |
| Frattini, cosocle | #493 | C22×C7 |
| prime radical | C7×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C42×C7 ◁ C7×#493 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#498 | C4 | semidirect |
| 10. C7×#322 | C4 | semidirect |
| 11. #7533 | C7 | direct (N × H) |
| 40. #498 | C4×C7 | semidirect |
| 46. #322 | C4×C7 | semidirect |
| 114. C7 | #7533 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#1715 | 2 |
| C7×#1408 | 2 |
| #7533 | 7 |