How to get \trianglelefteq similar to \leqslant?

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).

Here I stack a \rule atop the \leqslant in order to make \trianglelefteqslant, without the mathabx package. By using the scalerel package's features, it can be made to work across math styles.

\documentclass{article}
\usepackage{amssymb}
\usepackage{stackengine,scalerel}
\stackMath
\def\trianglelefteqslant{\ThisStyle{\mathrel{%
  \stackinset{r}{.75pt+.15\LMpt}{t}{.1\LMpt}{\rule{.3pt}{1.1\LMex+.2ex}}{\SavedStyle\leqslant}%
}}}
%\usepackage{mathabx}
\begin{document}
$x \leqslant y$\par
$x \trianglelefteqslant y$\par
$\scriptstyle x \trianglelefteqslant y$\par
$\scriptscriptstyle x \trianglelefteqslant y$
\end{document}