Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
672 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C2×C7×C22⋊Q8 | C2 |
| prime radical | C24×D7 | C22 |
| hypercenter | C2×C22⋊Q8 | D7 |
| hypocenter | C7 | #2561 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×C22⋊Q8 | C2 | semidirect |
| 3. #90003 | C2 | semidirect |
| 4. C4×C22⋊Dic7 | C2 | semidirect |
| 5. #89765 | C2 | semidirect |
| 6. #89762 | C2 | semidirect |
| 8. #89715 | C2 | semidirect |
| 10. C22×C7×Q8 | C22 | semidirect |
| 14. C2×C7×C4⋊C4 | C22 | semidirect |
| 15. C2×C7×C4⋊C4 | C22 | semidirect |
| 23. C2×C4×Dic7 | C22 | semidirect |
| 24. C7⋊#95 | C22 | semidirect |
| 25. C7⋊#95 | C22 | semidirect |
| 26. C7⋊#95 | C22 | semidirect |
| 55. C2×C22⋊Q8 | D7 | semidirect |
| 94. C22×Q8 | C2×D7 | semidirect |
| 98. C2×C4⋊C4 | C2×D7 | semidirect |
| 99. C2×C4⋊C4 | C2×D7 | semidirect |
| 194. C7 | #2561 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×C22⋊Q8 | 2 |
| C22×(C2×D7)⋊C4 | 2 |
| #90003 | 2 |
| C4×C22⋊Dic7 | 2 |
| #89765 | 2 |
| #89762 | 2 |
| #89727 | 2 |
| #89715 | 2 |
| #2561 | 7 |