Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 74] INSOLVABLE GROUPS OF LOW COMPOSITE ORfIERS 189

from the group of isomorphisms of the abelian group of order
8 and of type (1, 1, 1).
A simple group of order 168 contains 8 subgroups of order
7, and can therefore be represented as a transitive substitution
group G of degree 8. A maximal subgroup of G is of degree
7 and of order 21, and it therefore involves seven subgroups of
the form (abc.def). Each of these subgroups is transformed
into itself by six substitutions under G. Hence it may be
assumed that all the possible simple groups of order 168 contain a particular subgroup of degree 7 and order 21, and are
generated by this subgroup and a substitution of the form
ab cd ef.gh, which transforms into itself a particular subgroup
of the form (abc.def) contained in the given subgroup of
order 21.
If the given subgroup of order 6 is the symmetric group
of this order, the three possible substitutions of the form
ab cd ef.gh are completely determined by the subgroup of
the form (abc.def). That is, there is not more than one
transitive group of degree 8 which contains a particular subgroup of degree 7 and of order 21, and which is such that
its six substitutions which transform into itself a particular
subgroup of the form (abc def) constitute the symmetric
group of order 6. It is clear that there is one such group,
since the simple group of degree 7 and of order 168 transforms its eight subgroups of order 7 according to a transitive
group of degree 8.
To prove that there is only one simple group of order 168
it remains only to prove that a transitive group of degree 8 and
of order 168 cannot be simple if the subgroup of order 6 which
transforms a subgroup of the form (abc.def) into itself is
cyclic. In fact, such a transitive group contains 28 cyclic
subgroups of order 6, and hence it contains 48+56+56 substitutions of orders 7, 3, and 6 respectively. It can therefore
contain only one subgroup of order 8. Hence we have completed a proof of the theorem that there is only one simple group
of order 168, and hence there is also only one insolvable group of
this order.