Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
1212 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D173 |
| commutator, socle, cosocle | C22×C173 | C25 |
| Frattini | C22 | C24×D173 |
| Fitting | C22×C42×C173 | C2 |
| hypercenter | C22×C42 | D173 |
| hypocenter | C173 | #3444 |
| LCS | G ▷ C22×C173 ▷ C173 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26012048 | C2 | semidirect |
| 2. #26012041 | C2 | semidirect |
| 3. #26012025 | C2 | semidirect |
| 4. #26012002 | C2 | semidirect |
| 5. C4×#14583826 | C2 | semidirect |
| 6. C22×C42×C173 | C2 | semidirect |
| 8. #19149679 | C22 | semidirect |
| 9. #19149676 | C22 | semidirect |
| 10. #19149673 | C22 | semidirect |
| 11. #19149671 | C22 | semidirect |
| 12. #19149667 | C22 | semidirect |
| 17. C4×Dic346 | C22 | semidirect |
| 18. C4×D692 | C22 | semidirect |
| 21. #19149636 | C22 | semidirect |
| 22. C42×D173 | C22 | semidirect |
| 60. C2×C4×C173 | #41 | semidirect |
| 76. C173 | #3444 | semidirect |
| 83. C22×C42 | D173 | semidirect |
| 137. C2×C4 | #14583826 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26012048 | 2 |
| #26012041 | 2 |
| #26012025 | 2 |
| #26012002 | 2 |
| C4×#14583826 | 2 |
| C22×C42×C173 | 2 |
| #3444 | 173 |