Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D173 |
| commutator, socle, cosocle | C23×C173 | C24 |
| Fitting | #394×C173 | C2 |
| prime radical | C22×D4×C173 | C22 |
| hypercenter | #394 | D173 |
| hypocenter | C173 | #2398 |
| LCS | G ▷ C23×C173 ▷ C173 | |
| UCS | C1 ◁ C23 ◁ #394 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19149644 | C2 | semidirect |
| 3. #26011431 | C2 | semidirect |
| 4. #26011430 | C2 | semidirect |
| 5. #26011419 | C2 | semidirect |
| 7. #26011403 | C2 | semidirect |
| 8. C22⋊C4×Dic173 | C2 | semidirect |
| 9. C22⋊C4×Dic173 | C2 | semidirect |
| 10. #26011336 | C2 | semidirect |
| 134. C173 | #2398 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19149644 | 2 |
| C2×D4×Dic173 | 2 |
| #26011431 | 2 |
| #26011430 | 2 |
| #26011419 | 2 |
| #26011405 | 2 |
| #26011403 | 2 |
| C22⋊C4×Dic173 | 2 |
| #26011336 | 2 |
| #394×C173 | 2 |
| #2398 | 173 |