Finite Group DatabasebetaFinite Group DatabasebetaOrder 17344.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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 | #126×C271 | C2 |
| prime radical | C2×C22⋊D271 | C22 |
| hypercenter | #126 | D271 |
| hypocenter | C271 | #563 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C22 ◁ #126 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22576160 | C2 | semidirect |
| 2. #22576159 | C2 | semidirect |
| 3. #22576155 | C2 | semidirect |
| 4. #22576154 | C2 | semidirect |
| 5. #22576153 | C2 | semidirect |
| 6. #22576151 | C2 | semidirect |
| 7. #22576150 | C2 | semidirect |
| 11. #22576140 | C2 | semidirect |
| 12. #22576138 | C2 | semidirect |
| 13. #22576137 | C2 | semidirect |
| 14. #22576137 | C2 | semidirect |
| 15. #126×C271 | C2 | semidirect |
| 97. C271 | #563 | semidirect |
| 104. #126 | D271 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22576160 | 2 |
| #22576159 | 2 |
| #22576155 | 2 |
| #22576154 | 2 |
| #22576153 | 2 |
| #22576151 | 2 |
| #22576150 | 2 |
| #22576146 | 2 |
| #22576144 | 2 |
| #22576142 | 2 |
| #22576140 | 2 |
| #22576138 | 2 |
| #22576137 | 2 |
| #126×C271 | 2 |
| #563 | 271 |