Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
1112 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 575 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D7 |
| commutator, socle, cosocle | C22×C7 | C25 |
| Fitting | C7×Q82 | C2 |
| prime radical | C22×D7 | C24 |
| hypercenter | Q82 | D7 |
| hypocenter | C7 | C2×Q82 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×Q82 | C2 | semidirect |
| 3. #90413 | C2 | semidirect |
| 5. Q8×Dic14 | C2 | semidirect |
| 22. Q8×D7 | Q8 | direct (N × H) |
| 35. Q82 | D7 | direct (N × H) |
| 40. C7×Q8 | C2×Q8 | semidirect |
| 48. Dic14 | C2×Q8 | semidirect |
| 84. D7 | Q82 | direct (N × H) |
| 89. Q8 | Q8×D7 | direct (N × H) |
| 96. C7 | C2×Q82 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×Q82 | 2 |
| D7×C4⋊Q8 | 2 |
| #90413 | 2 |
| C4×Q8×D7 | 2 |
| Q8×Dic14 | 2 |
| C2×Q82 | 7 |