Group #140753 — subgroups

Subgroups — #140753

Order 512.

316 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₂×#2179	C₂	1	1	yes	maximal, characteristic
2	#13196	C₂	1	2	yes	maximal, complemented
3	C₂×#1369	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1336	C₂	1	2	yes	maximal, complemented
5	C₂×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#508	C₄	1	1	yes	characteristic
7	C₂×#443	C₂²	1	2	yes	—
8	#2179	—	2	2	no	—
9	#2179	—	2	2	no	—
10	C₂×#368	C₄	1	1	yes	characteristic
11	C₂×#364	C₂²	1	2	yes	—
12	C₂×#359	C₄	1	2	yes	—
13	C₂×#350	—	2	2	no	—
14	#2113	—	2	2	no	—
15	C₂×#346	C₂²	1	1	yes	characteristic
16	C₂×#346	C₂²	1	1	yes	characteristic
17	C₂×#322	C₂²	1	1	yes	characteristic
18	#1369	—	2	2	no	—
19	#1336	—	2	4	no	—
20	#1313	—	2	2	no	—
21	C₂×#135	C₂×C₄	1	2	yes	—
22	C₂×#131	—	2	2	no	—
23	#508	—	2	2	no	—
24	#508	—	2	2	no	—
25	C₂×Q₃₂	C₈	1	1	yes	characteristic, complemented
26	C₂×SD₃₂	C₈	1	2	yes	complemented
27	C₂×D₁₆	C₈	1	1	yes	characteristic, complemented
28	C₂×M₃₂	—	2	2	no	—
29	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	D₄	1	2	yes	—
32	#443	—	2	2	no	—
33	#443	—	2	2	no	—
34	#443	—	2	2	no	—
35	#443	—	2	2	no	—
36	C₂×#108	C₂×C₄	1	2	yes	—
37	C₂×#104	—	2	2	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#368	—	2	2	no	—
41	#364	—	2	2	no	—
42	#364	—	2	2	no	—
43	#359	—	2	4	no	—
44	#350	—	4	4	no	—
45	C₂²⋊C₁₆	—	4	4	no	—
46	#346	—	2	1	no	—
47	#346	—	2	1	no	—
48	#346	D₄	1	2	yes	—
49	#346	D₄	1	2	yes	—
50	#322	—	2	1	no	—
51	#322	D₄	1	2	yes	—
52	C₂×#41	—	2	2	no	—
53	#135	—	2	2	no	—
54	#135	—	2	2	no	—
55	#135	—	2	2	no	—
56	#135	—	2	2	no	—
57	C₂×Q₁₆	C₂×C₈	1	2	yes	—
58	C₂×SD₁₆	—	2	2	no	—
59	C₂×D₈	C₂×C₈	1	2	yes	—
60	#131	—	4	2	no	—
61	#131	—	4	2	no	—
62	#131	—	4	2	no	—
63	#131	—	4	2	no	—
64	C₂×M₁₆	—	2	2	no	—
65	C₂×M₁₆	—	4	2	no	—
66	C₂²×C₈	—	4	1	no	—
67	C₂²×C₈	—	4	2	no	—
68	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄²	#41	1	1	yes	characteristic
71	Q₃₂	—	2	2	no	—
72	SD₃₂	—	2	4	no	—
73	D₁₆	—	2	2	no	—
74	M₃₂	—	4	2	no	—
75	M₃₂	—	4	2	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
83	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
84	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
85	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
86	C₂×C₁₆	M₁₆	1	2	yes	—
87	#108	—	2	2	no	—
88	#108	—	2	2	no	—
89	#104	—	4	2	no	—
90	#104	—	4	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	4	1	no	—
93	C₄×C₈	C₂×D₄	1	1	yes	characteristic
94	C₄×C₈	C₂×D₄	1	1	yes	characteristic
95	C₄×C₈	C₂×D₄	1	1	yes	characteristic
96	C₄×C₈	#41	1	1	yes	characteristic
97	#41	—	4	2	no	—
98	#41	—	4	2	no	—
99	#41	—	4	2	no	—
100	#41	—	4	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	4	2	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	4	2	no	—
106	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
107	Q₁₆	—	2	2	no	—
108	Q₁₆	—	2	2	no	—
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	4	2	no	—
111	D₈	—	2	2	no	—
112	D₈	—	2	2	no	—
113	M₁₆	—	4	2	no	—
114	M₁₆	—	4	2	no	—
115	M₁₆	—	8	2	no	—
116	M₁₆	—	8	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	8	2	no	—
120	C₂×C₈	—	8	2	no	—
121	C₂×C₈	—	8	2	no	—
122	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₄×D₄	1	1	yes	characteristic
124	C₂×C₈	#121	1	1	yes	characteristic
125	C₂×C₈	#127	1	1	yes	characteristic
126	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₈	#131	1	1	yes	characteristic
129	C₂×C₈	#131	1	1	yes	characteristic
130	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
132	C₄²	—	2	1	no	—
133	C₄²	#121	1	1	yes	characteristic
134	C₄²	#124	1	1	yes	characteristic
135	C₁₆	—	2	2	no	—
136	C₁₆	—	4	2	no	—
137	C₁₆	—	8	2	no	—
138	C₁₆	C₂²⋊C₈	1	2	yes	—
139	C₁₆	C₂²⋊C₈	1	2	yes	—
140	C₂³	—	4	2	no	—
141	C₂³	#444	1	1	yes	characteristic
142	Q₈	—	4	2	no	—
143	Q₈	—	4	2	no	—
144	D₄	—	4	2	no	—
145	D₄	—	4	2	no	—
146	D₄	—	8	2	no	—
147	D₄	—	8	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	#390	1	1	yes	characteristic
155	C₂×C₄	C₈×D₄	1	1	yes	characteristic
156	C₂×C₄	#436	1	1	yes	characteristic
157	C₂×C₄	#442	1	1	yes	characteristic
158	C₂×C₄	#442	1	1	yes	characteristic
159	C₂×C₄	#444	1	1	yes	characteristic
160	C₈	—	8	2	no	—
161	C₈	—	8	2	no	—
162	C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₈	#408	1	1	yes	characteristic
164	C₈	C₈×D₄	1	1	yes	characteristic
165	C₈	#436	1	1	yes	characteristic
166	C₈	#469	1	2	yes	—
167	C₈	#496	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	—	8	4	no	—
170	C₂²	—	8	2	no	—
171	C₂²	#1580	1	1	yes	characteristic
172	C₂²	#1594	1	1	yes	characteristic
173	C₂²	#1975	1	1	yes	characteristic
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	8	2	no	—
177	C₄	#1580	1	1	yes	characteristic
178	C₄	#1594	1	1	yes	characteristic
179	C₄	#1954	1	1	yes	characteristic
180	C₄	#1972	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	16	2	no	—
184	C₂	#9659	1	1	yes	characteristic
185	C₂	#12068	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2179

