Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
836 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 836 conjugacy classes of subgroups — too many to draw. Only its 173 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 | C2×C7×C4⋊Q8 | C2 |
| prime radical | C22×#1161 | C2 |
| hypercenter | C2×C4⋊Q8 | D7 |
| hypocenter | C7 | #2432 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C7×C4⋊Q8 | C2 | semidirect |
| 3. C2×#6510 | C2 | semidirect |
| 4. C2×#6509 | C2 | semidirect |
| 5. #90002 | C2 | semidirect |
| 6. #89763 | C2 | semidirect |
| 7. #89761 | C2 | semidirect |
| 8. #89711 | C2 | semidirect |
| 9. C4×(C2×D7)⋊C4 | C2 | semidirect |
| 72. C2×C4⋊Q8 | D7 | semidirect |
| 104. C2×C4×C7 | #41 | semidirect |
| 240. C2×C4 | #1161 | semidirect |
| 243. C7 | #2432 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#1161 | 2 |
| C2×C7×C4⋊Q8 | 2 |
| C2×#6510 | 2 |
| C2×#6509 | 2 |
| #90002 | 2 |
| #89763 | 2 |
| #89761 | 2 |
| #89711 | 2 |
| C4×(C2×D7)⋊C4 | 2 |
| #2432 | 7 |