Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
666 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 666 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, socle | C2×C7 | #1782 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×Q8 | C23×C7 |
| prime radical | C2×C7×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C23×C7 ◁ C7×#519 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C7×#356 | C2 | semidirect |
| 4. C7×#1830 | C2 | semidirect |
| 23. #12730 | C7 | direct (N × H) |
| 71. C2×#356 | C2×C7 | semidirect |
| 72. #1830 | C2×C7 | semidirect |
| 305. C7 | #12730 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3509 | 2 |
| C7×#2133 | 2 |
| C2×C7×#356 | 2 |
| C7×#1830 | 2 |
| #12730 | 7 |