Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
238 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 238 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×D269 |
| commutator, socle, cosocle | C22×C269 | C24 |
| Fitting | #126×C269 | C2 |
| hypercenter | #126 | D269 |
| hypocenter | C269 | #553 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22575448 | C2 | semidirect |
| 2. #22575443 | C2 | semidirect |
| 3. #22575438 | C2 | semidirect |
| 4. #22575437 | C2 | semidirect |
| 5. #22575428 | C2 | semidirect |
| 6. #22575426 | C2 | semidirect |
| 7. #22575426 | C2 | semidirect |
| 8. #22575426 | C2 | semidirect |
| 9. C4×D1076 | C2 | semidirect |
| 10. #22575415 | C2 | semidirect |
| 11. #22575413 | C2 | semidirect |
| 12. #22575413 | C2 | semidirect |
| 13. #22575410 | C2 | semidirect |
| 14. C4⋊C4×D269 | C2 | semidirect |
| 15. #126×C269 | C2 | semidirect |
| 109. C269 | #553 | semidirect |
| 112. #126 | D269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22575448 | 2 |
| #22575443 | 2 |
| #22575438 | 2 |
| #22575437 | 2 |
| #22575428 | 2 |
| #22575426 | 2 |
| C4×D1076 | 2 |
| #22575415 | 2 |
| #22575413 | 2 |
| #22575410 | 2 |
| C4⋊C4×D269 | 2 |
| #126×C269 | 2 |
| #553 | 269 |