Please help me to draw the following diagram:
\begin{displaymath}
\xymatrix{
H\ar[r]&H .1={}_HI_Q\supset 1.Q&Q\ar[l]_\epsilon^{\cong} \\
&R\ar[u]\ar[r]&R\ar[u]
}
\end{displaymath}

Please help me to draw the following diagram:
\begin{displaymath}
\xymatrix{
H\ar[r]&H .1={}_HI_Q\supset 1.Q&Q\ar[l]_\epsilon^{\cong} \\
&R\ar[u]\ar[r]&R\ar[u]
}
\end{displaymath}

If you are willing to switch to tikz-cd, you could do like this:
% arara: pdflatex
\documentclass{article}
\usepackage{tikz-cd}
\tikzset{
symbol/.style={
draw=none,
every to/.append style={
edge node={node [sloped, allow upside down, auto=false]{$#1$}}}
}
}
\begin{document}
\[
\begin{tikzcd}[row sep= 2ex, column sep=0.8em]
H \arrow[twoheadrightarrow]{rrr} & & & H\cdot 1 \arrow[symbol={=}]{r} & {}_HI_Q \arrow[symbol=\supset]{r} & 1\cdot Q \arrow[symbol={=}]{d} & & & Q \arrow{lll}{\cong}[swap]{\varepsilon}\\
& & & & & \widetilde{E}(R) \arrow[symbol=\supset]{d} & & & \\
& & & & & R \arrow[symbol={=}]{rrr} & && R \arrow[rightarrowtail]{uu}
\end{tikzcd}
\]
\end{document}

In the down arrow with the equal sign, it is quite noticeable that the relational signs are set on different heights. If you do not like the misplacement, you could correct that (not that beautifully coded) with \arrow[symbol={\raisebox{-1.3ex}{=}}]{d}.
tikz-cdortikz commutative diagrams; if possible, it'd be great if you could post some attempt at what you've tried so far--once you have got part of the way there, you could update your question with the specific issue you have :) – cmhughes Dec 08 '14 at 16:55