Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
290 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 105 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 | #548 |
| 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. C2×#16864402 | C2 | semidirect |
| 2. C2×Q8×D269 | C2 | semidirect |
| 3. #22575442 | C2 | semidirect |
| 4. #22575439 | C2 | semidirect |
| 5. #22575431 | C2 | semidirect |
| 6. #22575428 | C2 | semidirect |
| 7. #22575426 | C2 | semidirect |
| 8. #22575418 | C2 | semidirect |
| 9. #22575415 | C2 | semidirect |
| 10. C4×C22⋊D269 | C2 | semidirect |
| 11. C22⋊Q8×C269 | C2 | semidirect |
| 40. Dic538 | D4 | semidirect |
| 68. C2×D269 | #41 | semidirect |
| 83. C269 | #548 | semidirect |
| 90. C22⋊Q8 | D269 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16864402 | 2 |
| C2×Q8×D269 | 2 |
| #22575442 | 2 |
| #22575439 | 2 |
| #22575431 | 2 |
| #22575428 | 2 |
| #22575426 | 2 |
| #22575418 | 2 |
| #22575415 | 2 |
| C4×C22⋊D269 | 2 |
| C22⋊Q8×C269 | 2 |
| #548 | 269 |