Netyjnd

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
usetikzlibrarychains, scopes

begindocument
begintikzpicture[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 $;
 
endtikzpicture
enddocument

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

documentclass[tikz]standalone
usetikzlibraryshapes,fit
usepackagemathptmx
begindocument
begintikzpicture[x=1.75cm,y=1.75cm]
beginscope[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) $e$;
endscope
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 $cong$;
endtikzpicture
enddocument

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

documentclass[tikz, margin=3mm]standalone
usetikzlibrarychains, positioning, shapes.geometric
makeatletter
tikzsetsuspend join/.code=deftikz@after@path
makeatother

begindocument
 begintikzpicture[
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);
 endtikzpicture
enddocument