Finite Group DatabasebetaFinite Group DatabasebetaOrder 34432.
4544 conjugacy classes of subgroups; only the 805 normal subgroups are shown.
This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 805 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D269 |
| commutator, cosocle | C22×C269 | C25 |
| Frattini | C22 | C24×D269 |
| Fitting | C26×C269 | C2 |
| socle | C24×C269 | C23 |
| hypercenter | C26 | D269 |
| hypocenter | C269 | C22×#120 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C24 ◁ C26 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C22⋊D269 | C2 | semidirect |
| 2. C2×#22575422 | C2 | direct (N × H) |
| 3. C22×C22⋊Dic269 | C2 | semidirect |
| 4. C26×C269 | C2 | semidirect |
| 5. C24×D269 | C22 | semidirect |
| 7. C22×C22⋊D269 | C22 | semidirect |
| 8. C23×Dic269 | C22 | semidirect |
| 9. #22575422 | C22 | direct (N × H) |
| 10. C2×C22⋊Dic269 | C22 | semidirect |
| 13. C25×C269 | C22 | semidirect |
| 15. C23×D269 | C23 | semidirect |
| 17. C2×C22⋊D269 | C23 | semidirect |
| 19. C22×Dic269 | C23 | semidirect |
| 20. C22⋊Dic269 | C23 | semidirect |
| 24. C24×C269 | D4 | semidirect |
| 27. C24×C269 | C23 | semidirect |
| 30. C22×D269 | C24 | semidirect |
| 33. C2×Dic269 | C24 | semidirect |
| 39. C23×C269 | C2×D4 | semidirect |
| 45. C22×C269 | #120 | semidirect |
| 47. C22×C269 | C22×D4 | semidirect |
| 53. C2×C269 | C2×#120 | semidirect |
| 56. C269 | C22×#120 | semidirect |
| 58. C26 | D269 | semidirect |
| 65. C25 | C2×D269 | semidirect |
| 76. C24 | C22⋊D269 | semidirect |
| 79. C24 | C22×D269 | semidirect |
| 93. C23 | C2×C22⋊D269 | semidirect |
| 104. C22 | #22575422 | direct (N × H) |
| 106. C22 | C22×C22⋊D269 | semidirect |
| 111. C2 | C2×#22575422 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C22⋊D269 | 2 |
| C2×#22575422 | 2 |
| C22×C22⋊Dic269 | 2 |
| C26×C269 | 2 |
| C22×#120 | 269 |