Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
444 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 200 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 | C24×C7 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×#391 | C2 | semidirect |
| 4. C7×#387 | C2 | semidirect |
| 21. #2607 | C7 | direct (N × H) |
| 27. C22×C4×C7 | Q8 | semidirect |
| 42. #391 | C2×C7 | semidirect |
| 43. #387 | C2×C7 | semidirect |
| 98. C22×C4 | C7×Q8 | semidirect |
| 136. C7 | #2607 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#399 | 2 |
| C7×#393 | 2 |
| C7×#391 | 2 |
| C7×#387 | 2 |
| C7×#381 | 2 |
| C4×C7×C22⋊C4 | 2 |
| #2607 | 7 |