Please help me to draw the following diagram:

H\ar[r]&H .1={}_HI_Q\supset 1.Q&Q\ar[l]_\epsilon^{\cong}  \\

enter image description here

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    Hi, welcome to TeX.SX!. Can you please add some code that you've tried so far and tell us what isn't working for you? – darthbith Dec 8 '14 at 16:53
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    Welcome to the site :) Search this site for tikz-cd or tikz 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 8 '14 at 16:55
  • Is your problem solved with my answer? If yes, please consider to accept it in order to close this post. If not, please tell us, what is missing. – LaRiFaRi May 19 '15 at 8:14

If you are willing to switch to , you could do like this:

% arara: pdflatex

    every to/.append style={
      edge node={node [sloped, allow upside down, auto=false]{$#1$}}}

\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}

enter image description here

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}.

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