Finite Group DatabasebetaFinite Group DatabasebetaOrder 10688.
342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 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 | C2×D4×D167 | C2 |
| hypercenter | #123 | D167 |
| hypocenter | C167 | C2×#123 |
| LCS | G ▷ C22×C167 ▷ C167 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4×D167 | C2 | semidirect |
| 3. #19128817 | C2 | semidirect |
| 4. #19128816 | C2 | semidirect |
| 5. #19128815 | C2 | semidirect |
| 6. #19128814 | C2 | semidirect |
| 7. #19128813 | C2 | semidirect |
| 8. C4⋊C4×D167 | C2 | semidirect |
| 9. C22⋊C4×D167 | C2 | semidirect |
| 10. C22⋊C4×D167 | C2 | semidirect |
| 11. #123×C167 | C2 | semidirect |
| 18. C22×Dic167 | C22 | semidirect |
| 23. C2×C4×D167 | C22 | semidirect |
| 24. C4⋊Dic167 | C22 | semidirect |
| 25. Dic167⋊C4 | C22 | semidirect |
| 26. Dic167⋊C4 | C22 | semidirect |
| 27. C22⋊Dic167 | C22 | semidirect |
| 28. C22⋊Dic167 | C22 | semidirect |
| 29. (C2×D167)⋊C4 | C22 | semidirect |
| 30. (C2×D167)⋊C4 | C22 | semidirect |
| 31. (C2×D167)⋊C4 | C22 | semidirect |
| 33. C22×C4×C167 | C22 | semidirect |
| 34. C4⋊C4×C167 | C22 | semidirect |
| 35. C22⋊C4×C167 | C22 | semidirect |
| 36. C22⋊C4×C167 | C22 | semidirect |
| 87. D167 | #123 | direct (N × H) |
| 88. C167 | C2×#123 | semidirect |
| 97. #123 | D167 | direct (N × H) |
| 114. C22×C4 | C2×D167 | semidirect |
| 118. C4⋊C4 | C2×D167 | semidirect |
| 124. C22⋊C4 | C2×D167 | semidirect |
| 125. C22⋊C4 | C2×D167 | semidirect |