Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
772 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#6516 |
| commutator, cosocle | C23×C7 | C24 |
| Frattini | C23 | C23×D7 |
| Fitting | C7×#585 | C2 |
| socle | C2×C7 | C2×#120 |
| hypercenter | #585 | D7 |
| hypocenter | C7 | #3027 |
| LCS | G ▷ C23×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#585 | C2 | semidirect |
| 2. #90521 | C2 | semidirect |
| 3. #90016 | C2 | semidirect |
| 4. #90012 | C2 | semidirect |
| 10. C22×D28 | C22 | semidirect |
| 13. #6500 | C22 | semidirect |
| 15. #6489 | C22 | semidirect |
| 16. #6408 | C22 | semidirect |
| 25. C2×C7×Q8 | D4 | semidirect |
| 28. C2×C7×D4 | D4 | semidirect |
| 31. C22×C4×C7 | D4 | semidirect |
| 40. #585 | D7 | semidirect |
| 94. C2×Q8 | C22⋊D7 | semidirect |
| 100. C2×D4 | C22⋊D7 | semidirect |
| 103. C22×C4 | C22⋊D7 | semidirect |
| 133. C7 | #3027 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#585 | 2 |
| #90521 | 2 |
| #90016 | 2 |
| #90012 | 2 |
| #3027 | 7 |