Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
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×D269 |
| commutator, socle, cosocle | C22×C269 | C24 |
| Fitting | #124×C269 | C2 |
| prime radical | #22575421 | C2 |
| hypercenter | #124 | D269 |
| hypocenter | C269 | #551 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22575446 | C2 | semidirect |
| 2. #22575435 | C2 | semidirect |
| 3. #22575434 | C2 | semidirect |
| 4. #22575429 | C2 | semidirect |
| 6. #22575413 | C2 | semidirect |
| 7. #22575410 | C2 | semidirect |
| 8. C22⋊C4×D269 | C2 | semidirect |
| 9. #124×C269 | C2 | semidirect |
| 12. C22×Dic269 | C22 | semidirect |
| 15. C4⋊Dic269 | C22 | semidirect |
| 16. #16864381 | C22 | semidirect |
| 17. #16864381 | C22 | semidirect |
| 18. #16864381 | C22 | semidirect |
| 21. C22⋊Dic269 | C22 | semidirect |
| 25. C4×Dic269 | C22 | semidirect |
| 26. C2×Q8×C269 | C22 | semidirect |
| 29. C22⋊C4×C269 | C22 | semidirect |
| 30. C42×C269 | C22 | semidirect |
| 68. C269 | #551 | semidirect |
| 71. #124 | D269 | semidirect |
| 80. C2×Q8 | C2×D269 | semidirect |
| 97. C22⋊C4 | C2×D269 | semidirect |
| 99. C42 | C2×D269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22575446 | 2 |
| #22575435 | 2 |
| #22575434 | 2 |
| #22575429 | 2 |
| #22575421 | 2 |
| #22575413 | 2 |
| #22575410 | 2 |
| C22⋊C4×D269 | 2 |
| #124×C269 | 2 |
| #551 | 269 |