Edit (Oct 12, 2015):

My original answer contained an error, as noticed by user47958.
I now record only those parts of what I wrote which
might be relevant to a future complete solution.
(What I am writing here is NOT a complete solution.)

I order topologies so that $\mathcal T\subseteq \mathcal T'$
means that $\mathcal T'$ is finer than $\mathcal T$.
I let $M_n$ denote the lattice of height three with $n$ atoms.

Background info:

If $N$ is any normal subgroup of a group $G$, then the set
$\mathcal T_N$ consisting of those subsets of $G$ that are unions
of cosets of $N$ is a group topology on $G$. The discrete
and indiscrete topologies have the form $\mathcal T_N$ for $N =\{e\}$
or $N=G$ respectively. If $M\neq N$,
then $\mathcal T_M\neq \mathcal T_N$. Since
$\mathcal T_M\vee \mathcal T_N = \mathcal T_{M\cap N}$ and
$\mathcal T_M\wedge \mathcal T_N = \mathcal T_{MN}$, the lattice of
normal subgroups of $G$ is dually embedded in the lattice of
group topologies via $N\mapsto \mathcal T_N$.

For any group topology $\mathcal T$ on $G$ the intersection
of the neighborhoods of the identity in $\mathcal T$
is a normal subgroup $N\lhd G$. Any set in $\mathcal T$ is a
union of cosets of $N$. The topology $\mathcal T_N$ from
Item 1 contains the starting topology $\mathcal T$.
In particular, if $\mathcal T$ is coatom in the lattice of group topologies,
and the intersection $N$ is nontrivial, then $\mathcal T = \mathcal T_N$.
(Briefly, non-Hausdorff coatoms in the lattice of group topologies have the form $\mathcal T_N$ for some normal subgroup $N$.)

The intersection $N$ from Item 2 equals $\{e\}$ iff $\mathcal T$
is $T_0$ (equivalently Hausdorff, equivalently Tychonoff).

If $G$ is finite, then all group topologies on
$G$ have the form $\mathcal T_N$.
The lattice of group topologies is therefore dually isomorphic
to the lattice of normal subgroups of $G$.

If a group $G$ has at least three pairwise complementary
normal subgroups, then $G$ is finite and in fact
$G\cong \mathbb Z_p\times \mathbb Z_p$
for some prime $p$. In this case the normal subgroup lattice
of $G$ is $M_{p+1}$.

The problem asks if there is a topological group whose lattice
of group topologies is $M_5$. Since $5$ is not a prime plus one,
such a group cannot be finite. In fact, no more than two of the
(co)atoms in the topology lattice can have the form $\mathcal T_N$
for $N\lhd G$. This means that (i) $G$ has at
most two proper nontrivial normal subgroups,
each of which is an atom and a coatom
in the normal subgroup lattice of $G$. Therefore (ii) at most two of the
coatoms in the group topology lattice can be non-Hausdorff.
Hence (iii) at least three of the atoms
in the lattice of group topologies on $G$
are pairwise complementary Hausdorff topologies.
Item (i) splits the full problem into these cases:

Case I. $G$ is simple and has 5 pairwise complementary proper
nontrivial Hausdorff group topologies.

Case II. $G$ has exactly one nontrivial proper normal subgroup $N$,
$G/N$ has no nontrivial proper group topology, but $G$ has four
pairwise complementary Hausdorff topologies
that are nontrivial and proper.

Case III. $G=S\times T$ is a product of two simple groups, where
neither $S$ nor $T$ has any nontrivial proper group topology,
but $G$ has three pairwise complementary Hausdorff topologies
that are nontrivial and proper.

I have learned that there is no known group that has even one
pair of complementary proper Hausdorff group topologies.
No abelian group can have complementary proper
Hausdorff group topologies, as proved in

*Relationship between the meet and join operators
in the lattice of group topologies,*
Bradd Clark and Victor Schneider,
Proceedings AMS 98(4) (1986) 681-682.

The general problem of whether there can exist complementary
proper Hausdorff group topologies is posed as Question 7.1(a) in

*On transversal group topologies,*
Dikran Dikranjan, Mikhail Tkachenko, Ivan Yaschenko,
Topology and its Applications 153 (2005) 786-817.

Here is a construction of a group topology that might
be applicable to derive properties of groups in
Cases I, II or III. It is a generalization of the $\mathcal T_N$
topology.

Let $H$ be a subgroup of $G$. Let $S_H$ be the set of subgroups
of $G$ that are finite intersections of conjugates of $H$.
The groups in $S_H$ form a system of neighborhoods of $e$
for a group topology $\mathcal T_H$ on $G$. The topology
$\mathcal T_H$ is nontrivial if $H$ is proper and is
proper if $\{e\}\notin S_H$. $\mathcal T_H$ is Hausdorff
if $H$ is core-free.

Here is an example that hints about how this construction might be applicable
to the problem. Suppose that $G = A_{\omega}$ is the alternating group
on $\omega$, $E\subseteq \omega$ is (say) the set of even natural
numbers, and $H$ is the stabilizer of $E$. Every group
in $S_H$ is the stabilizer of some set that is the symmetric
difference of $E$ and a finite subset of $\omega$. Therefore $\mathcal T_H$
is a nontrivial proper Hausdorff group topology that is tied to
the set $E$. You can produce lots more Hausdorff group
topologies on $A_{\omega}$ by varying the set $E$.

For this construction to fail to produce nontrivial proper Hausdorff group topologies, it must be that whenever $H\leq G$ is proper and core-free, then $\{e\}$ is the intersection of finitely many conjugates of $H$.

3more comments