  For a given graph $H$, its subdivisions carry the same topological structure.
The existence of $H$-subdivisions within a graph $G$ has deep connections with
topological, structural and extremal properties of $G$. One prominent example
of such a connection, due to Bollob\'{a}s and Thomason and independently
Koml\'os and Szemer\'edi, asserts that the average degree of $G$ being $d$
ensures a $K_{\Omega(\sqrt{d})}$-subdivision in $G$. Although this square-root
bound is best possible, various results showed that much larger clique
subdivisions can be found in a graph for many natural classes. We investigate
the connection between crux, a notion capturing the essential order of a graph,
and the existence of large clique subdivisions. This reveals the unifying cause
underpinning all those improvements for various classes of graphs studied.
Roughly speaking, when embedding subdivisions, natural space constraints arise;
and such space constraints can be measured via crux.
  Our main result gives an asymptotically optimal bound on the size of a
largest clique subdivision in a generic graph $G$, which is determined by both
its average degree and its crux size. As corollaries, we obtain (1) a
characterisation of extremal graphs for which the square-root bound above is
tight: they are essentially disjoint unions of graphs having crux size linear
in $d$; (2) a unifying approach to find a clique subdivision of almost optimal
size in graphs which do not contain a fixed bipartite graph as a subgraph; (3)
and that the clique subdivision size in random graphs $G(n,p)$ witnesses a
dichotomy: when $p = \omega(n^{-1/2})$, the barrier is the space, while when
$p=o( n^{-1/2})$, the bottleneck is the density.
