Toast the cleaned dried chiles in a dry skillet over medium heat for about 20 seconds per side until fragrant, without blackening. Put them in a bowl, cover with the measured boiling water and soak for 15 minutes.
Blend softened chiles with 1 cup soaking water, onion, garlic, oregano, cumin and salt until smooth. Strain through a fine sieve into a 6-quart slow cooker, pressing out the liquid and discarding tough skins.
Add broth and chicken. Cover and cook on low for 3 1/2 to 4 1/2 hours, checking from 3 1/2 hours and removing chicken as its thickest part reaches 165°F.
Add hominy to the broth, turn the cooker to high and cover for 20 to 25 minutes until hot. Meanwhile, remove every bone and cartilage piece from the chicken and shred the meat coarsely.
Return chicken to the broth and heat covered for 5 minutes until it reaches 165°F. Divide among 6 bowls, adding cabbage, radish slices and one lime wedge to each serving.