Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 580 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 | #393×C173 | C2 |
| prime radical | C24×D173 | C22 |
| hypercenter | #393 | D173 |
| hypocenter | C173 | #2840 |
| LCS | G ▷ C23×C173 ▷ C173 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26011404 | C2 | semidirect |
| 2. #26011403 | C2 | semidirect |
| 3. #26011402 | C2 | semidirect |
| 5. #393×C173 | C2 | semidirect |
| 7. C22×Dic346 | C22 | semidirect |
| 11. #19149586 | C22 | semidirect |
| 12. #19149586 | C22 | semidirect |
| 13. C22×Q8×C173 | C22 | semidirect |
| 15. #95×C173 | C22 | semidirect |
| 18. C23×D173 | Q8 | semidirect |
| 68. C173 | #2840 | semidirect |
| 73. #393 | D173 | semidirect |
| 77. C22×Q8 | C2×D173 | semidirect |
| 84. #95 | C2×D173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26011404 | 2 |
| #26011403 | 2 |
| #26011402 | 2 |
| #26011346 | 2 |
| #393×C173 | 2 |
| #2840 | 173 |