Group #10602867 — subgroups

Subgroups — #10602867

Order 640.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₂²×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4088	C₂	1	2	yes	maximal
3	C₂×#3968	C₂	1	2	yes	maximal, complemented
4	#65399	C₂	1	1	yes	maximal, characteristic
5	#64772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×Q₈	C₂²	1	2	yes	—
7	C₂×#709	C₂²	1	1	yes	characteristic
8	C₂×C₅×Q₁₆	—	2	4	no	—
9	C₂×C₅×Q₁₆	—	2	2	no	—
10	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#689	—	2	2	no	—
12	#4088	—	2	4	no	—
13	#4088	—	2	4	no	—
14	C₂×#688	—	2	2	no	—
15	C₂×#681	C₂²	1	1	yes	characteristic
16	C₂×#681	C₂²	1	2	yes	—
17	#3968	—	2	4	no	—
18	#3952	—	2	2	no	—
19	#2020	—	5	1	no	maximal, Sylow
20	C₂×C₅×Q₈	—	2	2	no	—
21	C₂×C₅×Q₈	—	2	2	no	—
22	C₂×C₅×Q₈	—	2	2	no	—
23	C₂×C₅×Q₈	—	2	2	no	—
24	C₂×C₅×Q₈	—	2	2	no	—
25	C₂×C₅×Q₈	D₄	1	4	yes	—
26	C₂²×C₄×C₅	—	2	2	no	—
27	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₅	—	2	1	no	—
29	#709	—	2	2	no	—
30	#709	—	2	2	no	—
31	C₂×D₂₀	D₄	1	1	yes	characteristic
32	C₂×C₄×D₅	—	2	1	no	—
33	C₂×Dic₁₀	D₄	1	1	yes	characteristic
34	C₅×Q₁₆	—	2	8	no	—
35	C₂×C₅×C₈	—	2	2	no	—
36	C₂×C₅×C₈	D₄	1	2	yes	—
37	#689	—	2	8	no	—
38	#688	—	2	8	no	—
39	#681	—	2	2	no	—
40	#681	—	2	2	no	—
41	#681	D₄	1	4	yes	—
42	C₂×#168	—	2	2	no	—
43	C₂×#168	—	2	1	no	—
44	C₂×#137	—	5	2	no	—
45	C₂²×Q₁₆	D₅	1	1	yes	characteristic, complemented
46	C₂×#101	—	5	2	no	—
47	#408	—	5	1	no	—
48	#341	—	5	1	no	—
49	C₂³×C₅	#41	1	1	yes	characteristic
50	C₂²×D₅	—	2	1	no	—
51	C₅×Q₈	—	2	4	no	—
52	C₅×Q₈	—	2	4	no	—
53	C₅×Q₈	—	4	2	no	—
54	C₅×Q₈	—	4	2	no	—
55	C₂×C₄×C₅	—	2	4	no	—
56	C₂×C₄×C₅	—	4	2	no	—
57	C₂×C₄×C₅	—	4	2	no	—
58	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊D₅	—	4	2	no	—
63	C₂×Dic₅	—	2	1	no	—
64	D₂₀	—	2	2	no	—
65	C₄×D₅	—	4	1	no	—
66	C₄×D₅	—	4	1	no	—
67	Dic₁₀	—	2	2	no	—
68	C₅×C₈	—	2	2	no	—
69	#168	—	2	4	no	—
70	#168	—	4	1	no	—
71	C₂×#41	—	5	1	no	—
72	C₂²×Q₈	C₂×D₅	1	2	yes	—
73	#137	—	10	4	no	—
74	#137	—	10	4	no	—
75	C₂×Q₁₆	—	2	4	no	—
76	C₂×Q₁₆	—	2	2	no	—
77	C₂×Q₁₆	—	10	2	no	—
78	C₂×SD₁₆	—	10	2	no	—
79	C₂×M₁₆	—	5	2	no	—
80	C₂×M₁₆	—	5	1	no	—
81	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
82	#101	—	10	4	no	—
83	C₂²⋊C₈	—	10	2	no	—
84	C₂²×C₅	—	2	2	no	—
85	C₂²×C₅	#121	1	1	yes	characteristic
86	C₂²×C₅	#121	1	2	yes	—
87	C₂×D₅	—	4	1	no	—
88	C₂×D₅	—	4	2	no	—
89	C₄×C₅	—	4	4	no	—
90	C₄×C₅	#120	1	1	yes	characteristic
91	C₄×C₅	#120	1	2	yes	—
92	C₄×C₅	#127	1	1	yes	characteristic
93	Dic₅	—	4	1	no	—
94	#41	—	10	2	no	—
95	#41	—	10	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	5	1	no	—
102	C₂×Q₈	C₂²⋊D₅	1	4	yes	—
103	C₂×D₄	—	5	1	no	—
104	C₂×D₄	—	10	1	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	10	1	no	—
107	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
108	Q₁₆	—	2	8	no	—
109	Q₁₆	—	10	8	no	—
110	SD₁₆	—	10	8	no	—
111	M₁₆	—	5	4	no	—
112	M₁₆	—	10	2	no	—
113	M₁₆	—	10	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	10	2	no	—
116	C₂×C₈	—	10	1	no	—
117	C₂×C₈	C₂²⋊D₅	1	2	yes	—
118	C₂×C₅	—	2	2	no	—
119	C₂×C₅	#392	1	1	yes	characteristic
120	C₂×C₅	#473	1	2	yes	—
121	D₅	—	8	1	no	—
122	C₂³	—	10	1	no	—
123	C₂³	#711	1	1	yes	characteristic
124	Q₈	—	2	4	no	—
125	Q₈	—	2	4	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	10	2	no	—
129	D₄	—	10	2	no	—
130	D₄	—	20	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	10	1	no	—
135	C₂×C₄	—	20	1	no	—
136	C₂×C₄	—	20	1	no	—
137	C₂×C₄	D₄×D₅	1	1	yes	characteristic
138	C₂×C₄	D₄×D₅	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	10	4	no	—
143	C₈	—	20	1	no	—
144	C₅	#2020	1	1	yes	characteristic, complemented, Sylow
145	C₂²	—	2	2	no	—
146	C₂²	—	20	2	no	—
147	C₂²	—	20	1	no	—
148	C₂²	#4084	1	1	yes	characteristic
149	C₂²	#4085	1	2	yes	—
150	C₄	—	4	4	no	—
151	C₄	—	20	1	no	—
152	C₄	#4083	1	2	yes	—
153	C₄	#4086	1	1	yes	characteristic
154	C₄	#4099	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	40	1	no	—
157	C₂	#65481	1	2	yes	—
158	C₂	#65512	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅×Q₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#4088

