Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
6578 conjugacy classes of subgroups; only the 6270 normal subgroups are shown.
This group has 6578 conjugacy classes of subgroups — too many to draw. Only its 6270 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 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C7 |
| prime radical | C24×C7 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3502 | C2 | semidirect |
| 5. C7×Q82 | C22 | semidirect |
| 6. C7×#557 | C22 | semidirect |
| 13. #62849 | C7 | direct (N × H) |
| 14. C22×C7×Q8 | Q8 | semidirect |
| 26. #3502 | C2×C7 | semidirect |
| 41. Q82 | C22×C7 | semidirect |
| 42. #557 | C22×C7 | semidirect |
| 58. C22×Q8 | C7×Q8 | semidirect |
| 99. C7 | #62849 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3502 | 2 |
| C7×#3458 | 2 |
| C2×C7×#519 | 2 |
| #62849 | 7 |