Subgroups — D₁₇₁ #66787

Order 342.

12 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₅₇	—	3	1	no	maximal
3	C₃×C₁₉	D₃	1	1	yes	characteristic
4	D₁₉	—	9	1	no	Hall
5	C₁₉	D₉	1	1	yes	characteristic, complemented, Sylow
6	D₉	—	19	1	no	maximal, Hall
7	C₉	D₁₉	1	1	yes	characteristic, complemented, Sylow
8	D₃	—	57	1	no	—
9	C₃	D₅₇	1	1	yes	characteristic
10	C₂	—	171	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₉	C₂
Frattini	C₃	D₅₇
socle	C₃×C₁₉	D₃
LCS	G ▷ C₉×C₁₉
UCS	C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.