Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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×#126 | C2 |
| prime radical | C22×C22⋊D7 | C22 |
| hypercenter | C2×#126 | D7 |
| hypocenter | C7 | #2823 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#126 | C2 | semidirect |
| 2. C2×#6439 | C2 | semidirect |
| 3. #89766 | C2 | semidirect |
| 4. #89756 | C2 | semidirect |
| 6. #89720 | C2 | semidirect |
| 7. #89707 | C2 | semidirect |
| 34. C42×C7 | D4 | semidirect |
| 43. C2×#126 | D7 | semidirect |
| 116. C42 | C22⋊D7 | semidirect |
| 143. C7 | #2823 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#126 | 2 |
| C2×#6439 | 2 |
| #89766 | 2 |
| #89756 | 2 |
| #89733 | 2 |
| #89720 | 2 |
| #89707 | 2 |
| #2823 | 7 |