Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
260 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 190 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×C8 | C23 |
| commutator | C22 | C2×C4×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×C4×C8 |
| prime radical | C23×C7 | C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1312 | C2 | semidirect |
| 4. C7×#1311 | C2 | semidirect |
| 5. C7×C8×C16 | C2 | semidirect |
| 15. #11412 | C7 | direct (N × H) |
| 29. #1312 | C2×C7 | semidirect |
| 30. #1311 | C2×C7 | semidirect |
| 31. C8×C16 | C2×C7 | semidirect |
| 108. C7 | #11412 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2162 | 2 |
| C7×#1450 | 2 |
| C7×#1312 | 2 |
| C7×#1311 | 2 |
| C7×C8×C16 | 2 |
| #11412 | 7 |