Subgroups — #4060

Order 160.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#713	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#688	C₂	1	1	yes	maximal, characteristic, complemented
5	#686	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
7	#676	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅	—	2	1	no	—
9	C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
10	C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²⋊D₅	—	2	1	no	—
12	D₂₀	—	2	1	no	—
13	D₂₀	C₂²	1	1	yes	characteristic
14	D₂₀	C₂²	1	1	yes	characteristic, complemented
15	C₄×D₅	—	2	1	no	—
16	C₄×D₅	C₂²	1	1	yes	characteristic
17	C₅×C₈	C₂²	1	1	yes	characteristic, complemented
18	#168	C₂²	1	1	yes	characteristic, complemented
19	AGL(1,ℤ/8)	—	5	1	no	maximal, Sylow
20	C₂²×C₅	—	2	1	no	—
21	C₂×D₅	—	2	1	no	—
22	C₂×D₅	—	2	1	no	—
23	C₂×D₅	—	4	2	no	—
24	C₂×D₅	D₄	1	1	yes	characteristic
25	C₄×C₅	—	2	1	no	—
26	C₄×C₅	C₂³	1	1	yes	characteristic
27	Dic₅	D₄	1	1	yes	characteristic
28	#41	—	5	1	no	—
29	C₂×D₄	—	5	1	no	—
30	SD₁₆	—	5	1	no	—
31	SD₁₆	D₅	1	1	yes	characteristic, complemented
32	D₈	—	5	1	no	—
33	D₈	—	5	1	no	—
34	M₁₆	—	5	1	no	—
35	C₂×C₅	—	4	1	no	—
36	C₂×C₅	C₂×D₄	1	1	yes	characteristic
37	D₅	—	2	1	no	—
38	D₅	—	4	1	no	—
39	D₅	—	4	1	no	—
40	C₂³	—	10	1	no	—
41	Q₈	C₂×D₅	1	1	yes	characteristic, complemented
42	D₄	—	5	1	no	—
43	D₄	—	5	1	no	—
44	D₄	—	10	1	no	—
45	D₄	—	10	1	no	—
46	D₄	C₂×D₅	1	1	yes	characteristic
47	C₂×C₄	—	5	1	no	—
48	C₂×C₄	—	10	1	no	—
49	C₈	—	5	1	no	—
50	C₈	C₂×D₅	1	1	yes	characteristic, complemented
51	C₅	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	2	1	no	—
53	C₂²	—	5	1	no	—
54	C₂²	—	10	1	no	—
55	C₂²	—	10	1	no	—
56	C₂²	—	20	2	no	—
57	C₄	—	2	1	no	—
58	C₄	—	5	1	no	—
59	C₄	C₂²×D₅	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	10	1	no	—
62	C₂	—	20	1	no	—
63	C₂	—	20	1	no	—
64	C₂	D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₅
commutator, cosocle	C₄×C₅	C₂³
Frattini	C₄	C₂²×D₅
Fitting	C₅×SD₁₆	C₂
socle	C₂×C₅	C₂×D₄
hypercenter	SD₁₆	D₅
hypocenter	C₅	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. #713	C₂	semidirect
2. D₄×D₅	C₂	semidirect
3. C₅×SD₁₆	C₂	semidirect
4. #688	C₂	semidirect
5. #686	C₂	semidirect
6. D₄₀	C₂	semidirect
7. #676	C₂	semidirect
9. C₅×Q₈	C₂²	semidirect
14. D₂₀	C₂²	semidirect
17. C₅×C₈	C₂²	semidirect
18. #168	C₂²	semidirect
31. SD₁₆	D₅	semidirect
41. Q₈	C₂×D₅	semidirect
50. C₈	C₂×D₅	semidirect
51. C₅	AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
#713	2
D₄×D₅	2
C₅×SD₁₆	2
#688	2
#686	2
D₄₀	2
#676	2
AGL(1,ℤ/8)	5