Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
840 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Frattini | C23 | C23×D7 |
| Fitting | C2×C4×C7×D4 | C2 |
| hypercenter | C2×C4×D4 | D7 |
| hypocenter | C7 | #2397 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4×C7×D4 | C2 | semidirect |
| 2. C2×C4×C22⋊D7 | C2 | semidirect |
| 3. C2×C4×D28 | C2 | semidirect |
| 4. #89995 | C2 | semidirect |
| 5. #89764 | C2 | semidirect |
| 6. #89732 | C2 | semidirect |
| 7. #89721 | C2 | semidirect |
| 8. #89704 | C2 | semidirect |
| 50. C42×C7 | D4 | semidirect |
| 56. C4×Dic7 | D4 | semidirect |
| 59. C2×C4×D4 | D7 | semidirect |
| 160. C42 | C22⋊D7 | semidirect |
| 196. C7 | #2397 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C7×D4 | 2 |
| C2×C4×C22⋊D7 | 2 |
| C2×C4×D28 | 2 |
| #89995 | 2 |
| #89764 | 2 |
| #89732 | 2 |
| #89721 | 2 |
| #89704 | 2 |
| #2397 | 7 |