C₂

yes

maximal

C₂×#3968

C₂

yes

maximal, complemented

#65399

C₂

yes

maximal, characteristic

#64772

C₂

yes

maximal, characteristic, complemented

C₂²×C₅×Q₈

C₂²

yes

—

C₂×#709

C₂²

yes

characteristic

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂²×C₅×C₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#681

C₂²

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

D₄

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂×C₂²⋊D₅

—

#709

—

#709

—

C₂×D₂₀

D₄

yes

characteristic

C₂×C₄×D₅

—

C₂×Dic₁₀

D₄

yes

characteristic

C₅×Q₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

D₄

yes

—

—

—

—

—

D₄

yes

—

C₂×#168

—

C₂×#168

—

C₂×#137

—

C₂²×Q₁₆

D₅

yes

characteristic, complemented

C₂×#101

—

#408

—

#341

—

C₂³×C₅

#41

yes

characteristic

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

C₅×C₈

—

#168

—

#168

—

C₂×#41

—

C₂²×Q₈

C₂×D₅

yes

—

#137

—

#137

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅

yes

characteristic

#101

—

C₂²⋊C₈

—

C₂²×C₅

—

C₂²×C₅

#121

yes

characteristic

C₂²×C₅

#121

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#120

yes

characteristic

C₄×C₅

#120

yes

—

C₄×C₅

#127

yes

characteristic

Dic₅

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂²⋊D₅

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂²×D₅

yes

characteristic

108

Q₁₆

—

109

Q₁₆

—

110

SD₁₆

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²⋊D₅

yes

—

118

C₂×C₅

—

119

C₂×C₅

#392

yes

characteristic

120

C₂×C₅

#473

yes

—

121

D₅

—

122

C₂³

—

123

C₂³

#711

yes

characteristic

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

D₄×D₅

yes

characteristic

138

C₂×C₄

D₄×D₅

yes

—

139

C₂×C₄

C₂×C₂²⋊D₅

yes

—

140

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₅

#2020

yes

characteristic, complemented, Sylow

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#4084

yes

characteristic

149

C₂²

#4085

yes

—

150

C₄

—

151

C₄

—

152

C₄

#4083

yes

—

153

C₄

#4086

yes

characteristic

154

C₄

#4099

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#65481

yes

—

158

C₂

#65512

yes

characteristic

	subgroup	G/H
center	C₂²	#4084
commutator, cosocle	C₂²×C₄×C₅	C₂³
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₂²×C₅×Q₁₆	C₂
socle	C₂²×C₅	#121
prime radical	C₂×#709	C₂²
hypercenter	C₂²×Q₁₆	D₅
hypocenter	C₅	#2020
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#4084

commutator, cosocle

C₂²×C₄×C₅

C₂³

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₂²×C₅×Q₁₆

C₂

C₂²×C₅

C₂²

C₂²×Q₁₆

D₅

hypocenter

C₅

#2020

LCS

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
1. C₂²×C₅×Q₁₆	C₂	semidirect
3. C₂×#3968	C₂	semidirect
5. #64772	C₂	semidirect
45. C₂²×Q₁₆	D₅	semidirect
144. C₅	#2020	semidirect

H ≅ G/N

type

1. C₂²×C₅×Q₁₆

C₂

semidirect

3. C₂×#3968

C₂

semidirect

5. #64772

C₂

semidirect

45. C₂²×Q₁₆

D₅

semidirect

144. C₅

#2020

semidirect

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

subgroup	index
C₂²×C₅×Q₁₆	2
C₂×#4088	2
C₂×#3968	2
#65399	2
#64772	2
#2020	5

subgroup

index

C₂²×C₅×Q₁₆

Subgroups — #10602867

Conjugacy classes of subgroups

Complements

Maximal subgroups