Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
544 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 185 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C4×C7×C22⋊C4 | C2 |
| prime radical | C24×C7 | C23 |
| hypercenter | C4×C22⋊C4 | D7 |
| hypocenter | C7 | #2549 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #89701 | C2 | semidirect |
| 7. #89700 | C2 | semidirect |
| 8. #89433 | C2 | semidirect |
| 9. #89428 | C2 | semidirect |
| 35. C22×C4×C7 | Q8 | semidirect |
| 51. C7×C22⋊C4 | Q8 | semidirect |
| 152. C22×C4 | Dic14 | semidirect |
| 164. C22⋊C4 | Dic14 | semidirect |
| 204. C7 | #2549 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6441 | 2 |
| C4×C7×C22⋊C4 | 2 |
| #89984 | 2 |
| C2×C7⋊#95 | 2 |
| #89722 | 2 |
| #89701 | 2 |
| #89700 | 2 |
| #89433 | 2 |
| #89428 | 2 |
| #2549 | 7 |