Finite Group DatabasebetaFinite Group DatabasebetaOrder 17344.
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×D271 |
| commutator, socle, cosocle | C22×C271 | C24 |
| Fitting | #123×C271 | C2 |
| prime radical | C2×#16864653 | C2 |
| hypercenter | #123 | D271 |
| hypocenter | C271 | #536 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Dic542 | C2 | semidirect |
| 3. #22576159 | C2 | semidirect |
| 4. #22576146 | C2 | semidirect |
| 5. #22576144 | C2 | semidirect |
| 6. #22576143 | C2 | semidirect |
| 7. #22576141 | C2 | semidirect |
| 8. #22576138 | C2 | semidirect |
| 9. #22576137 | C2 | semidirect |
| 10. #22576116 | C2 | semidirect |
| 11. #123×C271 | C2 | semidirect |
| 79. C271 | #536 | semidirect |
| 88. #123 | D271 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16864653 | 2 |
| C22×Dic542 | 2 |
| #22576159 | 2 |
| #22576146 | 2 |
| #22576144 | 2 |
| #22576143 | 2 |
| #22576141 | 2 |
| #22576138 | 2 |
| #22576137 | 2 |
| #22576116 | 2 |
| #123×C271 | 2 |
| #536 | 271 |