  We investigate a five-branes interpretation of hyper-K\"{a}hler geometry with
torsion (HKT). This geometry is obtained by conformal transformation of the
Taub-NUT space which represents a Kaluza-Klein five-brane. This HKT would
represent an NS5-brane on the Taub-NUT space. In order to explore the HKT
further, we compactify one transverse direction, and study the $O(2,2;{\mathbb
Z}) = SL(2,{\mathbb Z}) \times SL(2,{\mathbb Z})$ monodromy structure
associated with two-torus. Performing the conjugate transformation, we obtain a
new solution whose physical interpretation is a defect $(p,q)$ five-brane on
the ALG space. Throughout this analysis, we understand that the HKT represents
a coexistent state of two kinds of five-branes. This situation is different
from composite states such as $(p,q)$ five-branes or $(p,q)$ seven-branes in
type IIB theory. We also study the T-dualized system of the HKT. We again find
a new solution which also indicates another defect $(p,q)$ five-brane on the
ALG space.
