Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
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×D83 |
| commutator, socle, cosocle | C22×C83 | C25 |
| Frattini | C22 | C24×D83 |
| Fitting | C22×C42×C83 | C2 |
| hypercenter | C22×C42 | D83 |
| hypocenter | C83 | #3444 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C42 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19128258 | C2 | semidirect |
| 2. #19128251 | C2 | semidirect |
| 3. #19128235 | C2 | semidirect |
| 4. #19128212 | C2 | semidirect |
| 5. C4×#13050590 | C2 | semidirect |
| 6. C22×C42×C83 | C2 | semidirect |
| 8. #14581792 | C22 | semidirect |
| 9. #14581789 | C22 | semidirect |
| 10. #14581786 | C22 | semidirect |
| 11. #14581784 | C22 | semidirect |
| 12. #14581780 | C22 | semidirect |
| 17. C4×Dic166 | C22 | semidirect |
| 18. C4×D332 | C22 | semidirect |
| 21. #14581749 | C22 | semidirect |
| 22. C42×D83 | C22 | semidirect |
| 55. C2×C4×C83 | #41 | semidirect |
| 77. C83 | #3444 | semidirect |
| 83. C22×C42 | D83 | semidirect |
| 137. C2×C4 | #13050590 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19128258 | 2 |
| #19128251 | 2 |
| #19128235 | 2 |
| #19128212 | 2 |
| C4×#13050590 | 2 |
| C22×C42×C83 | 2 |
| #3444 | 83 |