Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
406 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 114 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 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C22×C7×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3162 | C2 | semidirect |
| 4. C7×#1386 | C2 | semidirect |
| 5. C7×#1377 | C2 | semidirect |
| 17. #13367 | C7 | direct (N × H) |
| 22. C2×C7×D8 | Q8 | semidirect |
| 43. #3162 | C2×C7 | semidirect |
| 46. #1386 | C2×C7 | semidirect |
| 47. #1377 | C2×C7 | semidirect |
| 119. C2×D8 | C7×Q8 | semidirect |
| 217. C7 | #13367 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3162 | 2 |
| C7×#2140 | 2 |
| C7×#1972 | 2 |
| C7×#1386 | 2 |
| C7×#1377 | 2 |
| #13367 | 7 |