Subgroups — #4024

Order 160.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#683	C₂	1	1	yes	maximal, characteristic, complemented
7	#683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×Dic₅	—	2	2	no	—
12	C₂×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×Dic₅	C₂²	1	1	yes	characteristic
16	Dic₁₀	—	2	2	no	—
17	C₂²⋊Q₈	—	5	1	no	maximal, Sylow
18	C₂²×C₅	—	2	1	no	—
19	C₂²×C₅	—	2	1	no	—
20	C₂²×C₅	Q₈	1	2	yes	complemented
21	C₂²×C₅	C₂³	1	1	yes	characteristic
22	C₄×C₅	—	2	1	no	—
23	C₄×C₅	—	2	1	no	—
24	Dic₅	—	2	1	no	—
25	Dic₅	—	2	1	no	—
26	Dic₅	—	2	1	no	—
27	Dic₅	D₄	1	2	yes	—
28	C₂×Q₈	—	5	1	no	—
29	C₂²×C₄	—	5	1	no	—
30	C₄⋊C₄	—	5	1	no	—
31	C₄⋊C₄	—	5	1	no	—
32	C₄⋊C₄	—	5	1	no	—
33	C₂²⋊C₄	—	5	1	no	—
34	C₂²⋊C₄	D₅	1	1	yes	characteristic
35	C₂×C₅	—	2	2	no	—
36	C₂×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂×C₅	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₅	#41	1	1	yes	characteristic
39	C₂³	C₂×D₅	1	1	yes	characteristic
40	Q₈	—	10	2	no	—
41	C₂×C₄	—	5	1	no	—
42	C₂×C₄	—	5	1	no	—
43	C₂×C₄	—	5	1	no	—
44	C₂×C₄	—	5	1	no	—
45	C₂×C₄	—	10	2	no	—
46	C₂×C₄	C₂×D₅	1	1	yes	characteristic
47	C₂×C₄	C₂×D₅	1	1	yes	characteristic
48	C₅	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	2	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	Dic₁₀	1	2	yes	complemented
52	C₂²	C₂²×D₅	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	5	2	no	—
56	C₄	—	10	1	no	—
57	C₄	—	10	1	no	—
58	C₄	—	10	1	no	—
59	C₂	—	2	2	no	—
60	C₂	C₂×Dic₁₀	1	1	yes	characteristic
61	C₂	D₄×D₅	1	1	yes	characteristic
62	C₂	#711	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅
commutator, socle, cosocle	C₂²×C₅	C₂³
Fitting	C₅×C₂²⋊C₄	C₂
prime radical	C₂³×C₅	C₂²
hypercenter	C₂²⋊C₄	D₅
hypocenter	C₅	C₂²⋊Q₈
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
2. C₂×Dic₁₀	C₂	semidirect
5. C₄⋊Dic₅	C₂	semidirect
6. #683	C₂	semidirect
7. #683	C₂	semidirect
20. C₂²×C₅	Q₈	semidirect
48. C₅	C₂²⋊Q₈	semidirect
51. C₂²	Dic₁₀	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅	2
C₂×Dic₁₀	2
C₅×C₂²⋊C₄	2
C₂²⋊Dic₅	2
C₄⋊Dic₅	2
#683	2
C₂²⋊Q₈	5