Subgroups — #10623753

Order 720.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₅×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#6648	—	3	4	no	maximal
5	C₂×C₃⋊D₁₅	C₄	1	1	yes	characteristic
6	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃⋊Dic₁₅	C₄	1	1	yes	characteristic
8	C₈×C₃⋊D₃	—	5	1	no	maximal, Hall
9	C₄×D₁₅	—	3	4	no	—
10	C₃×#168	D₃	1	4	yes	complemented
11	C₅×C₃⋊C₈	—	3	4	no	—
12	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
13	C₃⋊D₁₅	C₈	1	2	yes	complemented
14	C₈×D₅	—	9	1	no	Hall
15	C₄×C₃⋊D₃	—	5	1	no	—
16	C₃²×C₈	—	5	1	no	—
17	C₃⋊(C₃⋊C₈)	D₅	1	1	yes	characteristic, complemented
18	C₂×D₁₅	—	3	4	no	—
19	C₃×C₄×C₅	C₂×D₃	1	4	yes	—
20	Dic₁₅	—	3	4	no	—
21	D₃×C₈	—	15	4	no	—
22	C₃²×C₅	C₂×C₈	1	1	yes	characteristic, complemented, Hall
23	C₄×D₅	—	9	1	no	—
24	C₅×C₈	—	9	1	no	—
25	#168	C₃⋊D₃	1	1	yes	characteristic, complemented
26	C₂×C₃⋊D₃	—	5	1	no	—
27	C₃²×C₄	C₂×D₅	1	1	yes	characteristic
28	C₃⋊Dic₃	—	5	1	no	—
29	C₂×C₃×C₅	C₄×D₃	1	4	yes	—
30	D₁₅	—	3	8	no	—
31	C₄×D₃	—	15	4	no	—
32	C₃×C₈	—	5	4	no	—
33	C₃⋊C₈	—	3	4	no	—
34	C₂×D₅	—	9	1	no	—
35	C₄×C₅	C₂×C₃⋊D₃	1	1	yes	characteristic
36	Dic₅	—	9	1	no	—
37	C₂×C₃²	C₄×D₅	1	1	yes	characteristic
38	C₃⋊D₃	—	5	2	no	—
39	C₂×C₈	—	45	1	no	Sylow
40	C₃×C₅	D₃×C₈	1	4	yes	complemented
41	C₂×D₃	—	15	4	no	—
42	C₃×C₄	D₃×D₅	1	4	yes	—
43	Dic₃	—	15	4	no	—
44	C₂×C₅	C₄×C₃⋊D₃	1	1	yes	characteristic
45	D₅	—	9	2	no	—
46	C₃²	C₈×D₅	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄	—	45	1	no	—
48	C₈	—	5	1	no	—
49	C₈	—	9	1	no	—
50	C₂×C₃	#1200	1	4	yes	—
51	D₃	—	15	8	no	—
52	C₅	C₈×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	45	1	no	—
54	C₄	—	45	1	no	—
55	C₄	D₅×C₃⋊D₃	1	1	yes	characteristic
56	C₃	#6648	1	4	yes	complemented
57	C₂	—	45	2	no	—
58	C₂	#67137	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₅×C₃⋊D₃
commutator, hypocenter	C₃²×C₅	C₂×C₈
Fitting, cosocle	C₃²×C₄×C₅	C₂²
socle	C₂×C₃²×C₅	C₂×C₄
prime radical	C₂×C₃⋊D₁₅	C₄
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₅×C₃⋊(C₃⋊C₈)	C₂	semidirect
3. C₃²×#168	C₂	semidirect
10. C₃×#168	D₃	semidirect
13. C₃⋊D₁₅	C₈	semidirect
17. C₃⋊(C₃⋊C₈)	D₅	semidirect
22. C₃²×C₅	C₂×C₈	semidirect
25. #168	C₃⋊D₃	semidirect
40. C₃×C₅	D₃×C₈	semidirect
46. C₃²	C₈×D₅	semidirect
52. C₅	C₈×C₃⋊D₃	semidirect
56. C₃	#6648	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃⋊D₁₅	2
C₅×C₃⋊(C₃⋊C₈)	2
C₃²×#168	2
#6648	3
C₈×C₃⋊D₃	5