Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
290 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 107 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 | C4×D4×C269 | C2 |
| hypercenter | C4×D4 | D269 |
| hypocenter | C269 | #548 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16864402 | C2 | semidirect |
| 2. C22×Dic538 | C2 | semidirect |
| 3. #22575441 | C2 | semidirect |
| 4. #22575439 | C2 | semidirect |
| 5. #22575432 | C2 | semidirect |
| 6. #22575431 | C2 | semidirect |
| 7. #22575427 | C2 | semidirect |
| 8. #22575425 | C2 | semidirect |
| 9. C4×Dic538 | C2 | semidirect |
| 10. #22575413 | C2 | semidirect |
| 11. C4×D4×C269 | C2 | semidirect |
| 40. Dic538 | D4 | semidirect |
| 64. C22×C269 | #41 | semidirect |
| 83. C269 | #548 | semidirect |
| 95. C4×D4 | D269 | semidirect |
| 151. C22 | #16864402 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16864402 | 2 |
| C22×Dic538 | 2 |
| #22575441 | 2 |
| #22575439 | 2 |
| #22575432 | 2 |
| #22575431 | 2 |
| #22575427 | 2 |
| #22575425 | 2 |
| C4×Dic538 | 2 |
| #22575413 | 2 |
| C4×D4×C269 | 2 |
| #548 | 269 |