Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
472 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 117 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×D83 |
| commutator, socle, cosocle | C23×C83 | C24 |
| Fitting, prime radical | #392×C83 | C2 |
| hypercenter | #392 | D83 |
| hypocenter | C83 | #2824 |
| LCS | G ▷ C23×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #19127642 | C2 | semidirect |
| 2. #19127629 | C2 | semidirect |
| 3. #19127615 | C2 | semidirect |
| 4. #19127615 | C2 | semidirect |
| 5. #19127613 | C2 | semidirect |
| 6. #19127613 | C2 | semidirect |
| 7. #19127581 | C2 | semidirect |
| 8. #19127573 | C2 | semidirect |
| 127. C83 | #2824 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19127642 | 2 |
| #19127629 | 2 |
| #19127615 | 2 |
| #19127613 | 2 |
| #19127581 | 2 |
| #19127573 | 2 |
| #392×C83 | 2 |
| #2824 | 83 |