Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
728 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 165 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 | C2×#117×C83 | C2 |
| hypercenter | C2×#117 | D83 |
| hypocenter | C83 | #2619 |
| LCS | G ▷ C23×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ C2×#117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14581789 | C2 | semidirect |
| 2. C2×#14581766 | C2 | semidirect |
| 3. #19127618 | C2 | semidirect |
| 4. #19127610 | C2 | semidirect |
| 5. #19127603 | C2 | semidirect |
| 6. #19127567 | C2 | semidirect |
| 7. C2×#117×C83 | C2 | semidirect |
| 38. C22×C4×C83 | D4 | semidirect |
| 46. C42×C83 | D4 | semidirect |
| 98. C83 | #2619 | semidirect |
| 101. C2×#117 | D83 | semidirect |
| 133. C22×C4 | D332 | semidirect |
| 144. C42 | C22⋊D83 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#14581789 | 2 |
| C2×#14581766 | 2 |
| #19127618 | 2 |
| #19127610 | 2 |
| #19127603 | 2 |
| #19127567 | 2 |
| C2×#117×C83 | 2 |
| #2619 | 83 |