Finite Group DatabasebetaFinite Group DatabasebetaOrder 10240.
At least 7356 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has at least 7356 conjugacy classes of subgroups — too many to draw. Only its 14 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #72268127 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #72268062 | D5 | 1 | 1 | yes | characteristic, complemented |
| 3 | C26 | (C24⋊C5)⋊C2 | 1 | 1 | yes | characteristic |
| 4 | C2×2−1+4 | (C24⋊C5)⋊C2 | 1 | 4 | yes | complemented |
| 5 | C25 | #66244 | 1 | 1 | yes | characteristic, complemented |
| 6 | C22 | #72209701 | 1 | 1 | yes | characteristic |
| 7 | C2 | #72210032 | 1 | 1 | yes | characteristic |
| 8 | C2 | #72211305 | 1 | 1 | yes | characteristic |
| 9 | C2 | #72227629 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | #72209701 |
| commutator, cosocle, hypocenter | #72268127 | C2 |
| Fitting | #72268062 | D5 |
| LCS | G ▷ #72268127 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72268127 | C2 | semidirect |
| 2. #72268062 | D5 | semidirect |
| 4. C2×2−1+4 | (C24⋊C5)⋊C2 | semidirect |
| 5. C25 | #66244 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72268216 | 5 |
| #72268127 | 2 |
| C22×(C24⋊C5)⋊C2 | 16 |
| C2×#66244 | 16 |