Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
126 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D7 |
| commutator, socle, prime radical | C22×C7 | C22×C8 |
| Fitting | C7×C4⋊C16 | C2 |
| cosocle | C2×C7×C8 | C23 |
| hypercenter | C4⋊C16 | D7 |
| hypocenter | C7 | #2183 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C8 ◁ C4⋊C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 34. C7⋊C16 | Q8 | semidirect |
| 106. C7 | #2183 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×Dic14 | 2 |
| C7×C4⋊C16 | 2 |
| #89297 | 2 |
| Dic7⋊C16 | 2 |
| C16×Dic7 | 2 |
| #89256 | 2 |
| #2183 | 7 |