Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
800 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 213 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 | #2434 |
| 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 |
|---|---|---|
| 3. C2×#6432 | C2 | semidirect |
| 6. #89701 | C2 | semidirect |
| 7. #89428 | C2 | semidirect |
| 45. C7×C22⋊C4 | Q8 | semidirect |
| 70. C2×C4×C7 | C2×Q8 | semidirect |
| 102. C22×C7 | C4⋊Q8 | semidirect |
| 136. C22⋊C4 | Dic14 | semidirect |
| 163. C2×C4 | C2×Dic14 | semidirect |
| 167. C7 | #2434 | semidirect |
| 171. C22 | #6432 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Dic14 | 2 |
| C2×#6487 | 2 |
| C2×#6432 | 2 |
| C4×C7×C22⋊C4 | 2 |
| #89722 | 2 |
| #89701 | 2 |
| #89428 | 2 |
| #2434 | 7 |