Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 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 | #392×C173 | C2 |
| hypercenter | #392 | D173 |
| hypocenter | C173 | #2388 |
| LCS | G ▷ C23×C173 ▷ C173 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14583825 | C2 | semidirect |
| 2. C2×D4×Dic173 | C2 | semidirect |
| 3. C2×#19149639 | C2 | semidirect |
| 4. #26011412 | C2 | semidirect |
| 5. #26011407 | C2 | semidirect |
| 6. #26011405 | C2 | semidirect |
| 7. #26011403 | C2 | semidirect |
| 8. #26011399 | C2 | semidirect |
| 9. C22⋊C4×Dic173 | C2 | semidirect |
| 10. #26011338 | C2 | semidirect |
| 11. #392×C173 | C2 | semidirect |
| 53. C22×Dic173 | D4 | semidirect |
| 60. C2×C4×D173 | D4 | semidirect |
| 104. C2×Dic173 | #41 | semidirect |
| 117. C23×C173 | #41 | semidirect |
| 127. C2×C4×C173 | #41 | semidirect |
| 166. C173 | #2388 | semidirect |
| 176. #392 | D173 | semidirect |
| 265. C23 | #14583825 | semidirect |
| 294. C2×C4 | #14583825 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14583825 | 2 |
| C2×D4×Dic173 | 2 |
| C2×#19149639 | 2 |
| #26011412 | 2 |
| #26011407 | 2 |
| #26011405 | 2 |
| #26011403 | 2 |
| #26011399 | 2 |
| C22⋊C4×Dic173 | 2 |
| #26011338 | 2 |
| #392×C173 | 2 |
| #2388 | 173 |