Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
708 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 197 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 | C2×#6493 | C2 |
| hypercenter | C4×C22⋊C4 | D7 |
| hypocenter | C7 | #2501 |
| 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 |
|---|---|---|
| 2. C22×C4⋊Dic7 | C2 | semidirect |
| 3. C2×#6449 | C2 | semidirect |
| 4. C4×C7×C22⋊C4 | C2 | semidirect |
| 5. #89711 | C2 | semidirect |
| 6. #89707 | C2 | semidirect |
| 7. #89449 | C2 | semidirect |
| 40. C7×C22⋊C4 | D4 | semidirect |
| 54. C4×C22⋊C4 | D7 | semidirect |
| 109. C4×C7 | #123 | semidirect |
| 133. C22⋊C4 | D28 | semidirect |
| 165. C7 | #2501 | semidirect |
| 182. C4 | #6449 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6493 | 2 |
| C22×C4⋊Dic7 | 2 |
| C2×#6449 | 2 |
| C4×C7×C22⋊C4 | 2 |
| #89711 | 2 |
| #89707 | 2 |
| #89449 | 2 |
| #2501 | 7 |