Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
190 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #397 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#424 | C2 | semidirect |
| 5. C7×#344 | C2 | semidirect |
| 6. C7×#341 | C2 | semidirect |
| 7. C7×H(3,ℤ/4) | C2 | semidirect |
| 21. #2080 | C7 | direct (N × H) |
| 49. #424 | C2×C7 | semidirect |
| 53. #344 | C2×C7 | semidirect |
| 54. #341 | C2×C7 | semidirect |
| 55. H(3,ℤ/4) | C2×C7 | semidirect |
| 160. C7 | #2080 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#424 | 2 |
| C7×#421 | 2 |
| C7×#416 | 2 |
| C7×#413 | 2 |
| C7×#344 | 2 |
| C7×#341 | 2 |
| C7×H(3,ℤ/4) | 2 |
| #2080 | 7 |