Finite Group DatabasebetaFinite Group DatabasebetaOrder 17344.
236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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 | #121×C271 | C2 |
| hypercenter | #121 | D271 |
| hypocenter | C271 | #555 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22576159 | C2 | semidirect |
| 2. #22576147 | C2 | semidirect |
| 3. #22576146 | C2 | semidirect |
| 4. #22576132 | C2 | semidirect |
| 5. #22576131 | C2 | semidirect |
| 6. D4×Dic271 | C2 | semidirect |
| 7. C4×C22⋊D271 | C2 | semidirect |
| 8. #22576119 | C2 | semidirect |
| 9. #22576118 | C2 | semidirect |
| 10. #22576116 | C2 | semidirect |
| 11. #121×C271 | C2 | semidirect |
| 78. C271 | #555 | semidirect |
| 85. #121 | D271 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22576159 | 2 |
| #22576147 | 2 |
| #22576146 | 2 |
| #22576132 | 2 |
| #22576131 | 2 |
| D4×Dic271 | 2 |
| C4×C22⋊D271 | 2 |
| #22576119 | 2 |
| #22576118 | 2 |
| #22576116 | 2 |
| #121×C271 | 2 |
| #555 | 271 |