Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
246 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C42 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×#97 |
| prime radical | C23×C7 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#1313 | C2 | semidirect |
| 12. #11415 | C7 | direct (N × H) |
| 27. #1313 | C2×C7 | semidirect |
| 120. C7 | #11415 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#2163 | 2 |
| C2×C7×#322 | 2 |
| C7×#1313 | 2 |
| #11415 | 7 |