Subgroups — C₃⋊(C₃⋊#672) #10623624

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#672	—	3	4	no	maximal
3	C₃²×C₄×C₅	C₄	1	1	yes	characteristic
4	C₃⋊(C₃⋊C₁₆)	—	5	1	no	maximal, Hall
5	C₃×C₅×C₈	D₃	1	4	yes	—
6	C₂×C₃²×C₅	C₈	1	1	yes	characteristic
7	#672	—	9	1	no	Hall
8	C₃²×C₈	D₅	1	1	yes	characteristic
9	C₃×C₄×C₅	Dic₃	1	4	yes	—
10	C₃⋊C₁₆	—	15	4	no	—
11	C₃²×C₅	C₁₆	1	1	yes	characteristic, complemented, Hall
12	C₅×C₈	C₃⋊D₃	1	1	yes	characteristic
13	C₃²×C₄	Dic₅	1	1	yes	characteristic
14	C₂×C₃×C₅	C₃⋊C₈	1	4	yes	—
15	C₃×C₈	D₁₅	1	4	yes	—
16	C₄×C₅	C₃⋊Dic₃	1	1	yes	characteristic
17	C₂×C₃²	#168	1	1	yes	characteristic
18	C₁₆	—	45	1	no	Sylow
19	C₃×C₅	C₃⋊C₁₆	1	4	yes	complemented
20	C₃×C₄	Dic₁₅	1	4	yes	—
21	C₂×C₅	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
22	C₃²	#672	1	1	yes	characteristic, complemented, Sylow
23	C₈	C₃⋊D₁₅	1	1	yes	characteristic
24	C₂×C₃	C₃⋊#168	1	4	yes	—
25	C₅	C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic, complemented, Sylow
26	C₄	C₃⋊Dic₁₅	1	1	yes	characteristic
27	C₃	C₃⋊#672	1	4	yes	complemented
28	C₂	C₃⋊(C₃⋊#168)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₈	C₃⋊D₁₅
commutator, hypocenter	C₃²×C₅	C₁₆
Fitting, cosocle	C₃²×C₅×C₈	C₂
socle, prime radical	C₂×C₃²×C₅	C₈
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₈

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.