Group #10601975 — subgroups

Subgroups — #10601975

Order 640.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₅×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	#65381	C₂	1	1	yes	maximal, characteristic, complemented
3	#65377	C₂	1	1	yes	maximal, characteristic, complemented
4	#65362	C₂	1	1	yes	maximal, characteristic
5	#65359	C₂	1	1	yes	maximal, characteristic, complemented
6	#64983	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₅×#127	C₂²	1	1	yes	characteristic
10	C₂×#689	—	2	2	no	—
11	C₂×#686	—	2	2	no	—
12	#4020	C₂²	1	1	yes	characteristic
13	#4015	C₂²	1	1	yes	characteristic
14	C₅×#102	—	2	2	no	—
15	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#3964	—	2	2	no	—
17	#3953	—	2	2	no	—
18	#3941	—	2	2	no	—
19	C₄×#168	C₂²	1	1	yes	characteristic
20	C₄×#168	C₂²	1	1	yes	characteristic
21	#1709	—	5	1	no	maximal, Sylow
22	C₂×C₅×Q₈	—	2	1	no	—
23	C₂×C₅×D₄	—	2	1	no	—
24	C₂×C₅×D₄	—	2	1	no	—
25	C₂×C₅×D₄	—	2	1	no	—
26	C₂×D₂₀	—	2	1	no	—
27	C₂×D₂₀	—	2	1	no	—
28	C₂×D₂₀	—	2	1	no	—
29	C₂×Dic₁₀	—	2	1	no	—
30	C₂×C₅×C₈	D₄	1	2	yes	—
31	C₅×C₄⋊C₄	—	2	1	no	—
32	C₅×C₄⋊C₄	—	2	1	no	—
33	C₄²×C₅	C₂³	1	1	yes	characteristic
34	#689	—	4	4	no	—
35	#686	—	4	4	no	—
36	C₄⋊Dic₅	—	2	1	no	—
37	C₄⋊Dic₅	—	2	1	no	—
38	C₂×#168	D₄	1	2	yes	—
39	C₂×#168	D₄	1	2	yes	—
40	#494	—	5	1	no	—
41	#493	—	5	1	no	—
42	#486	—	5	1	no	—
43	#486	—	5	1	no	—
44	#486	—	5	1	no	—
45	#486	D₅	1	1	yes	characteristic, complemented
46	C₈²	—	5	1	no	—
47	C₂³×C₅	—	4	1	no	—
48	C₂²×D₅	—	4	1	no	—
49	C₅×Q₈	—	4	2	no	—
50	C₅×D₄	—	4	2	no	—
51	C₅×D₄	—	4	2	no	—
52	C₅×D₄	—	4	2	no	—
53	C₂×C₄×C₅	—	4	1	no	—
54	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₅	—	4	1	no	—
58	D₂₀	—	4	2	no	—
59	D₂₀	—	4	2	no	—
60	D₂₀	—	4	2	no	—
61	Dic₁₀	—	4	2	no	—
62	C₅×C₈	—	2	2	no	—
63	#168	—	2	2	no	—
64	#168	D₈	1	4	yes	—
65	C₂×Q₁₆	—	10	2	no	—
66	C₂×D₈	—	10	2	no	—
67	C₄⋊Q₈	—	5	1	no	—
68	C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
69	#127	—	5	1	no	—
70	#127	C₂×D₅	1	1	yes	characteristic
71	#102	—	2	2	no	—
72	#102	—	10	2	no	—
73	#102	—	10	2	no	—
74	#102	—	10	2	no	—
75	C₄×C₈	—	5	1	no	—
76	C₄×C₈	—	5	1	no	—
77	C₄×C₈	C₂×D₅	1	1	yes	characteristic
78	C₂²×C₅	—	8	1	no	—
79	C₂²×C₅	—	8	1	no	—
80	C₂²×C₅	—	8	1	no	—
81	C₂²×C₅	#127	1	1	yes	characteristic
82	C₂×D₅	—	8	1	no	—
83	C₂×D₅	—	8	1	no	—
84	C₂×D₅	—	8	1	no	—
85	C₄×C₅	—	8	1	no	—
86	C₄×C₅	C₂×D₈	1	2	yes	—
87	C₄×C₅	#135	1	2	yes	—
88	C₄×C₅	#135	1	2	yes	—
89	Dic₅	—	8	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	10	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	10	1	no	—
96	C₂×D₄	—	10	1	no	—
97	C₂×D₄	—	10	1	no	—
98	Q₁₆	—	20	4	no	—
99	D₈	—	20	4	no	—
100	C₂×C₈	—	5	2	no	—
101	C₂×C₈	—	5	2	no	—
102	C₂×C₈	C₂²⋊D₅	1	2	yes	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	10	1	no	—
106	C₄⋊C₄	—	10	1	no	—
107	C₄²	C₂²×D₅	1	1	yes	characteristic
108	C₂×C₅	—	16	1	no	—
109	C₂×C₅	#493	1	1	yes	characteristic
110	C₂×C₅	#495	1	1	yes	characteristic
111	C₂×C₅	#495	1	1	yes	characteristic
112	D₅	—	16	1	no	—
113	C₂³	—	4	1	no	—
114	C₂³	—	20	1	no	—
115	Q₈	—	4	2	no	—
116	Q₈	—	20	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	20	2	no	—
121	D₄	—	20	2	no	—
122	D₄	—	20	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	20	1	no	—
125	C₂×C₄	D₄×D₅	1	1	yes	characteristic
126	C₂×C₄	D₄×D₅	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	5	4	no	—
130	C₈	—	10	2	no	—
131	C₅	#1709	1	1	yes	characteristic, complemented, Sylow
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	40	1	no	—
136	C₂²	—	40	1	no	—
137	C₂²	—	40	1	no	—
138	C₂²	#4086	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	40	1	no	—
141	C₄	D₅×D₈	1	2	yes	—
142	C₄	#4062	1	2	yes	—
143	C₄	#4096	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	80	1	no	—
146	C₂	#65348	1	1	yes	characteristic
147	C₂	#65440	1	1	yes	characteristic
148	C₂	#65457	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#486

