Subgroups — C₂×#4021 #65810

Order 320.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#685	C₂	1	4	yes	maximal, complemented
5	#4021	C₂	1	8	yes	maximal, direct product factor
6	C₂³×D₅	C₂²	1	2	yes	—
7	C₂²×C₄×C₅	C₂²	1	2	yes	—
8	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅	C₂²	1	2	yes	—
10	C₂×D₂₀	—	2	4	no	—
11	C₂×D₂₀	C₂²	1	4	yes	—
12	C₂×Dic₁₀	—	2	4	no	—
13	C₂×Dic₁₀	C₂²	1	4	yes	complemented
14	C₄²×C₅	C₂²	1	4	yes	complemented
15	#685	C₂²	1	16	yes	complemented
16	C₂×#124	—	5	1	no	maximal, Sylow
17	C₂³×C₅	C₂³	1	1	yes	characteristic
18	C₂²×D₅	—	2	4	no	—
19	C₂²×D₅	—	2	8	no	—
20	C₂²×D₅	C₂³	1	4	yes	—
21	C₂×C₄×C₅	—	2	4	no	—
22	C₂×C₄×C₅	D₄	1	4	yes	—
23	C₂×C₄×C₅	C₂³	1	4	yes	—
24	C₂×C₄×C₅	C₂³	1	2	yes	—
25	C₂×Dic₅	—	2	4	no	—
26	C₂×Dic₅	C₂³	1	4	yes	—
27	D₂₀	—	2	8	no	—
28	Dic₁₀	—	2	8	no	—
29	C₂²×Q₈	—	5	1	no	—
30	C₂²×D₄	—	5	1	no	—
31	#124	—	5	8	no	—
32	C₂×C₂²⋊C₄	—	5	4	no	—
33	C₂×C₄²	D₅	1	1	yes	characteristic, complemented
34	C₂²×C₅	C₂×D₄	1	2	yes	—
35	C₂²×C₅	#41	1	4	yes	—
36	C₂²×C₅	C₂⁴	1	1	yes	characteristic
37	C₂×D₅	—	2	4	no	—
38	C₂×D₅	—	2	8	no	—
39	C₂×D₅	—	4	8	no	—
40	C₄×C₅	—	2	4	no	—
41	C₄×C₅	C₂×D₄	1	4	yes	—
42	Dic₅	—	2	4	no	—
43	C₂⁴	—	5	2	no	—
44	C₂×Q₈	—	5	4	no	—
45	C₂×Q₈	—	10	4	no	—
46	C₂×D₄	—	5	4	no	—
47	C₂×D₄	—	10	4	no	—
48	C₂²×C₄	—	5	2	no	—
49	C₂²×C₄	C₂×D₅	1	2	yes	—
50	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
51	C₂²⋊C₄	—	5	16	no	—
52	C₄²	C₂×D₅	1	4	yes	complemented
53	C₂×C₅	#124	1	4	yes	complemented
54	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₅	C₂×#41	1	2	yes	—
56	D₅	—	4	4	no	—
57	C₂³	—	5	4	no	—
58	C₂³	—	10	4	no	—
59	C₂³	—	10	8	no	—
60	C₂³	C₂²×D₅	1	1	yes	characteristic
61	Q₈	—	10	8	no	—
62	D₄	—	10	8	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	5	4	no	—
65	C₂×C₄	—	10	4	no	—
66	C₂×C₄	D₂₀	1	4	yes	—
67	C₂×C₄	C₂²×D₅	1	4	yes	—
68	C₂×C₄	C₂²×D₅	1	2	yes	—
69	C₅	C₂×#124	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	10	4	no	—
71	C₂²	—	10	8	no	—
72	C₂²	—	20	8	no	—
73	C₂²	C₂×D₂₀	1	2	yes	—
74	C₂²	#709	1	4	yes	—
75	C₂²	C₂³×D₅	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	10	4	no	—
78	C₄	C₂×D₂₀	1	4	yes	—
79	C₂	—	20	4	no	—
80	C₂	#4021	1	4	yes	direct product factor
81	C₂	C₂²×D₂₀	1	1	yes	characteristic
82	C₂	C₂×#709	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴
Frattini	C₂²	C₂³×D₅
Fitting	C₂×C₄²×C₅	C₂
socle	C₂³×C₅	C₂³
prime radical	C₂²×D₂₀	C₂
hypercenter	C₂×C₄²	D₅
hypocenter	C₅	C₂×#124
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₁₀	C₂	semidirect
3. C₂×C₄²×C₅	C₂	semidirect
4. C₂×#685	C₂	semidirect
5. #4021	C₂	direct (N × H)
13. C₂×Dic₁₀	C₂²	semidirect
14. C₄²×C₅	C₂²	semidirect
15. #685	C₂²	semidirect
33. C₂×C₄²	D₅	semidirect
52. C₄²	C₂×D₅	semidirect
53. C₂×C₅	#124	semidirect
69. C₅	C₂×#124	semidirect
80. C₂	#4021	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×D₂₀	2
C₂²×Dic₁₀	2
C₂×C₄²×C₅	2
C₂×#685	2
#4021	2
C₂×#124	5

Subgroups — C2×#4021 #65810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4021 #65810