Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
344 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C2×D4 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C24×C7 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C7×#340 | C2 | semidirect |
| 10. C2×C7×C4⋊C4 | C4 | semidirect |
| 16. #1824 | C7 | direct (N × H) |
| 37. #340 | C2×C7 | semidirect |
| 63. C2×C4⋊C4 | C4×C7 | semidirect |
| 127. C7 | #1824 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×C4⋊C4 | 2 |
| C2×C7×C22⋊C8 | 2 |
| C7×#380 | 2 |
| C7×#340 | 2 |
| #1824 | 7 |