Group #10604010 — subgroups

Subgroups — #10604010

Order 640.

212 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65680	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#65122	C₂	1	1	yes	maximal, characteristic
4	#170⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₈×#170	C₂	1	2	yes	maximal, complemented
6	C₂×D₅⋊C₈	C₄	1	2	yes	—
7	C₄²×D₅	C₂²	1	1	yes	characteristic
8	#4003	C₄	1	2	yes	—
9	C₄⋊#170	C₄	1	2	yes	—
10	C₄×#170	C₄	1	2	yes	—
11	C₄×#170	C₂²	1	1	yes	characteristic
12	C₄×#170	C₂²	1	2	yes	—
13	C₄×#170	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
15	#3952	C₄	1	2	yes	—
16	C₈×Dic₅	C₂²	1	2	yes	—
17	C₄⋊#168	C₄	1	1	yes	characteristic
18	#1450	—	5	1	no	maximal, Sylow
19	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂×#170	C₂×C₄	1	4	yes	—
23	C₂×#170	C₂×C₄	1	2	yes	—
24	C₂×#170	C₂×C₄	1	2	yes	—
25	D₅⋊C₈	C₈	1	8	yes	complemented
26	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅	C₂³	1	1	yes	characteristic
31	C₂×#168	C₂×C₄	1	2	yes	—
32	#431	—	5	1	no	—
33	#431	—	5	1	no	—
34	C₂×C₄×C₈	—	5	1	no	—
35	#322	—	5	2	no	—
36	C₈²	—	5	2	no	—
37	C₂²×D₅	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₅	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₅	C₄²	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₅	C₄²	1	1	yes	characteristic
42	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
44	C₄×D₅	—	2	1	no	—
45	C₄×D₅	—	2	1	no	—
46	C₄×D₅	—	2	2	no	—
47	C₄×D₅	C₂×C₈	1	4	yes	—
48	#170	—	2	4	no	—
49	#170	C₂×C₈	1	4	yes	—
50	#170	C₂×C₈	1	4	yes	—
51	C₅×C₈	—	2	2	no	—
52	#168	—	2	2	no	—
53	C₂²×C₈	—	5	2	no	—
54	C₂×C₄²	—	5	1	no	—
55	C₄⋊C₈	—	5	2	no	—
56	C₄⋊C₈	—	5	1	no	—
57	C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂²⋊C₈	—	5	2	no	—
59	C₂²⋊C₈	—	5	2	no	—
60	C₄×C₈	—	5	1	no	—
61	C₄×C₈	—	5	2	no	—
62	C₄×C₈	—	5	1	no	—
63	C₄×C₈	—	5	2	no	—
64	C₄×C₈	—	5	2	no	—
65	C₂²×C₅	C₂×C₄²	1	1	yes	characteristic
66	C₂×D₅	—	2	1	no	—
67	C₂×D₅	—	2	1	no	—
68	C₂×D₅	C₄×C₈	1	2	yes	—
69	C₄×C₅	—	2	1	no	—
70	C₄×C₅	C₄×C₈	1	2	yes	—
71	C₄×C₅	#131	1	1	yes	characteristic
72	C₄×C₅	#131	1	1	yes	characteristic
73	Dic₅	—	2	1	no	—
74	Dic₅	C₂²×C₈	1	2	yes	—
75	Dic₅	#131	1	2	yes	—
76	C₂²×C₄	—	5	1	no	—
77	C₂²×C₄	—	5	1	no	—
78	C₂²×C₄	—	5	1	no	—
79	C₂×C₈	—	5	2	no	—
80	C₂×C₈	—	5	2	no	—
81	C₂×C₈	—	5	8	no	—
82	C₂×C₈	—	5	4	no	—
83	C₂×C₈	—	5	2	no	—
84	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
85	C₄²	—	5	1	no	—
86	C₄²	—	5	1	no	—
87	C₄²	—	5	1	no	—
88	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
89	C₂×C₅	C₂×C₄×C₈	1	1	yes	characteristic
90	C₂×C₅	#405	1	1	yes	characteristic
91	C₂×C₅	#405	1	1	yes	characteristic
92	D₅	—	2	2	no	—
93	C₂³	—	5	1	no	—
94	C₂×C₄	—	5	1	no	—
95	C₂×C₄	—	5	1	no	—
96	C₂×C₄	—	5	4	no	—
97	C₂×C₄	—	5	1	no	—
98	C₂×C₄	—	10	2	no	—
99	C₂×C₄	—	10	1	no	—
100	C₂×C₄	—	10	1	no	—
101	C₂×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₂×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	5	4	no	—
106	C₈	—	5	4	no	—
107	C₈	—	10	4	no	—
108	C₈	—	10	2	no	—
109	C₅	#1450	1	1	yes	characteristic, complemented, Sylow
110	C₂²	—	5	2	no	—
111	C₂²	—	10	1	no	—
112	C₂²	—	10	1	no	—
113	C₂²	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	5	2	no	—
116	C₄	—	5	2	no	—
117	C₄	—	10	1	no	—
118	C₄	C₈×AGL(1,𝔽₅)	1	2	yes	—
119	C₄	#4131	1	1	yes	characteristic
120	C₄	#4133	1	1	yes	characteristic
121	C₂	—	10	2	no	—
122	C₂	#65701	1	1	yes	characteristic
123	C₂	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₂	#65728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65680

