Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 146 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 | C7×C16 | D4 |
| commutator | C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C8×D4 |
| prime radical | C7×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C16 ◁ C2×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#504 | C2 | semidirect |
| 3. C7×#364 | C2 | semidirect |
| 4. C7×#350 | C2 | semidirect |
| 5. C4×C7×C16 | C2 | semidirect |
| 16. #2170 | C7 | direct (N × H) |
| 31. #504 | C2×C7 | semidirect |
| 33. #364 | C2×C7 | semidirect |
| 34. #350 | C2×C7 | semidirect |
| 35. C4×C16 | C2×C7 | semidirect |
| 91. C7 | #2170 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#504 | 2 |
| C7×#443 | 2 |
| C7×#364 | 2 |
| C7×#350 | 2 |
| C4×C7×C16 | 2 |
| #2170 | 7 |