Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #21443638 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C61×C64 | C4 | 1 | 1 | yes | characteristic |
| 3 | C32×C61 | C8 | 1 | 1 | yes | characteristic |
| 4 | C16×C61 | C16 | 1 | 1 | yes | characteristic |
| 5 | C8×C61 | C32 | 1 | 1 | yes | characteristic |
| 6 | C256 | — | 61 | 1 | no | maximal, Sylow |
| 7 | C4×C61 | C64 | 1 | 1 | yes | characteristic |
| 8 | C128 | — | 61 | 1 | no | — |
| 9 | C2×C61 | C128 | 1 | 1 | yes | characteristic |
| 10 | C64 | #6923 | 1 | 1 | yes | characteristic |
| 11 | C61 | C256 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | C32 | #92420 | 1 | 1 | yes | characteristic |
| 13 | C16 | #11758187 | 1 | 1 | yes | characteristic |
| 14 | C8 | #14519827 | 1 | 1 | yes | characteristic |
| 15 | C4 | #16840601 | 1 | 1 | yes | characteristic |
| 16 | C2 | #21460634 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C64 | #6923 |
| commutator, hypocenter | C61 | C256 |
| Fitting | C61×C64 | C4 |
| socle, prime radical | C2×C61 | C128 |
| cosocle | #21443638 | C2 |
| LCS | G ▷ C61 | |
| UCS | C1 ◁ C64 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. C61 | C256 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21443638 | 2 |
| C256 | 61 |