Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
816 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C7 |
| prime radical | C22×C7×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×C22⋊Q8 | C2 | semidirect |
| 3. C7×#395 | C2 | semidirect |
| 4. C7×#394 | C2 | semidirect |
| 5. C7×#380 | C2 | semidirect |
| 17. #2616 | C7 | direct (N × H) |
| 35. C2×C22⋊Q8 | C2×C7 | semidirect |
| 36. #395 | C2×C7 | semidirect |
| 37. #394 | C2×C7 | semidirect |
| 38. #380 | C2×C7 | semidirect |
| 116. C7 | #2616 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×#41 | 2 |
| C2×C7×C22⋊Q8 | 2 |
| C7×#395 | 2 |
| C7×#394 | 2 |
| C7×#380 | 2 |
| #2616 | 7 |