I'm not so sure this is better looking (but I dislike \leqslant as well).
Probably you'll think to a better name than \slunlhd. ;-)
\documentclass{article}
\usepackage{amssymb}
\makeatletter
\newcommand{\slunlhd}{%
\mathrel{\mathpalette\sl@unlhd\relax}%
}
\newcommand{\sl@unlhd}[2]{%
\sbox\z@{$#1\lhd$}%
\sbox\tw@{$#1\leqslant$}%
\dimen@=\ht\tw@
\advance\dimen@-\ht\z@
\ifx#1\displaystyle
\advance\dimen@ .2pt
\else
\ifx#1\textstyle
\advance\dimen@ .2pt
\fi
\fi
\ooalign{\raisebox{\dimen@}{$\m@th#1\lhd$}\cr$\m@th#1\leqslant$\cr}%
}
\makeatother
\begin{document}
$N\slunlhd G_{\slunlhd}$
\Large
$N\slunlhd G_{\slunlhd}$
\end{document}
Basically, I superimpose \leqslant to \lhd (with a small correction in text and display styles).
