Finite Group DatabasebetaFinite Group DatabasebetaOrder 3392.
248 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 95 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×D53 |
| commutator, socle, cosocle | C22×C53 | C24 |
| Fitting | #126×C53 | C2 |
| prime radical | #13133187 | C2 |
| hypercenter | #126 | D53 |
| hypocenter | C53 | #551 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13133210 | C2 | semidirect |
| 2. #13133203 | C2 | semidirect |
| 3. #13133202 | C2 | semidirect |
| 4. #13133202 | C2 | semidirect |
| 5. #13133199 | C2 | semidirect |
| 6. #13133197 | C2 | semidirect |
| 7. #13133195 | C2 | semidirect |
| 9. C4×D212 | C2 | semidirect |
| 10. #13133179 | C2 | semidirect |
| 11. #13133175 | C2 | semidirect |
| 12. #13133174 | C2 | semidirect |
| 13. C22⋊C4×D53 | C2 | semidirect |
| 14. C22⋊C4×D53 | C2 | semidirect |
| 15. #126×C53 | C2 | semidirect |
| 24. C2×C4×D53 | C22 | semidirect |
| 25. C2×Dic106 | C22 | semidirect |
| 26. C4⋊C4×C53 | C22 | semidirect |
| 27. C4⋊C4×C53 | C22 | semidirect |
| 28. C4⋊C4×C53 | C22 | semidirect |
| 32. C42×C53 | C22 | semidirect |
| 41. #11719921 | C22 | semidirect |
| 42. #11719921 | C22 | semidirect |
| 43. #11719921 | C22 | semidirect |
| 44. C4⋊Dic53 | C22 | semidirect |
| 45. C4⋊Dic53 | C22 | semidirect |
| 46. #11719919 | C22 | semidirect |
| 47. #11719919 | C22 | semidirect |
| 48. #11719919 | C22 | semidirect |
| 49. C4×Dic53 | C22 | semidirect |
| 50. C4×Dic53 | C22 | semidirect |
| 111. C53 | #551 | semidirect |
| 113. #126 | D53 | semidirect |
| 142. C4⋊C4 | C2×D53 | semidirect |
| 143. C4⋊C4 | C2×D53 | semidirect |
| 144. C4⋊C4 | C2×D53 | semidirect |
| 161. C42 | C2×D53 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13133210 | 2 |
| #13133203 | 2 |
| #13133202 | 2 |
| #13133199 | 2 |
| #13133197 | 2 |
| #13133195 | 2 |
| #13133187 | 2 |
| C4×D212 | 2 |
| #13133179 | 2 |
| #13133175 | 2 |
| #13133174 | 2 |
| C22⋊C4×D53 | 2 |
| #126×C53 | 2 |
| #551 | 53 |