  Cohen, Lenstra, and Martinet have given conjectures for the distribution of
class groups of extensions of number fields, but Achter and Malle have given
theoretical and numerical evidence that these conjectures are wrong regarding
the Sylow $p$-subgroups of the class group when the base number field contains
$p$th roots of unity. We give complete conjectures of the distribution of Sylow
$p$-subgroups of class groups of extensions of a number field when $p$ does not
divide the degree of the Galois closure of the extension. These conjectures are
based on $q\rightarrow\infty$ theorems on these distributions in the function
field analog and use recent work of the authors on explicitly giving a
distribution of modules from its moments. Our conjecture matches many, but not
all, of the previous conjectures that were made in special cases taking into
account roots of unity.
