Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
228 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 115 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 |
| prime radical | C2×D4×C269 | C22 |
| hypercenter | C4×D4 | D269 |
| hypocenter | C269 | #554 |
| 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. #22575453 | C2 | semidirect |
| 2. #22575444 | C2 | semidirect |
| 3. #22575432 | C2 | semidirect |
| 4. #22575430 | C2 | semidirect |
| 5. C4×Dic538 | C2 | semidirect |
| 7. C2×C4⋊Dic269 | C2 | semidirect |
| 32. D4×C269 | Q8 | semidirect |
| 53. C269 | #554 | semidirect |
| 82. D4 | Dic538 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22575453 | 2 |
| #22575444 | 2 |
| #22575432 | 2 |
| #22575430 | 2 |
| C4×Dic538 | 2 |
| D4×Dic269 | 2 |
| C2×C4⋊Dic269 | 2 |
| C4×D4×C269 | 2 |
| #554 | 269 |