Subgroups — C₂×#709 #4141

Order 160.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
3	#709	C₂	1	8	yes	maximal, direct product factor
4	C₂×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₅	C₂	1	2	yes	maximal, complemented
6	C₂×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅	C₂²	1	1	yes	characteristic
8	C₂²×D₅	C₂²	1	2	yes	—
9	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅	C₂²	1	4	yes	complemented
12	C₂²⋊D₅	C₂²	1	8	yes	complemented
13	C₂×Dic₅	C₂²	1	2	yes	—
14	D₂₀	C₂²	1	4	yes	complemented
15	C₄×D₅	C₂²	1	8	yes	complemented
16	Dic₁₀	C₂²	1	4	yes	complemented
17	C₂×#41	—	5	1	no	maximal, Sylow
18	C₂²×C₅	—	2	2	no	—
19	C₂²×C₅	C₂³	1	2	yes	—
20	C₂²×C₅	C₂³	1	1	yes	characteristic
21	C₂×D₅	—	2	4	no	—
22	C₂×D₅	C₂³	1	4	yes	—
23	C₄×C₅	C₂³	1	2	yes	—
24	C₄×C₅	C₂³	1	2	yes	—
25	Dic₅	C₂³	1	4	yes	—
26	#41	—	5	8	no	—
27	C₂×Q₈	—	5	1	no	—
28	C₂×D₄	—	5	2	no	—
29	C₂×D₄	—	5	1	no	—
30	C₂²×C₄	—	5	2	no	—
31	C₂²×C₄	D₅	1	1	yes	characteristic, complemented
32	C₂×C₅	—	2	2	no	—
33	C₂×C₅	#41	1	2	yes	complemented
34	C₂×C₅	C₂⁴	1	1	yes	characteristic
35	D₅	—	2	4	no	—
36	C₂³	—	5	2	no	—
37	C₂³	C₂×D₅	1	1	yes	characteristic
38	Q₈	—	5	4	no	—
39	D₄	—	5	8	no	—
40	D₄	—	5	4	no	—
41	C₂×C₄	—	5	2	no	—
42	C₂×C₄	—	5	8	no	—
43	C₂×C₄	C₂×D₅	1	1	yes	characteristic
44	C₂×C₄	C₂×D₅	1	1	yes	characteristic
45	C₂×C₄	C₂×D₅	1	4	yes	complemented
46	C₅	C₂×#41	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	2	2	no	—
48	C₂²	—	5	4	no	—
49	C₂²	—	10	4	no	—
50	C₂²	C₂²×D₅	1	2	yes	—
51	C₂²	C₂²×D₅	1	1	yes	characteristic
52	C₄	—	5	4	no	—
53	C₄	C₂²×D₅	1	2	yes	—
54	C₄	C₂²×D₅	1	2	yes	—
55	C₂	—	2	2	no	—
56	C₂	—	10	4	no	—
57	C₂	#709	1	2	yes	direct product factor
58	C₂	C₂³×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅
commutator, cosocle	C₂×C₅	C₂⁴
Frattini	C₂	C₂³×D₅
Fitting	C₂²×C₄×C₅	C₂
socle	C₂²×C₅	C₂³
hypercenter	C₂²×C₄	D₅
hypocenter	C₅	C₂×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×C₅	C₂	semidirect
2. C₂×C₂²⋊D₅	C₂	semidirect
3. #709	C₂	direct (N × H)
4. C₂×D₂₀	C₂	semidirect
5. C₂×C₄×D₅	C₂	semidirect
6. C₂×Dic₁₀	C₂	semidirect
11. C₂×C₄×C₅	C₂²	semidirect
12. C₂²⋊D₅	C₂²	semidirect
14. D₂₀	C₂²	semidirect
15. C₄×D₅	C₂²	semidirect
16. Dic₁₀	C₂²	semidirect
31. C₂²×C₄	D₅	semidirect
33. C₂×C₅	#41	semidirect
45. C₂×C₄	C₂×D₅	semidirect
46. C₅	C₂×#41	semidirect
57. C₂	#709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅	2
C₂×C₂²⋊D₅	2
#709	2
C₂×D₂₀	2
C₂×C₄×D₅	2
C₂×Dic₁₀	2
C₂×#41	5

Subgroups — C2×#709 #4141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#709 #4141