Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
612 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 126 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 | C2×#99 |
| commutator | C24 | C22×C4×C7 |
| Frattini, cosocle | C22×D4 | C23×C7 |
| prime radical | C22×C7×D4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C23×C7×Q8 | C4 | semidirect |
| 15. C7×#385 | C4 | semidirect |
| 17. C7×#380 | C4 | semidirect |
| 20. #13293 | C7 | direct (N × H) |
| 45. C7×#100 | Q8 | semidirect |
| 103. C23×Q8 | C4×C7 | semidirect |
| 113. #385 | C4×C7 | semidirect |
| 115. #380 | C4×C7 | semidirect |
| 187. #100 | C7×Q8 | semidirect |
| 320. C7 | #13293 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2409 | 2 |
| C7×#1799 | 2 |
| C4×C7×#100 | 2 |
| C7×#1390 | 2 |
| #13293 | 7 |