Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
410 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C7 |
| prime radical | C24×C7 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#483 | C2 | semidirect |
| 3. C7×#470 | C2 | semidirect |
| 4. C7×#451 | C2 | semidirect |
| 8. C2×C7×Q16 | C22 | semidirect |
| 17. C7×#106 | C22 | semidirect |
| 18. C7×#103 | C22 | semidirect |
| 20. #3195 | C7 | direct (N × H) |
| 42. #483 | C2×C7 | semidirect |
| 43. #470 | C2×C7 | semidirect |
| 44. #451 | C2×C7 | semidirect |
| 66. C2×Q16 | C22×C7 | semidirect |
| 75. #106 | C22×C7 | semidirect |
| 76. #103 | C22×C7 | semidirect |
| 132. C7 | #3195 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×C22⋊Q8 | 2 |
| C7×#483 | 2 |
| C7×#470 | 2 |
| C7×#451 | 2 |
| C7×#407 | 2 |
| #3195 | 7 |