Finite Group DatabasebetaFinite Group DatabasebetaOrder 61440.
At least 19058 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has at least 19058 conjugacy classes of subgroups — too many to draw. Only its 9 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 | C2×#6342371 | A5 | 1 | 1 | yes | maximal normal, characteristic |
| 2 | C29 | SL(2,𝔽5) | 1 | 1 | yes | characteristic, complemented |
| 3 | C26 | #11757162 | 1 | 1 | yes | characteristic |
| 4 | C25 | #14515300 | 1 | 1 | yes | characteristic |
| 5 | C25 | #14515301 | 1 | 1 | yes | characteristic |
| 6 | C25 | #14515302 | 1 | 1 | yes | characteristic |
| 7 | C24 | #72209604 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini | C26 | #11757162 |
| Fitting, cosocle | C2×#6342371 | A5 |
| socle | C24 | #72209604 |
| LCS | G | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C29 | SL(2,𝔽5) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#72215789 | 10 |
| C2×#72227520 | 6 |
| #72239357 | 5 |
| C2×#14515303 | 16 |
| #72235305 | 16 |
| #72235209 | 16 |