The digits $1,2,3,4,5,6$ are each used once to write a six-digit number $ABCDEF$. The three-digit number $ABC$ is divisible by $4$, the three-digit number $BCD$ is divisible by $3$, the three-digit number $CDE$ is divisible by $5$, and the three-digit number $DEF$ is divisible by $6$. What is $C$?