已知公差大于零的等差数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}(n∈N^{*})\)，且满足\(a_{3}·a_{4}=117\)，\(a_{2}+a_{5}=22\)．

\((1)\)求集合\(\{n|2\leqslant a_{n}\leqslant 23\}\)的子集的个数；

\((2)\)求证：数列\(\left\{ \dfrac{{{S}_{n}}}{n+c} \right\}(c\ne 0)\)为等差数列的充要条件为\(c=-\dfrac{1}{2}\)．