Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#391 | C2 |
| prime radical | C22×D28 | C22 |
| hypercenter | #391 | D7 |
| hypocenter | C7 | #2861 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#6453 | C2 | semidirect |
| 3. C7×#391 | C2 | semidirect |
| 4. #90002 | C2 | semidirect |
| 7. C42×Dic7 | C2 | semidirect |
| 8. #89442 | C2 | semidirect |
| 9. #89441 | C2 | semidirect |
| 61. #391 | D7 | semidirect |
| 194. C7 | #2861 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6510 | 2 |
| C2×#6453 | 2 |
| C7×#391 | 2 |
| #90002 | 2 |
| #89760 | 2 |
| #89711 | 2 |
| C42×Dic7 | 2 |
| #89442 | 2 |
| #89441 | 2 |
| #2861 | 7 |