Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D83 |
| commutator, cosocle | C22×C83 | C25 |
| Frattini | C22 | C24×D83 |
| Fitting | C2×#125×C83 | C2 |
| socle | C23×C83 | C24 |
| hypercenter | C2×#125 | D83 |
| hypocenter | C83 | C2×#540 |
| LCS | G ▷ C22×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19128307 | C2 | direct (N × H) |
| 2. C2×#14581789 | C2 | semidirect |
| 3. C2×#14581777 | C2 | semidirect |
| 4. C2×#14581767 | C2 | semidirect |
| 5. C2×#14581767 | C2 | semidirect |
| 6. C2×#14581754 | C2 | semidirect |
| 7. C2×#14581749 | C2 | semidirect |
| 8. C2×#125×C83 | C2 | semidirect |
| 14. #14581789 | C22 | semidirect |
| 15. #14581777 | C22 | semidirect |
| 16. #14581767 | C22 | semidirect |
| 17. #14581767 | C22 | semidirect |
| 18. #14581754 | C22 | semidirect |
| 19. #14581749 | C22 | semidirect |
| 25. #125×C83 | C22 | semidirect |
| 92. C2×C83 | #540 | semidirect |
| 98. C83 | C2×#540 | semidirect |
| 101. C2×#125 | D83 | semidirect |
| 113. #125 | C2×D83 | semidirect |
| 192. C2 | #19128307 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19128307 | 2 |
| C2×#14581789 | 2 |
| C2×#14581777 | 2 |
| C2×#14581767 | 2 |
| C2×#14581754 | 2 |
| C2×#14581749 | 2 |
| C2×#125×C83 | 2 |
| C2×#540 | 83 |