Cfxtrjtrk

I am using chains to re-draw the structure (shown below) in Second isomorphism theorem(wiki).

The code and the resulting figure is shown as follows.

How to get rid of the dummy node (added for alignment) and the unnecessary join lines between it and the node hn and hcapn, respectively?

documentclass[tikz]{standalone}

usetikzlibrary{chains, scopes}



begin{document}

begin{tikzpicture}[every on chain/.style = join, every join/.style = -, node distance = 0.8cm and 1.0cm]

  { 

    [start chain = G going below] % G chain

    node (g) [on chain] {$G$};

    node (hn) [on chain, below = 0.50cm of g] {$HN$}; 

    { [start branch = H going below left] % H branch

      node (h) [on chain] {$H$};

    }

    { [start branch = N going below right] % N branch

      node [on chain] {$N$};

    }

    node [on chain] {}; % dummy node % continue G chain

    node (hcapn) [on chain, join = with G/H-end, join = with G/N-end] {$H cap N$};

    node (e) [on chain, below = 0.50cm of hcapn] {${ e }$};

  }

end{tikzpicture}

end{document}

with chain, but two nodes are excluded in join macro and for it the connection is drawn separately:

documentclass[tikz, margin=3mm]{standalone}

usetikzlibrary{chains, positioning, shapes.geometric}

makeatletter

tikzset{suspend join/.code={deftikz@after@path{}}}

makeatother



begin{document}

    begin{tikzpicture}[

node distance = 8mm and 10 mm,

  start chain = going below,

     N/.style = {ellipse, draw, inner sep=2pt, on chain, join=by -}]

node (g)       [N] {$G$};

node (hn)      [N] {$HN$};

node (h)       [N, below left=of hn]   {$H$};

node (hcapn)   [N, below=of h -| hn]   {$H cap N$};

node (e)       [N] {${ e }$};

%

node (n)       [N,suspend join,

                 below right=of hn]     {$N$};

node [below=of hn] {$cong$};

draw (hn) -- (n)   (n) -- (hcapn);

    end{tikzpicture}

end{document}

If you want to replicate the linked picture, I propose this

documentclass[tikz]{standalone}

usetikzlibrary{shapes,fit}

usepackage{mathptmx}

begin{document}

begin{tikzpicture}[x=1.75cm,y=1.75cm]

begin{scope}[every node/.style={draw,circle,minimum size=1cm}]

node (g) at (0,2) {$G$};

node (sn) at (0,1) {$SN$};

node (n) at (-1,0) {$N$};

node (s) at (1,0) {$S$};

node[ellipse,draw,minimum height=1cm] (scn) at (0,-1) {$Scap N$};

node (e) at (0,-2) {${e}$};

end{scope}

draw (g)--(sn)--(n)--(scn)--(e) (scn)--(s)--(sn);

node[rotate=-45,ellipse,draw,dashed,inner xsep=-7mm,inner ysep=-1mm,fit=(sn)(n)] {};

node[rotate=-45,ellipse,draw,dashed,inner xsep=-9mm,inner ysep=1mm,fit=(scn)(s)] {};

node {$cong$};

end{tikzpicture}

end{document}