Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
676 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 157 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 | C7×#390 | C2 |
| prime radical | C2×#6503 | C2 |
| hypercenter | #390 | D7 |
| hypocenter | C7 | #2548 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#6499 | C2 | semidirect |
| 3. C2×#6450 | C2 | semidirect |
| 4. C2×#6438 | C2 | semidirect |
| 5. C7×#390 | C2 | semidirect |
| 6. #89760 | C2 | semidirect |
| 7. #89731 | C2 | semidirect |
| 8. #89729 | C2 | semidirect |
| 9. C4×Dic7⋊C4 | C2 | semidirect |
| 53. C2×Dic14 | D4 | semidirect |
| 72. #390 | D7 | semidirect |
| 150. Dic7 | #124 | semidirect |
| 225. C7 | #2548 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6503 | 2 |
| C2×#6499 | 2 |
| C2×#6450 | 2 |
| C2×#6438 | 2 |
| C7×#390 | 2 |
| #89760 | 2 |
| #89731 | 2 |
| #89729 | 2 |
| C4×Dic7⋊C4 | 2 |
| #2548 | 7 |