Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
170 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #117 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | #2163 |
| prime radical | C23×C7 | M32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C7×#346 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1422 | C2 | semidirect |
| 4. C7×#1398 | C2 | semidirect |
| 13. #13642 | C7 | direct (N × H) |
| 29. #1422 | C2×C7 | semidirect |
| 31. #1398 | C2×C7 | semidirect |
| 123. C7 | #13642 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C7×M32 | 2 |
| C7×#1422 | 2 |
| C7×#1400 | 2 |
| C7×#1398 | 2 |
| #13642 | 7 |