C₂

yes

maximal, characteristic, complemented

#65381

C₂

yes

maximal, characteristic, complemented

#65377

C₂

yes

maximal, characteristic, complemented

#65362

C₂

yes

maximal, characteristic

#65359

C₂

yes

maximal, characteristic, complemented

#64983

C₂

yes

maximal, characteristic, complemented

C₈×#168

C₂

yes

maximal, characteristic, complemented

C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₅×#127

C₂²

yes

—

—

C₂²

yes

characteristic

#4015

C₂²

yes

characteristic

C₅×#102

—

C₄×C₅×C₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#168

C₂²

yes

characteristic

#1709

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂³

yes

characteristic

#689

—

#686

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

D₄

yes

—

C₂×#168

D₄

yes

—

—

—

—

—

—

D₅

yes

characteristic, complemented

C₈²

—

C₂³×C₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

#168

D₈

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅

yes

characteristic

#127

—

#127

C₂×D₅

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#127

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

C₂×D₈

yes

—

C₄×C₅

#135

yes

—

C₄×C₅

#135

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²⋊D₅

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

C₂²×D₅

yes

characteristic

108

C₂×C₅

—

109

C₂×C₅

#493

yes

characteristic

110

C₂×C₅

#495

yes

characteristic

111

C₂×C₅

#495

yes

characteristic

112

D₅

—

113

C₂³

—

114

C₂³

—

115

Q₈

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

D₄×D₅

yes

characteristic

126

C₂×C₄

D₄×D₅

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₅

#1709

yes

characteristic, complemented, Sylow

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#4086

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

D₅×D₈

yes

—

142

C₄

#4062

yes

—

143

C₄

#4096

yes

—

144

C₂

—

145

C₂

—

146

C₂

#65348

yes

characteristic

147

C₂

#65440

yes

characteristic

148

C₂

#65457

yes

characteristic

	subgroup	G/H
center	C₂²	#4086
commutator, cosocle	C₄²×C₅	C₂³
Frattini	C₄²	C₂²×D₅
Fitting	C₅×#486	C₂
socle	C₂²×C₅	#127
prime radical	#65362	C₂
hypercenter	#486	D₅
hypocenter	C₅	#1709
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#4086

commutator, cosocle

C₄²×C₅

C₂³

Frattini

C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. C₅×#486	C₂	semidirect
2. #65381	C₂	semidirect
3. #65377	C₂	semidirect
5. #65359	C₂	semidirect
6. #64983	C₂	semidirect
7. C₈×#168	C₂	semidirect
45. #486	D₅	semidirect
131. C₅	#1709	semidirect

H ≅ G/N

type

1. C₅×#486

C₂

semidirect

2. #65381

C₂

semidirect

3. #65377

C₂

semidirect

5. #65359

C₂

semidirect

6. #64983

C₂

semidirect

7. C₈×#168

C₂

semidirect

45. #486

D₅

semidirect

131. C₅

#1709

semidirect

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

subgroup	index
C₅×#486	2
#65381	2
#65377	2
#65362	2
#65359	2
#64983	2
C₈×#168	2
#1709	5

subgroup

index

Subgroups — #10601975

Conjugacy classes of subgroups

Complements

Maximal subgroups