Finite Group DatabasebetaFinite Group DatabasebetaOrder 32412.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C37×C73 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | #22549193 | — | 4 | 1 | no | maximal, Hall |
| 3 | C2×C37×C73 | — | 3 | 1 | no | — |
| 4 | C37×C73 | A4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C22⋊(C73⋊C3) | — | 37 | 1 | no | maximal, Hall |
| 6 | C22⋊(C37⋊C3) | — | 73 | 1 | no | maximal, Hall |
| 7 | C22×C73 | C37⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | C73⋊C3 | — | 148 | 1 | no | Hall |
| 9 | C22×C37 | C73⋊C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | C2×C73 | — | 3 | 1 | no | — |
| 11 | C37⋊C3 | — | 292 | 1 | no | Hall |
| 12 | C2×C37 | — | 3 | 1 | no | — |
| 13 | C73 | C22⋊(C37⋊C3) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C37 | C22⋊(C73⋊C3) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | A4 | — | 2701 | 1 | no | Hall |
| 16 | C22 | #22549193 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C3 | — | 10804 | 1 | no | Sylow |
| 18 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C37×C73 | C3 |
| LCS | G ▷ C22×C37×C73 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C37×C73 | C3 | semidirect |
| 4. C37×C73 | A4 | semidirect |
| 7. C22×C73 | C37⋊C3 | semidirect |
| 9. C22×C37 | C73⋊C3 | semidirect |
| 13. C73 | C22⋊(C37⋊C3) | semidirect |
| 14. C37 | C22⋊(C73⋊C3) | semidirect |
| 16. C22 | #22549193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C37×C73 | 3 |
| #22549193 | 4 |
| C22⋊(C73⋊C3) | 37 |
| C22⋊(C37⋊C3) | 73 |