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 | #124×C53 | C2 |
| prime radical | #13133185 | C2 |
| hypercenter | #124 | D53 |
| hypocenter | C53 | #551 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13133210 | C2 | semidirect |
| 2. #13133199 | C2 | semidirect |
| 3. #13133198 | C2 | semidirect |
| 4. #13133193 | C2 | semidirect |
| 6. #13133177 | C2 | semidirect |
| 7. #13133174 | C2 | semidirect |
| 8. C22⋊C4×D53 | C2 | semidirect |
| 9. #124×C53 | C2 | semidirect |
| 11. C2×Q8×C53 | C22 | semidirect |
| 14. C22×Dic53 | C22 | semidirect |
| 18. C22⋊C4×C53 | C22 | semidirect |
| 19. C42×C53 | C22 | semidirect |
| 22. C22⋊Dic53 | C22 | semidirect |
| 26. C4⋊Dic53 | C22 | semidirect |
| 27. #11719919 | C22 | semidirect |
| 28. #11719919 | C22 | semidirect |
| 29. #11719919 | C22 | semidirect |
| 30. C4×Dic53 | C22 | semidirect |
| 69. C53 | #551 | semidirect |
| 71. #124 | D53 | semidirect |
| 80. C2×Q8 | C2×D53 | semidirect |
| 97. C22⋊C4 | C2×D53 | semidirect |
| 99. 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 |
| #13133199 | 2 |
| #13133198 | 2 |
| #13133193 | 2 |
| #13133185 | 2 |
| #13133177 | 2 |
| #13133174 | 2 |
| C22⋊C4×D53 | 2 |
| #124×C53 | 2 |
| #551 | 53 |