You can find many relevant results in the book: Pisier, Gilles Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. See, for example, Theorem 10.6 and Corollary 10.8.
I type this as an answer because I doubt that anyone will be willing to write for you everything that is known on this matter. The mentioned book can be used as a good starting point for learning results of this type.