That may depend on the parents genetics as well as what colors you are looking for. The only mutations I know of in GCC's (from Gencalc) are Grey, Cinnamon, Opaline (Yellowside), Turquoise (parblue), Misty, and Recessive Pied. The most common mutations are Cinnamon and Opaline, with Turquoise gaining popularity.
Therefore, you'd probably find Cinnamons and Opalines most frequent and maybe if you look a bit harder, the Turquoise ones as well... but the others you are less likely to come across. Therefore, you need to know what colors you'd like to get in order to achieve the desired result. All are stunning mutations in their own right.
Not only that but some birds have more reds, yellows, blues or greens than others... therefore affecting the total outcome of colors in offspring...
You might be interested in this chart on GCC breeding mutations (Normals, Cinnamons, Yellowside, and Pineapple)
http://www.conure.org/Resources/GCC_col ... 202005.pdf
Click on the images to view a larger pic
Thanks to Outback Aviaries for displaying these pics on their site!
Turquoise Green-cheek hen with normal male
Turquoise mutation Green-cheek
Cinnamon Green-cheeked
Pineapple Green-cheeked (cinnamon yellowside)
Turquoise Pineapple Green-cheeked (turquoise cinnamon yellowside)
Turquoise Yellow-side Green-cheeked
Picture from Feathered Imps
From Left to Right
Normal, Yellowsided, Cinnamon, Pineapple
Picture from Ruffled Feathers Aviary
Cinnamon Turquoise
Picture from DJ Feathers
Pinapple (left) and Cinnamon (right)
Picture from Centurion Aviaries
Cinnamon turquoise Greencheek
More pictures of these mutations can be seen at
http://conures.co.uk/gallery.htm