Finite Group DatabasebetaFinite Group DatabasebetaOrder 22112.
62 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #26010850 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×C691⋊C8 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #41×C691 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C691⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 5 | C691⋊C8 | C22 | 1 | 3 | yes | — |
| 6 | Q8×C691 | C4 | 1 | 1 | yes | characteristic |
| 7 | D4×C691 | C4 | 1 | 3 | yes | — |
| 8 | C2×C4×C691 | C22 | 1 | 3 | yes | — |
| 9 | C22×C691 | C2×C4 | 1 | 3 | yes | — |
| 10 | C4×C691 | C2×C4 | 1 | 3 | yes | — |
| 11 | C4×C691 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×C691 | — | 2 | 3 | no | — |
| 13 | C2×C691 | C22×C4 | 1 | 1 | yes | characteristic |
| 14 | C691 | #131 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | #131 | — | 691 | 1 | no | maximal, Sylow |
| 16 | #41 | D691 | 1 | 1 | yes | characteristic |
| 17 | M16 | — | 691 | 3 | no | — |
| 18 | C2×C8 | — | 691 | 3 | no | — |
| 19 | Q8 | Dic691 | 1 | 1 | yes | characteristic |
| 20 | D4 | Dic691 | 1 | 3 | yes | — |
| 21 | C2×C4 | C2×D691 | 1 | 3 | yes | — |
| 22 | C8 | — | 691 | 1 | no | — |
| 23 | C8 | — | 691 | 3 | no | — |
| 24 | C22 | C2×Dic691 | 1 | 3 | yes | — |
| 25 | C4 | C2×Dic691 | 1 | 3 | yes | — |
| 26 | C4 | C22×D691 | 1 | 1 | yes | characteristic |
| 27 | C2 | — | 2 | 3 | no | — |
| 28 | C2 | C22×Dic691 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D691 |
| commutator, socle | C2×C691 | C22×C4 |
| Fitting, prime radical | #41×C691 | C2 |
| cosocle | C4×C691 | C23 |
| hypercenter | #41 | D691 |
| hypocenter | C691 | #131 |
| LCS | G ▷ C2×C691 ▷ C691 | |
| UCS | C1 ◁ C4 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26010850 | C2 | semidirect |
| 2. C2×C691⋊C8 | C2 | semidirect |
| 14. C691 | #131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26010850 | 2 |
| C2×C691⋊C8 | 2 |
| #41×C691 | 2 |
| #131 | 691 |