Finite Group DatabasebetaFinite Group DatabasebetaOrder 20313.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C37×C61 | C3 | 1 | 1 | yes | maximal, characteristic |
| 2 | C37×C61 | C9 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | C9×C61 | — | 37 | 1 | no | maximal, Hall |
| 4 | #66524 | C61 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | C3×C61 | C37⋊C3 | 1 | 1 | yes | characteristic |
| 6 | C3×C37 | C3×C61 | 1 | 1 | yes | characteristic |
| 7 | C61 | #66524 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 8 | C37 | C9×C61 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C9 | — | 37 | 1 | no | Sylow |
| 10 | C3 | C61×C37⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C61 | C37⋊C3 |
| commutator, hypocenter | C37 | C9×C61 |
| Frattini | C3 | C61×C37⋊C3 |
| Fitting, socle, prime radical | C3×C37×C61 | C3 |
| cosocle | C3×C37 | C3×C61 |
| LCS | G ▷ C37 | |
| UCS | C1 ◁ C3×C61 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C37×C61 | C9 | semidirect |
| 4. #66524 | C61 | direct (N × H) |
| 7. C61 | #66524 | direct (N × H) |
| 8. C37 | C9×C61 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C37×C61 | 3 |
| C9×C61 | 37 |
| #66524 | 61 |