The Eratosthenes Shuffle

Mathematica, 61 bytes

#[[SortBy[Range@Length@#,FactorInteger[#][[1,1]]PrimeQ@#&]]]&

Unnamed function taking a list of characters as input and returning a list of characters.

FactorInteger[#][[1,1]] yields the smallest prime factor of # (and returns 1 if # equals 1). Therefore FactorInteger[#][[1,1]]PrimeQ@# yields a strange expression: [#'s smallest prime factor] False if # is not prime, and # True if # is prime (these are unevaluated products of a number and a boolean).

Range@Length@# yields a list of the numbers up to the length of the input. Then SortBy sorts those numbers by the funny function described above. Mathematica is really type-sensitive in many ways, but cheerfully mixes them in other ways: expressions of the form [number] False are alphabetically sorted before expressions of the form [number] True, while ties are broken by sorting the numbers numerically. That produces exactly the permutation we want here, and #[[...]] permutes the characters of the input accordingly.

Jelly, 10 bytes

JÆfṂ$ÞÆPÞị

TryItOnline

How?

JÆfṂ$ÞÆPÞị - Main link: theString
J          - range(length), the 1-based indexes of theString
     Þ     - sort these by
    $      -     last two links as a monad
 Æf        -         prime factorization array (e.g. 20 -> [2,2,5])
   Ṃ       -         minimum
        Þ  - sort these by
      ÆP   - isPrime, i.e. move all the primes to the right
         ị - index into theString

C, 164 bytes

This takes input as the first command parameter and prints back to stdout. As we process each character, we clear it, allowing the final pass.

#define p putchar
main(c,s,i,j,t)char**s,*t;{c=strlen(t=s[1]);p(*t);for(;j++<c;)if(t[j])for(i=2*j+1;i<=c;i+=j+1)!t[i]?:p(t[i]),t[i]=0;for(j=0;j++<c;)!*++t?:p(*t);}