Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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 | C23×D167 |
| commutator, socle, cosocle | C22×C167 | C24 |
| Fitting | #123×C167 | C2 |
| prime radical | #19128800 | C2 |
| hypercenter | #123 | D167 |
| hypocenter | C167 | #551 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19128827 | C2 | semidirect |
| 2. #19128827 | C2 | semidirect |
| 3. #19128818 | C2 | semidirect |
| 4. #19128816 | C2 | semidirect |
| 5. #19128814 | C2 | semidirect |
| 6. #19128812 | C2 | semidirect |
| 7. #19128809 | C2 | semidirect |
| 9. D4×Dic167 | C2 | semidirect |
| 10. (C22⋊D167)⋊C4 | C2 | semidirect |
| 11. #19128791 | C2 | semidirect |
| 12. #19128790 | C2 | semidirect |
| 13. C2×(C2×D167)⋊C4 | C2 | semidirect |
| 14. C22⋊C4×D167 | C2 | semidirect |
| 15. #123×C167 | C2 | semidirect |
| 17. C2×Dic334 | C22 | semidirect |
| 21. C22×Dic167 | C22 | semidirect |
| 24. C4⋊Dic167 | C22 | semidirect |
| 25. C4⋊Dic167 | C22 | semidirect |
| 26. C4⋊Dic167 | C22 | semidirect |
| 27. C4⋊Dic167 | C22 | semidirect |
| 28. Dic167⋊C4 | C22 | semidirect |
| 29. Dic167⋊C4 | C22 | semidirect |
| 32. C22⋊Dic167 | C22 | semidirect |
| 41. C4×Dic167 | C22 | semidirect |
| 42. C4×Dic167 | C22 | semidirect |
| 43. C4×Dic167 | C22 | semidirect |
| 46. C4⋊C4×C167 | C22 | semidirect |
| 47. C4⋊C4×C167 | C22 | semidirect |
| 49. C22⋊C4×C167 | C22 | semidirect |
| 50. C22⋊C4×C167 | C22 | semidirect |
| 110. C167 | #551 | semidirect |
| 117. #123 | D167 | semidirect |
| 143. C4⋊C4 | C2×D167 | semidirect |
| 144. C4⋊C4 | C2×D167 | semidirect |
| 157. C22⋊C4 | C2×D167 | semidirect |
| 158. C22⋊C4 | C2×D167 | semidirect |