Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
696 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 165 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 | #391×C173 | C2 |
| prime radical | C2×#19149654 | C2 |
| hypercenter | #391 | D173 |
| hypocenter | C173 | #2850 |
| LCS | G ▷ C23×C173 ▷ C173 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19149668 | C2 | semidirect |
| 2. C2×#19149655 | C2 | semidirect |
| 4. C2×C42×D173 | C2 | semidirect |
| 5. #26011393 | C2 | semidirect |
| 6. #26011380 | C2 | semidirect |
| 7. #26011369 | C2 | semidirect |
| 8. #26011356 | C2 | semidirect |
| 9. #391×C173 | C2 | semidirect |
| 54. C4×Dic173 | D4 | semidirect |
| 113. C2×D173 | #124 | semidirect |
| 128. C173 | #2850 | semidirect |
| 134. #391 | D173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19149668 | 2 |
| C2×#19149655 | 2 |
| C2×#19149654 | 2 |
| C2×C42×D173 | 2 |
| #26011393 | 2 |
| #26011380 | 2 |
| #26011369 | 2 |
| #26011356 | 2 |
| #391×C173 | 2 |
| #2850 | 173 |