Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
164 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 71 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 | C22×C7 | C22×C8 |
| Fitting | C7×#346 | C2 |
| prime radical | C2×D28 | C8 |
| cosocle | C2×C7×C8 | C23 |
| hypercenter | #346 | D7 |
| hypocenter | C7 | #2168 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C8 ◁ #346 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6372 | C2 | semidirect |
| 2. C2×D7×C16 | C2 | semidirect |
| 4. C7×#346 | C2 | semidirect |
| 5. (C2×D7)⋊C16 | C2 | semidirect |
| 6. (C2×D7)⋊C16 | C2 | semidirect |
| 7. C4⋊(C7⋊C16) | C2 | semidirect |
| 38. C7×C16 | D4 | semidirect |
| 47. #346 | D7 | semidirect |
| 84. C2×D7 | M32 | semidirect |
| 109. C16 | D28 | semidirect |
| 132. C7 | #2168 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6372 | 2 |
| C2×D7×C16 | 2 |
| C8×D28 | 2 |
| C7×#346 | 2 |
| (C2×D7)⋊C16 | 2 |
| C4⋊(C7⋊C16) | 2 |
| #2168 | 7 |