Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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 | C22⋊Q8×C269 | C2 |
| hypercenter | C22⋊Q8 | D269 |
| hypocenter | C269 | #562 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22575443 | C2 | semidirect |
| 2. #22575442 | C2 | semidirect |
| 3. #22575439 | C2 | semidirect |
| 4. #22575438 | C2 | semidirect |
| 5. #22575438 | C2 | semidirect |
| 6. #22575436 | C2 | semidirect |
| 7. #22575435 | C2 | semidirect |
| 8. #22575433 | C2 | semidirect |
| 9. #22575431 | C2 | semidirect |
| 10. #22575429 | C2 | semidirect |
| 11. #22575428 | C2 | semidirect |
| 12. #22575427 | C2 | semidirect |
| 13. #22575426 | C2 | semidirect |
| 14. #22575426 | C2 | semidirect |
| 15. C22⋊Q8×C269 | C2 | semidirect |
| 109. C269 | #562 | semidirect |
| 121. C22⋊Q8 | D269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22575443 | 2 |
| #22575442 | 2 |
| #22575439 | 2 |
| #22575438 | 2 |
| #22575436 | 2 |
| #22575435 | 2 |
| #22575433 | 2 |
| #22575431 | 2 |
| #22575429 | 2 |
| #22575428 | 2 |
| #22575427 | 2 |
| #22575426 | 2 |
| C22⋊Q8×C269 | 2 |
| #562 | 269 |