Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
362 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #418 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1824 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#1388 | C2 | semidirect |
| 5. C7×#1381 | C2 | semidirect |
| 18. #12878 | C7 | direct (N × H) |
| 44. #1388 | C2×C7 | semidirect |
| 45. #1381 | C2×C7 | semidirect |
| 181. C7 | #12878 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2916 | 2 |
| C7×#2113 | 2 |
| C7×#1813 | 2 |
| C7×#1388 | 2 |
| C7×#1381 | 2 |
| #12878 | 7 |