For an electric dryer, you should be counting on 30 amps for the dryer alone.
I don't know how much current your dryer's heating element actually draws, but the cord and receptacle for an electric dryer will be rated at 220 Volts, 30 amps. So, if it wuz me, I'd use double 30 amp breakers in the sub-panel and run cable rated at 30 amps as well.
In the little bit of wiring I did to put range cords on my 21 electric stoves and 3 electric dryers, ALL of the wiring I was dealing with was stranded. That's because if it were solid, it would be far too stiff to bend in the confines of an electrical panel or receptacle box.