Subgroups — Q₈×D₂₀ #65909

Order 320.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₅	C₂	1	2	yes	maximal, complemented
2	C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#4043	C₂	1	6	yes	maximal, complemented
4	C₄×D₂₀	C₂	1	3	yes	maximal
5	#4015	C₂	1	3	yes	maximal, complemented
6	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
7	Q₈×D₅	—	2	8	no	—
8	C₂×D₂₀	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₅	C₂²	1	6	yes	—
10	C₂×Dic₁₀	C₂²	1	6	yes	—
11	C₅×C₄⋊C₄	C₂²	1	3	yes	—
12	C₄²×C₅	C₂²	1	3	yes	—
13	#685	C₂²	1	6	yes	—
14	C₄⋊Dic₅	C₂²	1	3	yes	—
15	C₄⋊Dic₅	C₂²	1	6	yes	—
16	D₄×Q₈	—	5	1	no	maximal, Sylow
17	C₂²×D₅	C₂³	1	2	yes	—
18	C₅×Q₈	D₄	1	4	yes	complemented
19	C₂×C₄×C₅	C₂³	1	3	yes	—
20	C₂×C₄×C₅	C₂³	1	3	yes	—
21	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂×Dic₅	C₂³	1	6	yes	—
23	D₂₀	Q₈	1	4	yes	direct product factor
24	C₄×D₅	—	2	12	no	—
25	Dic₁₀	—	2	12	no	—
26	C₂²×Q₈	—	5	2	no	—
27	C₄⋊Q₈	—	5	3	no	—
28	C₂²⋊Q₈	—	5	6	no	—
29	C₄×Q₈	D₅	1	1	yes	characteristic, complemented
30	C₄×D₄	—	5	3	no	—
31	C₂²×C₅	C₂⁴	1	1	yes	characteristic
32	C₂×D₅	—	2	2	no	—
33	C₂×D₅	—	2	2	no	—
34	C₂×D₅	C₂×Q₈	1	4	yes	complemented
35	C₄×C₅	—	2	3	no	—
36	C₄×C₅	C₂×D₄	1	6	yes	—
37	C₄×C₅	C₂×Q₈	1	2	yes	complemented
38	Dic₅	—	2	6	no	—
39	C₂×Q₈	—	5	6	no	—
40	C₂×Q₈	—	10	8	no	—
41	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
42	C₂×D₄	—	5	1	no	—
43	C₂²×C₄	—	5	6	no	—
44	C₄⋊C₄	—	5	3	no	—
45	C₄⋊C₄	—	5	6	no	—
46	C₄⋊C₄	C₂×D₅	1	3	yes	—
47	C₂²⋊C₄	—	5	6	no	—
48	C₄²	C₂×D₅	1	3	yes	—
49	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₅	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
52	D₅	—	2	4	no	—
53	C₂³	—	5	2	no	—
54	Q₈	—	10	12	no	—
55	Q₈	D₂₀	1	4	yes	direct product factor
56	D₄	—	5	4	no	—
57	C₂×C₄	—	5	6	no	—
58	C₂×C₄	—	10	12	no	—
59	C₂×C₄	C₂²×D₅	1	3	yes	—
60	C₂×C₄	C₂²×D₅	1	3	yes	—
61	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
62	C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	5	4	no	—
64	C₂²	—	10	2	no	—
65	C₂²	—	10	2	no	—
66	C₂²	C₂³×D₅	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	10	6	no	—
69	C₄	C₂×D₂₀	1	6	yes	—
70	C₄	Q₈×D₅	1	2	yes	complemented
71	C₂	—	10	4	no	—
72	C₂	C₂²×D₂₀	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₅	1	1	yes	characteristic
74	C₂	#4150	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₄×C₅×Q₈	C₂
prime radical	C₂×D₂₀	C₂²
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	D₄×Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₅	C₂	semidirect
2. C₄×C₅×Q₈	C₂	semidirect
3. #4043	C₂	semidirect
5. #4015	C₂	semidirect
18. C₅×Q₈	D₄	semidirect
23. D₂₀	Q₈	direct (N × H)
29. C₄×Q₈	D₅	semidirect
34. C₂×D₅	C₂×Q₈	semidirect
37. C₄×C₅	C₂×Q₈	semidirect
55. Q₈	D₂₀	direct (N × H)
62. C₅	D₄×Q₈	semidirect
70. C₄	Q₈×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₅	2
C₄×C₅×Q₈	2
#4043	2
C₄×D₂₀	2
#4015	2
D₄×Q₈	5

Subgroups — Q8×D20 #65909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₂₀ #65909