Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 216 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, prime radical | #121×C269 | C2 |
| hypercenter | #121 | D269 |
| hypocenter | C269 | #542 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22575452 | C2 | semidirect |
| 2. #22575446 | C2 | semidirect |
| 3. #22575440 | C2 | semidirect |
| 4. #22575440 | C2 | semidirect |
| 5. #22575434 | C2 | semidirect |
| 6. #22575432 | C2 | semidirect |
| 7. #22575430 | C2 | semidirect |
| 8. #22575418 | C2 | semidirect |
| 9. D4×Dic269 | C2 | semidirect |
| 10. #22575412 | C2 | semidirect |
| 73. C269 | #542 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22575452 | 2 |
| #22575446 | 2 |
| #22575440 | 2 |
| #22575434 | 2 |
| #22575432 | 2 |
| #22575430 | 2 |
| #22575418 | 2 |
| D4×Dic269 | 2 |
| #22575412 | 2 |
| #121×C269 | 2 |
| #542 | 269 |