If G is a
group, a series of
subgroups
G=G1>G2>...>Gn>Gn+1={1}
(where G>H means H is a subgroup of G and {1} denotes the trivial group) is said to be normal if for each i=1,...,n,n+1, Gi is normal in G.
The series is said to be subnormal if for each i=1,...,n, Gi+1 is normal in Gi. So a normal series is a subnormal series, but the converse is not true in general.
See also composition series, Jordan-Holder theorem, soluble group.