Subgroups — #4096

Order 160.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	#709	C₂	1	1	yes	maximal, characteristic, complemented
3	#689	C₂	1	1	yes	maximal, characteristic, complemented
4	#688	C₂	1	1	yes	maximal, characteristic, complemented
5	#687	C₂	1	1	yes	maximal, characteristic, complemented
6	#686	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₅×D₄	—	2	1	no	—
10	C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅	—	2	1	no	—
12	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂²⋊D₅	—	2	1	no	—
14	D₂₀	C₂²	1	1	yes	characteristic
15	C₄×D₅	—	2	1	no	—
16	Dic₁₀	C₂²	1	1	yes	characteristic
17	#168	C₂²	1	1	yes	characteristic
18	#168	C₂²	1	1	yes	characteristic
19	#135	—	5	1	no	maximal, Sylow
20	C₂²×C₅	—	2	1	no	—
21	C₂²×C₅	D₄	1	1	yes	characteristic
22	C₂×D₅	—	2	1	no	—
23	C₄×C₅	—	2	1	no	—
24	C₄×C₅	D₄	1	1	yes	characteristic
25	C₄×C₅	C₂³	1	1	yes	characteristic
26	Dic₅	—	2	1	no	—
27	#41	—	5	1	no	—
28	#41	D₅	1	1	yes	characteristic, complemented
29	Q₁₆	—	5	1	no	—
30	SD₁₆	—	5	1	no	—
31	SD₁₆	—	5	1	no	—
32	D₈	—	5	1	no	—
33	C₂×C₈	—	5	1	no	—
34	C₂×C₅	—	2	1	no	—
35	C₂×C₅	—	4	1	no	—
36	C₂×C₅	C₂×D₄	1	1	yes	characteristic
37	D₅	—	4	1	no	—
38	Q₈	—	5	1	no	—
39	Q₈	C₂×D₅	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	—	5	1	no	—
42	D₄	—	10	1	no	—
43	D₄	C₂×D₅	1	1	yes	characteristic
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	10	1	no	—
46	C₂×C₄	C₂×D₅	1	1	yes	characteristic
47	C₈	—	5	1	no	—
48	C₈	—	5	1	no	—
49	C₅	#135	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	2	1	no	—
51	C₂²	—	10	1	no	—
52	C₂²	C₂²⋊D₅	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	10	1	no	—
55	C₄	C₂²⋊D₅	1	1	yes	characteristic
56	C₄	C₂²×D₅	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	—	4	1	no	—
59	C₂	—	20	1	no	—
60	C₂	C₂×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂³
Frattini	C₄	C₂²×D₅
Fitting	C₅×#41	C₂
socle	C₂×C₅	C₂×D₄
hypercenter	#41	D₅
hypocenter	C₅	#135
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₅×#41	C₂	semidirect
2. #709	C₂	semidirect
3. #689	C₂	semidirect
4. #688	C₂	semidirect
5. #687	C₂	semidirect
6. #686	C₂	semidirect
7. C₂×#168	C₂	semidirect
28. #41	D₅	semidirect
49. C₅	#135	semidirect

Maximal subgroups

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

subgroup	index
C₅×#41	2
#709	2
#689	2
#688	2
#687	2
#686	2
C₂×#168	2
#135	5