Subgroups — #4028

Order 160.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₂₀	C₂	1	2	yes	maximal, complemented
4	C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#685	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅	C₂²	1	1	yes	characteristic
7	C₂²×D₅	—	2	2	no	—
8	C₂²×D₅	—	2	4	no	—
9	C₂²×D₅	C₂²	1	2	yes	—
10	C₂²×D₅	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₄×C₅	C₂²	1	2	yes	complemented
12	C₂²⋊D₅	—	2	2	no	—
13	C₂×Dic₅	C₂²	1	1	yes	characteristic, complemented
14	D₂₀	—	2	4	no	—
15	#120	—	5	1	no	maximal, Sylow
16	C₂²×C₅	—	2	2	no	—
17	C₂²×C₅	D₄	1	2	yes	complemented
18	C₂²×C₅	C₂³	1	1	yes	characteristic
19	C₂×D₅	—	2	8	no	—
20	C₂×D₅	—	2	2	no	—
21	C₂×D₅	—	2	2	no	—
22	C₂×D₅	—	2	2	no	—
23	C₂×D₅	—	2	1	no	—
24	C₂×D₅	D₄	1	4	yes	complemented
25	C₄×C₅	—	2	2	no	—
26	Dic₅	—	2	1	no	—
27	C₂⁴	—	5	1	no	—
28	C₂×D₄	—	5	1	no	—
29	C₂×D₄	—	5	2	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	C₂×D₄	1	1	yes	characteristic
34	C₂×C₅	C₂×D₄	1	2	yes	—
35	D₅	—	2	4	no	—
36	D₅	—	4	1	no	—
37	C₂³	—	5	2	no	—
38	C₂³	—	5	1	no	—
39	C₂³	—	10	4	no	—
40	C₂³	—	10	2	no	—
41	C₂³	C₂×D₅	1	1	yes	characteristic
42	D₄	—	10	4	no	—
43	D₄	—	10	2	no	—
44	C₂×C₄	—	5	1	no	—
45	C₂×C₄	C₂×D₅	1	2	yes	complemented
46	C₅	#120	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	2	2	no	—
48	C₂²	—	5	4	no	—
49	C₂²	—	10	8	no	—
50	C₂²	—	10	2	no	—
51	C₂²	—	10	2	no	—
52	C₂²	—	10	1	no	—
53	C₂²	—	10	2	no	—
54	C₂²	D₂₀	1	2	yes	complemented
55	C₂²	C₂²×D₅	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	10	1	no	—
58	C₂	—	2	2	no	—
59	C₂	—	10	4	no	—
60	C₂	—	20	1	no	—
61	C₂	C₂×D₂₀	1	1	yes	characteristic
62	C₂	D₄×D₅	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅
commutator, socle, cosocle	C₂²×C₅	C₂³
Fitting	C₅×C₂²⋊C₄	C₂
hypercenter	C₂²⋊C₄	D₅
hypocenter	C₅	#120
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₅	C₂	semidirect
2. C₂×C₂²⋊D₅	C₂	semidirect
3. C₂×D₂₀	C₂	semidirect
4. C₅×C₂²⋊C₄	C₂	semidirect
5. #685	C₂	semidirect
10. C₂²×D₅	C₂²	semidirect
11. C₂×C₄×C₅	C₂²	semidirect
13. C₂×Dic₅	C₂²	semidirect
17. C₂²×C₅	D₄	semidirect
24. C₂×D₅	D₄	semidirect
31. C₂²⋊C₄	D₅	semidirect
45. C₂×C₄	C₂×D₅	semidirect
46. C₅	#120	semidirect
54. C₂²	D₂₀	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₅	2
C₂×C₂²⋊D₅	2
C₂×D₂₀	2
C₅×C₂²⋊C₄	2
#685	2
#120	5