C₂

yes

maximal, characteristic

#13196

C₂

yes

maximal, complemented

C₂×#1369

C₂

yes

maximal, characteristic, complemented

C₂×#1336

C₂

yes

maximal, complemented

C₂×#1313

C₂

yes

maximal, characteristic, complemented

C₂×#508

C₄

yes

characteristic

C₂×#443

C₂²

yes

—

#2179

—

#2179

—

C₂×#368

C₄

yes

characteristic

C₂×#364

C₂²

yes

—

C₂×#359

C₄

yes

—

C₂×#350

—

#2113

—

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₂×#322

C₂²

yes

—

—

—

C₂×C₄

yes

—

C₂×#131

—

#508

—

#508

—

C₂×Q₃₂

C₈

yes

characteristic, complemented

C₂×SD₃₂

C₈

yes

complemented

C₂×D₁₆

C₈

yes

characteristic, complemented

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#104

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

C₂²⋊C₁₆

—

#346

—

#346

—

#346

D₄

yes

—

#346

D₄

yes

—

#322

—

#322

D₄

yes

—

C₂×#41

—

—

—

—

—

C₂×Q₁₆

C₂×C₈

yes

—

C₂×SD₁₆

—

C₂×D₈

C₂×C₈

yes

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄²

#41

yes

characteristic

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

M₁₆

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

—

—

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₄

yes

characteristic

107

Q₁₆

—

108

Q₁₆

—

109

SD₁₆

—

110

SD₁₆

—

111

D₈

—

112

D₈

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

123

C₂×C₈

C₄×D₄

yes

characteristic

124

C₂×C₈

#121

yes

characteristic

125

C₂×C₈

#127

yes

characteristic

126

C₂×C₈

C₂²×C₈

yes

characteristic

127

C₂×C₈

C₂×M₁₆

yes

characteristic

128

C₂×C₈

#131

yes

characteristic

129

C₂×C₈

#131

yes

characteristic

130

C₂×C₈

AGL(1,ℤ/8)

yes

—

131

C₂×C₈

AGL(1,ℤ/8)

yes

—

132

C₄²

—

133

C₄²

#121

yes

characteristic

134

C₄²

#124

yes

characteristic

135

C₁₆

—

136

C₁₆

—

137

C₁₆

—

138

C₁₆

C₂²⋊C₈

yes

—

139

C₁₆

C₂²⋊C₈

yes

—

140

C₂³

—

141

C₂³

#444

yes

characteristic

142

Q₈

—

143

Q₈

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#390

yes

characteristic

155

C₂×C₄

C₈×D₄

yes

characteristic

156

C₂×C₄

#436

yes

characteristic

157

C₂×C₄

#442

yes

characteristic

158

C₂×C₄

#442

yes

characteristic

159

C₂×C₄

#444

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

C₂×C₂²⋊C₈

yes

characteristic

163

C₈

#408

yes

characteristic

164

C₈

C₈×D₄

yes

characteristic

165

C₈

#436

yes

characteristic

166

C₈

#469

yes

—

167

C₈

#496

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#1580

yes

characteristic

172

C₂²

#1594

yes

characteristic

173

C₂²

#1975

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#1580

yes

characteristic

178

C₄

#1594

yes

characteristic

179

C₄

#1954

yes

characteristic

180

C₄

#1972

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#9659

yes

characteristic

185

C₂

#12068

yes

—

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1580
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#442

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₄

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13196	C₂	semidirect
3. C₂×#1369	C₂	semidirect
4. C₂×#1336	C₂	semidirect
5. C₂×#1313	C₂	semidirect
25. C₂×Q₃₂	C₈	semidirect
26. C₂×SD₃₂	C₈	semidirect
27. C₂×D₁₆	C₈	semidirect

H ≅ G/N

type

2. #13196

C₂

semidirect

3. C₂×#1369

C₂

semidirect

4. C₂×#1336

C₂

semidirect

5. C₂×#1313

C₂

semidirect

25. C₂×Q₃₂

C₈

semidirect

26. C₂×SD₃₂

C₈

semidirect

27. C₂×D₁₆

C₈

semidirect

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

subgroup	index
C₂×#2179	2
#13196	2
C₂×#1369	2
C₂×#1336	2
C₂×#1313	2

subgroup

index

Subgroups — #140753

Conjugacy classes of subgroups

Complements

Maximal subgroups