Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 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 | #396×C173 | C2 |
| prime radical | C24×D173 | C22 |
| hypercenter | #396 | D173 |
| hypocenter | C173 | #2842 |
| LCS | G ▷ C23×C173 ▷ C173 | |
| UCS | C1 ◁ C23 ◁ #396 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26011428 | C2 | semidirect |
| 2. #26011426 | C2 | semidirect |
| 3. #26011415 | C2 | semidirect |
| 4. #26011414 | C2 | semidirect |
| 5. #26011413 | C2 | semidirect |
| 7. #396×C173 | C2 | semidirect |
| 9. C2×C4⋊Dic173 | C22 | semidirect |
| 10. C2×#14583805 | C22 | semidirect |
| 11. C2×#14583805 | C22 | semidirect |
| 17. #19149586 | C22 | semidirect |
| 18. C2×C4⋊C4×C173 | C22 | semidirect |
| 20. #95×C173 | C22 | semidirect |
| 89. C173 | #2842 | semidirect |
| 96. #396 | D173 | semidirect |
| 102. C2×C4⋊C4 | C2×D173 | semidirect |
| 110. #95 | C2×D173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26011428 | 2 |
| #26011426 | 2 |
| #26011415 | 2 |
| #26011414 | 2 |
| #26011413 | 2 |
| #26011346 | 2 |
| #396×C173 | 2 |
| #2842 | 173 |