Finite Group DatabasebetaFinite Group DatabasebetaOrder 17344.
292 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 107 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 | #547 |
| 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. C2×#16864653 | C2 | semidirect |
| 2. #22576159 | C2 | semidirect |
| 3. #22576142 | C2 | semidirect |
| 4. #22576141 | C2 | semidirect |
| 5. #22576137 | C2 | semidirect |
| 6. #22576131 | C2 | semidirect |
| 7. D4×Dic271 | C2 | semidirect |
| 8. C4×C22⋊D271 | C2 | semidirect |
| 9. C2×C4⋊Dic271 | C2 | semidirect |
| 10. C4⋊C4×D271 | C2 | semidirect |
| 11. #121×C271 | C2 | semidirect |
| 39. C22⋊D271 | D4 | semidirect |
| 69. C4×C271 | #41 | semidirect |
| 81. C271 | #547 | semidirect |
| 89. #121 | D271 | semidirect |
| 156. C4 | #16864653 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16864653 | 2 |
| #22576159 | 2 |
| #22576142 | 2 |
| #22576141 | 2 |
| #22576137 | 2 |
| #22576131 | 2 |
| D4×Dic271 | 2 |
| C4×C22⋊D271 | 2 |
| C2×C4⋊Dic271 | 2 |
| C4⋊C4×D271 | 2 |
| #121×C271 | 2 |
| #547 | 271 |