Finite Group DatabasebetaFinite Group DatabasebetaOrder 22010.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C31×C71 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D2201 | — | 5 | 1 | no | maximal, Hall |
| 3 | C31×C71 | D5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | D355 | — | 31 | 1 | no | maximal, Hall |
| 5 | C5×C71 | D31 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | D155 | — | 71 | 1 | no | maximal, Hall |
| 7 | C5×C31 | D71 | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | D71 | — | 155 | 1 | no | Hall |
| 9 | C71 | D155 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | D31 | — | 355 | 1 | no | Hall |
| 11 | C31 | D355 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | D5 | — | 2201 | 1 | no | Hall |
| 13 | C5 | D2201 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 11005 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C31×C71 | C2 |
| LCS | G ▷ C5×C31×C71 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C31×C71 | C2 | semidirect |
| 3. C31×C71 | D5 | semidirect |
| 5. C5×C71 | D31 | semidirect |
| 7. C5×C31 | D71 | semidirect |
| 9. C71 | D155 | semidirect |
| 11. C31 | D355 | semidirect |
| 13. C5 | D2201 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C31×C71 | 2 |
| D2201 | 5 |
| D355 | 31 |
| D155 | 71 |