Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×D269 |
| commutator, socle, cosocle | C22×C269 | C24 |
| Fitting | #127×C269 | C2 |
| hypercenter | #127 | D269 |
| hypocenter | C269 | #535 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16864401 | C2 | semidirect |
| 2. C2×D4×D269 | C2 | semidirect |
| 3. #22575449 | C2 | semidirect |
| 4. #22575441 | C2 | semidirect |
| 5. #22575440 | C2 | semidirect |
| 6. #22575425 | C2 | semidirect |
| 7. #22575421 | C2 | semidirect |
| 8. #22575410 | C2 | semidirect |
| 9. #127×C269 | C2 | semidirect |
| 12. C2×Dic538 | C22 | semidirect |
| 16. C2×C22⋊D269 | C22 | semidirect |
| 18. C22×Dic269 | C22 | semidirect |
| 20. #16864381 | C22 | semidirect |
| 21. C22⋊Dic269 | C22 | semidirect |
| 23. #16864379 | C22 | semidirect |
| 24. C4×Dic269 | C22 | semidirect |
| 26. C2×D4×C269 | C22 | semidirect |
| 27. C2×D4×C269 | C22 | semidirect |
| 28. C42×C269 | C22 | semidirect |
| 42. C4×D269 | D4 | semidirect |
| 71. C269 | #535 | semidirect |
| 76. #127 | D269 | semidirect |
| 91. C2×D4 | C2×D269 | semidirect |
| 92. C2×D4 | C2×D269 | semidirect |
| 100. C42 | C2×D269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16864401 | 2 |
| C2×D4×D269 | 2 |
| #22575449 | 2 |
| #22575441 | 2 |
| #22575440 | 2 |
| #22575425 | 2 |
| #22575421 | 2 |
| #22575410 | 2 |
| #127×C269 | 2 |
| #535 | 269 |