Finite Group DatabasebetaFinite Group DatabasebetaOrder 17344.
270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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, prime radical | C2×#41×C271 | C2 |
| hypercenter | C2×#41 | D271 |
| hypocenter | C271 | #536 |
| LCS | G ▷ C22×C271 ▷ C271 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Dic542 | C2 | semidirect |
| 2. #22576158 | C2 | semidirect |
| 3. #22576147 | C2 | semidirect |
| 4. #22576141 | C2 | semidirect |
| 5. #22576139 | C2 | semidirect |
| 6. #22576119 | C2 | semidirect |
| 52. C271 | #536 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Dic542 | 2 |
| #22576158 | 2 |
| #22576147 | 2 |
| #22576141 | 2 |
| #22576139 | 2 |
| #22576119 | 2 |
| C2×#41×C271 | 2 |
| #536 | 271 |