C₂

yes

maximal, characteristic

C₄×D₅⋊C₈

C₂

yes

maximal, characteristic

#65122

C₂

yes

maximal, characteristic

#170⋊C₈

C₂

yes

maximal, complemented

C₈×#170

C₂

yes

maximal, complemented

C₂×D₅⋊C₈

C₄

yes

—

C₄²×D₅

C₂²

yes

characteristic

#4003

C₄

yes

—

C₄⋊#170

C₄

yes

—

C₄×#170

C₄

yes

—

C₄×#170

C₂²

yes

characteristic

C₄×#170

C₂²

yes

—

C₄×#170

C₂²

yes

characteristic

C₅×C₄⋊C₈

C₄

yes

characteristic

#3952

C₄

yes

—

C₈×Dic₅

C₂²

yes

—

C₄⋊#168

C₄

yes

characteristic

#1450

—

maximal, Sylow

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×#170

C₂×C₄

yes

—

C₂×#170

C₂×C₄

yes

—

C₂×#170

C₂×C₄

yes

—

D₅⋊C₈

C₈

yes

complemented

C₂×C₅×C₈

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂³

yes

characteristic

C₂×#168

C₂×C₄

yes

—

#431

—

#431

—

C₂×C₄×C₈

—

#322

—

C₈²

—

C₂²×D₅

C₄²

yes

characteristic

C₂×C₄×C₅

C₄²

yes

characteristic

C₂×C₄×C₅

C₄²

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₄²

yes

characteristic

C₂×Dic₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂²×C₄

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×C₈

yes

—

#170

—

#170

C₂×C₈

yes

—

#170

C₂×C₈

yes

—

C₅×C₈

—

#168

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

AGL(1,𝔽₅)

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₂×C₄²

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

C₄×C₈

yes

—

C₄×C₅

—

C₄×C₅

C₄×C₈

yes

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

C₂²×C₈

yes

—

Dic₅

#131

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₅

C₂×C₄×C₈

yes

C₂×C₅

yes

C₂×C₅

yes

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×AGL(1,𝔽₅)

yes

characteristic

102

C₂×C₄

C₄×AGL(1,𝔽₅)

yes

characteristic

103

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₅

#1450

yes

characteristic, complemented, Sylow

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₈×AGL(1,𝔽₅)

yes

—

119

C₄

#4131

yes

characteristic

120

C₄

#4133

yes

characteristic

121

C₂

—

122

C₂

#65701

yes

characteristic

123

C₂

C₂×C₈×AGL(1,𝔽₅)

yes

characteristic

124

C₂

#65728

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄×C₈
Fitting	C₅×C₄⋊C₈	C₄
socle	C₂²×C₅	C₂×C₄²
prime radical	C₂²×D₅	C₄²
cosocle	C₄×Dic₅	C₂³
hypercenter	C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1450
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

commutator

C₂×C₅

C₂×C₄×C₈

Fitting

C₅×C₄⋊C₈

C₄

socle

C₂²×C₅

C₂×C₄²

prime radical

C₂²×D₅

C₄²

cosocle

C₄×Dic₅

C₂³

C₄⋊C₈

C₅

G ▷ C₂×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
4. #170⋊C₈	C₂	semidirect
5. C₈×#170	C₂	semidirect
25. D₅⋊C₈	C₈	semidirect
109. C₅	#1450	semidirect

H ≅ G/N

type

4. #170⋊C₈

C₂

semidirect

5. C₈×#170

C₂

semidirect

25. D₅⋊C₈

C₈

semidirect

109. C₅

#1450

semidirect

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

subgroup	index
#65680	2
C₄×D₅⋊C₈	2
#65122	2
#170⋊C₈	2
C₈×#170	2
#1450	5

subgroup

index

C₄×D₅⋊C₈

Subgroups — #10604010

Conjugacy classes of subgroups

Complements

Maximal subgroups