Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #6516 |
| commutator, cosocle | C22×C4×C7 | C23 |
| Frattini | C22×C4 | C22×D7 |
| Fitting | C7×#537 | C2 |
| socle | C22×C7 | #120 |
| hypercenter | #537 | D7 |
| hypocenter | C7 | #2006 |
| LCS | G ▷ C22×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #537 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#537 | C2 | semidirect |
| 2. C2×#6512 | C2 | semidirect |
| 3. C2×#6411 | C2 | semidirect |
| 4. #89760 | C2 | semidirect |
| 5. #89325 | C2 | semidirect |
| 37. C2×D28 | D4 | semidirect |
| 46. #1136 | D4 | semidirect |
| 49. #537 | D7 | semidirect |
| 177. C7 | #2006 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#537 | 2 |
| C2×#6512 | 2 |
| C2×#6411 | 2 |
| #89760 | 2 |
| #89325 | 2 |
| #2006 | 7 |