Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
234 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 92 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 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C42 | C23×C7 |
| socle | C22×C7 | C2×C22⋊C4 |
| prime radical | C7×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C7×M16 | C2 | semidirect |
| 3. C2×C7×#97 | C2 | semidirect |
| 4. C7×#330 | C2 | semidirect |
| 5. C7×#329 | C2 | semidirect |
| 23. #1524 | C7 | direct (N × H) |
| 52. C4×M16 | C2×C7 | semidirect |
| 53. C2×#97 | C2×C7 | semidirect |
| 54. #330 | C2×C7 | semidirect |
| 55. #329 | C2×C7 | semidirect |
| 161. C7 | #1524 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#529 | 2 |
| C4×C7×M16 | 2 |
| C2×C7×#97 | 2 |
| C7×#330 | 2 |
| C7×#329 | 2 |
| #1524 | 7 |