Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
356 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C7×#3293 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#2214 | C2 | semidirect |
| 3. C7×#1713 | C2 | semidirect |
| 4. C7×#1358 | C2 | semidirect |
| 5. C7×#1335 | C2 | semidirect |
| 17. #12313 | C7 | direct (N × H) |
| 39. #2214 | C2×C7 | semidirect |
| 40. #1713 | C2×C7 | semidirect |
| 41. #1358 | C2×C7 | semidirect |
| 42. #1335 | C2×C7 | semidirect |
| 166. C7 | #12313 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#3293 | 2 |
| C7×#2214 | 2 |
| C7×#1713 | 2 |
| C7×#1358 | 2 |
| C7×#1335 | 2 |
| #12313 | 7 |