Monolithic groups

A finite group G is monolithic when it has exactly one minimal normal subgroup. That subgroup is the monolith of G.

Equivalent conditions, for G ≠ {e}:

The socle soc(G) — the product of all minimal normal subgroups — is a minimal normal subgroup.
Every non-trivial normal subgroup of G contains the monolith N: each one contains some minimal normal subgroup, and there is only the one.
N = ⋂ {K ⊴ G : K ≠ {e}} is non-trivial — G has a least non-trivial normal subgroup.

The trivial group is not monolithic: it has no non-trivial normal subgroup to be minimal.

What a monolith looks like

A minimal normal subgroup of a finite group is characteristically simple, so it is a direct power of a simple group:

N ≅ S × ⋯ × S,    S simple.

Two cases:

S = Cₚ — N is elementary abelian Cₚⁿ.
S non-abelian — N is a direct power of one non-abelian simple group; when N = S is that simple group itself, G is almost simple, the extreme where S ≤ G ≤ Aut(S).

Directly indecomposable. In A × B with both factors non-trivial, a minimal normal subgroup of A and one of B are two distinct minimal normal subgroups of the product, so no non-trivial direct product is monolithic.
μ(G) = μ_t(G). The minimal faithful degree is attained transitively: the kernels of a faithful family of actions on cosets intersect trivially, so one of them fails to contain N — and a normal subgroup that does not contain the monolith is trivial, making that single orbit already faithful. See minimal transitive degree.
Not a subdirect product of proper quotients. If ⋂ Kᵢ = {e} for normal subgroups Kᵢ, some Kᵢ is trivial — otherwise every Kᵢ contains N, and so does the intersection.

Every simple group, Cₚ included: G is its own only non-trivial normal subgroup, so the monolith is G.
Abelian: exactly the cyclic groups of prime-power order C_pᵏ, whose monolith is their unique subgroup of order p. Every subgroup is normal, so monolithic means having a unique subgroup of prime order.
p-groups: exactly those with cyclic center. A non-trivial normal subgroup of a p-group meets the center, so the minimal normal subgroups are the central subgroups of order p — one of them exactly when Z(G) is cyclic.
Nilpotent: exactly the p-groups with cyclic center — a nilpotent group is the direct product of its Sylow subgroups, and two non-trivial factors give two minimal normal subgroups.
Almost simple, with the simple socle as monolith.

C₄ — monolith C₂, quotient C₂. C₈, C₉, C₂₅ behave the same; C₆ does not.
D₄ and Q₈ — order 8, both with center C₂ as monolith and quotient C₂ × C₂. The two other non-cyclic groups of that order, C₄ × C₂ and C₂³, are not monolithic.
S₄ — order 24, monolith the Klein four-group V = C₂ × C₂, quotient S₃. Its normal subgroups are {e} ⊂ V ⊂ A₄ ⊂ S₄, a chain, so the least non-trivial one is V.
A₅ — simple, so it is its own monolith; S₅ is almost simple with monolith A₅.
2⁴:A₅ — order 960, with A₅ acting irreducibly on 2⁴. The monolith is that elementary abelian 2⁴: it is self-centralizing, so no second minimal normal subgroup has room to commute with it.

C₂ × C₂ — three subgroups of order 2, all normal, all minimal.
C₆ — C₂ and C₃. Any nilpotent group whose order has two prime divisors fails the same way.
A₅ × A₅ — one minimal normal subgroup in each factor.