Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1378 conjugacy classes of subgroups; only the 334 normal subgroups are shown.
This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 334 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator, Frattini, cosocle | C24 | C24×C7 |
| socle | C22×C7 | C2×#120 |
| prime radical | C25×C7 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#3458 | C2 | semidirect |
| 3. C7×#2030 | C2 | semidirect |
| 5. C7×#1740 | C2 | semidirect |
| 17. #23467 | C7 | direct (N × H) |
| 22. C23×C4×C7 | Q8 | semidirect |
| 40. #3458 | C2×C7 | semidirect |
| 42. #2030 | C2×C7 | semidirect |
| 44. #1740 | C2×C7 | semidirect |
| 117. C23×C4 | C7×Q8 | semidirect |
| 213. C7 | #23467 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3458 | 2 |
| C7×#2280 | 2 |
| C7×#2030 | 2 |
| C7×#2029 | 2 |
| C7×#1740 | 2 |
| #23467 | 7 |