Subgroups — D₁₈₉ #67352

Order 378.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₆₃	—	3	1	no	maximal
3	C₇×C₉	D₃	1	1	yes	characteristic
4	D₂₇	—	7	1	no	maximal, Hall
5	D₂₁	—	9	1	no	—
6	C₂₇	D₇	1	1	yes	characteristic, complemented, Sylow
7	C₃×C₇	D₉	1	1	yes	characteristic
8	D₉	—	21	1	no	—
9	D₇	—	27	1	no	Hall
10	C₉	D₂₁	1	1	yes	characteristic
11	C₇	D₂₇	1	1	yes	characteristic, complemented, Sylow
12	D₃	—	63	1	no	—
13	C₃	D₆₃	1	1	yes	characteristic
14	C₂	—	189	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₇	C₂
Frattini	C₉	D₂₁
socle	C₃×C₇	D₉
LCS	G ▷ C₇×C₂₇
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.