Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D7 |
| commutator, socle, cosocle | C2×C7 | C26 |
| Fitting | C2×C7×2−1+4 | C2 |
| hypercenter | C2×2−1+4 | D7 |
| hypocenter | C7 | 2−1+6 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×2−1+4 | C2 | semidirect |
| 2. #90620 | C2 | semidirect |
| 3. D7×2−1+4 | C2 | semidirect |
| 4. #90616 | C2 | semidirect |
| 5. C2×#6551 | C2 | semidirect |
| 6. C7×2−1+4 | C22 | semidirect |
| 7. C2×C7×#41 | C22 | semidirect |
| 8. C22×C7×Q8 | C22 | semidirect |
| 9. #6554 | C22 | semidirect |
| 10. #6553 | C22 | semidirect |
| 11. D7×#41 | C22 | semidirect |
| 12. #6551 | C22 | semidirect |
| 13. #6551 | C22 | semidirect |
| 14. C2×#1161 | C22 | semidirect |
| 15. C2×Q8×D7 | C22 | semidirect |
| 16. #6548 | C22 | semidirect |
| 17. C2×#1157 | C22 | semidirect |
| 38. C2×2−1+4 | D7 | semidirect |
| 54. 2−1+4 | C2×D7 | semidirect |
| 60. C2×#41 | C2×D7 | semidirect |
| 62. C22×Q8 | C2×D7 | semidirect |
| 101. C7 | 2−1+6 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×2−1+4 | 2 |
| #90620 | 2 |
| D7×2−1+4 | 2 |
| #90616 | 2 |
| C2×#6551 | 2 |
| 2−1+6 | 7 |