Subgroups — C₃⋊(C₃⋊#168) #67107

Order 360.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#168	—	3	4	no	maximal
3	C₂×C₃²×C₅	C₄	1	1	yes	characteristic
4	C₃⋊(C₃⋊C₈)	—	5	1	no	maximal, Hall
5	C₃×C₄×C₅	D₃	1	4	yes	—
6	C₃²×C₅	C₈	1	1	yes	characteristic, complemented, Hall
7	#168	—	9	1	no	Hall
8	C₃²×C₄	D₅	1	1	yes	characteristic
9	C₂×C₃×C₅	Dic₃	1	4	yes	—
10	C₃⋊C₈	—	15	4	no	—
11	C₄×C₅	C₃⋊D₃	1	1	yes	characteristic
12	C₂×C₃²	Dic₅	1	1	yes	characteristic
13	C₃×C₅	C₃⋊C₈	1	4	yes	complemented
14	C₃×C₄	D₁₅	1	4	yes	—
15	C₂×C₅	C₃⋊Dic₃	1	1	yes	characteristic
16	C₃²	#168	1	1	yes	characteristic, complemented, Sylow
17	C₈	—	45	1	no	Sylow
18	C₂×C₃	Dic₁₅	1	4	yes	—
19	C₅	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic, complemented, Sylow
20	C₄	C₃⋊D₁₅	1	1	yes	characteristic
21	C₃	C₃⋊#168	1	4	yes	complemented
22	C₂	C₃⋊Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	C₃⋊D₁₅
commutator, hypocenter	C₃²×C₅	C₈
Fitting, cosocle	C₃²×C₄×C₅	C₂
socle, prime radical	C₂×C₃²×C₅	C₄
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₄

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.