Subgroups — C₂×#64886 #10604701

Order 640.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#674	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#64886	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₅	C₄	1	1	yes	characteristic
6	C₂²×#168	C₂²	1	1	yes	characteristic
7	C₂×#674	C₄	1	8	yes	complemented
8	C₂×#674	C₂²	1	4	yes	—
9	C₈×Dic₅	C₄	1	2	yes	—
10	C₈×Dic₅	C₂²	1	2	yes	—
11	C₂×#346	—	5	1	no	maximal, Sylow
12	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×Dic₅	C₈	1	2	yes	—
14	C₂×C₅×C₈	C₈	1	4	yes	—
15	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
16	C₄×Dic₅	C₈	1	2	yes	—
17	C₄×Dic₅	C₂×C₄	1	2	yes	—
18	C₂×#168	C₂×C₄	1	1	yes	characteristic
19	C₂×#168	C₂×C₄	1	4	yes	—
20	C₂×#168	C₂³	1	1	yes	characteristic
21	#674	C₂×C₄	1	8	yes	complemented
22	C₂²×C₁₆	—	5	2	no	—
23	C₂×C₄×C₈	—	5	1	no	—
24	#346	—	5	4	no	—
25	C₂³×C₅	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅	C₄²	1	2	yes	—
27	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₅	—	2	4	no	—
31	C₂×Dic₅	C₂×C₈	1	4	yes	—
32	C₅×C₈	C₂×C₈	1	4	yes	—
33	#168	C₄²	1	2	yes	—
34	#168	C₂²×C₄	1	2	yes	—
35	C₂²×C₈	—	5	1	no	—
36	C₂²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
37	C₂×C₄²	—	5	1	no	—
38	C₂×C₁₆	—	5	4	no	—
39	C₂×C₁₆	—	5	8	no	—
40	C₄×C₈	—	5	2	no	—
41	C₄×C₈	—	5	2	no	—
42	C₂²×C₅	C₄×C₈	1	2	yes	—
43	C₂²×C₅	M₃₂	1	4	yes	—
44	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
45	C₄×C₅	C₄×C₈	1	2	yes	—
46	C₄×C₅	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₅	C₂²×C₈	1	1	yes	characteristic
48	Dic₅	—	2	4	no	—
49	C₂²×C₄	—	5	2	no	—
50	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₂×C₈	—	5	4	no	—
52	C₂×C₈	—	5	1	no	—
53	C₂×C₈	—	5	1	no	—
54	C₂×C₈	#170	1	4	yes	—
55	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
56	C₄²	—	5	2	no	—
57	C₄²	—	5	2	no	—
58	C₁₆	—	5	8	no	—
59	C₂×C₅	#346	1	4	yes	complemented
60	C₂×C₅	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₅	C₂×M₃₂	1	2	yes	—
62	C₂³	D₅⋊C₈	1	1	yes	characteristic
63	C₂×C₄	—	5	4	no	—
64	C₂×C₄	—	10	4	no	—
65	C₂×C₄	D₅⋊C₈	1	1	yes	characteristic
66	C₂×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
67	C₂×C₄	C₂×#170	1	2	yes	—
68	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₈	—	5	2	no	—
70	C₈	—	5	2	no	—
71	C₈	C₂×#170	1	4	yes	—
72	C₅	C₂×#346	1	1	yes	characteristic, complemented, Sylow
73	C₂²	#3990	1	4	yes	—
74	C₂²	C₄×#170	1	2	yes	—
75	C₂²	C₂×D₅⋊C₈	1	1	yes	characteristic
76	C₄	—	10	4	no	—
77	C₄	C₄×#170	1	2	yes	—
78	C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₄	C₂²×#170	1	1	yes	characteristic
80	C₂	#64886	1	4	yes	direct product factor
81	C₂	C₂×#3990	1	2	yes	—
82	C₂	C₂×C₄×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₅×C₈	C₄
socle, prime radical	C₂³×C₅	C₂×C₈
cosocle	C₂×#168	C₂³
hypercenter	C₂²×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#346
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #64886	C₂	direct (N × H)
7. C₂×#674	C₄	semidirect
21. #674	C₂×C₄	semidirect
59. C₂×C₅	#346	semidirect
72. C₅	C₂×#346	semidirect
80. C₂	#64886	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×#674	2
C₂×C₈×Dic₅	2
#64886	2
C₂×#346	5

Subgroups — C2×#64886 #10604701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#64886 #10604701