Group #10713676 — subgroups

Subgroups — #10713676

Order 768.

566 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 143 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×C₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#4563	C₂	1	4	yes	maximal, complemented
3	C₄×#819	C₂	1	1	yes	maximal, characteristic, complemented
4	#69743	C₂	1	1	yes	maximal, characteristic, complemented
5	#13257	—	3	1	no	maximal, Sylow
6	C₂²×C₃×D₈	—	2	1	no	—
7	C₂²×C₃×D₈	C₂²	1	2	yes	—
8	C₂×C₃×#127	—	2	1	no	—
9	C₃×#493	—	2	2	no	—
10	C₃×#493	C₄	1	4	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#819	C₂²	1	2	yes	—
13	C₂×#812	C₂²	1	2	yes	—
14	C₄×#208	—	2	1	no	—
15	#4563	—	2	8	no	—
16	#4552	—	2	1	no	—
17	C₄⋊(C₃⋊C₁₆)	—	2	2	no	—
18	C₂×#493	D₃	1	1	yes	characteristic
19	#2150	—	3	1	no	—
20	C₂×#361	—	3	4	no	—
21	C₄×#108	—	3	1	no	—
22	C₂²×C₃×D₄	—	2	2	no	—
23	C₂²×C₃×D₄	—	4	2	no	—
24	C₂×C₃×D₈	—	2	4	no	—
25	C₂×C₃×D₈	—	2	4	no	—
26	C₂×C₃×D₈	—	2	2	no	—
27	C₂×C₃×D₈	—	2	4	no	—
28	C₂×C₃×D₈	—	4	4	no	—
29	C₂×C₃×D₈	C₂×C₄	1	4	yes	—
30	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
32	C₃×#127	—	2	4	no	—
33	C₃×#127	—	4	1	no	—
34	C₃×#127	—	4	1	no	—
35	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
36	C₂×#208	—	2	2	no	—
37	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄×C₈	D₄	1	2	yes	—
39	#819	—	2	2	no	—
40	#819	D₄	1	4	yes	complemented
41	#812	—	2	4	no	—
42	C₂×C₃⋊C₁₆	—	2	2	no	—
43	#803	—	4	1	no	—
44	C₄×C₃⋊C₈	—	4	1	no	—
45	C₂²×D₈	—	2	1	no	—
46	C₂²×D₈	C₂×D₃	1	2	yes	—
47	C₂×#127	—	2	1	no	—
48	C₂×M₃₂	—	3	2	no	—
49	#493	—	2	2	no	—
50	#493	Dic₃	1	4	yes	—
51	C₂×#108	—	3	2	no	—
52	C₄×M₁₆	—	6	1	no	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₄⋊C₁₆	—	6	2	no	—
55	#361	—	6	8	no	—
56	#341	—	6	1	no	—
57	C₂⁴×C₃	—	4	2	no	—
58	C₂×C₃×D₄	—	2	4	no	—
59	C₂×C₃×D₄	—	4	4	no	—
60	C₂×C₃×D₄	—	4	4	no	—
61	C₂×C₃×D₄	—	4	4	no	—
62	C₂×C₃×D₄	—	4	2	no	—
63	C₂×C₃×D₄	—	4	4	no	—
64	C₂×C₃×D₄	—	8	8	no	—
65	C₂²×C₃×C₄	SD₁₆	1	2	yes	—
66	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₃×D₈	—	2	8	no	—
68	C₃×D₈	—	4	8	no	—
69	C₃×D₈	—	4	2	no	—
70	C₃×D₈	—	4	2	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₈	#41	1	2	yes	—
77	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
78	C₃×C₄²	D₈	1	1	yes	characteristic
79	C₃×C₄²	D₈	1	1	yes	characteristic
80	#208	—	2	4	no	—
81	#208	—	4	2	no	—
82	C₂×C₃⋊C₈	—	4	2	no	—
83	C₃⋊C₁₆	—	4	2	no	—
84	C₂²×D₄	—	2	2	no	—
85	C₂²×D₄	—	4	2	no	—
86	C₂×D₈	—	2	4	no	—
87	C₂×D₈	—	2	4	no	—
88	C₂×D₈	—	2	4	no	—
89	C₂×D₈	—	2	2	no	—
90	C₂×D₈	—	4	4	no	—
91	C₂×D₈	C₂×Dic₃	1	4	yes	—
92	C₂×M₁₆	—	6	2	no	—
93	C₂²×C₈	#67	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
95	#127	—	2	4	no	—
96	#127	—	4	1	no	—
97	#127	—	4	1	no	—
98	C₂×C₄²	#67	1	1	yes	characteristic
99	M₃₂	—	6	4	no	—
100	C₂×C₁₆	—	6	2	no	—
101	#108	—	3	4	no	—
102	#108	—	6	2	no	—
103	#97	—	12	1	no	—
104	C₄×C₈	—	12	1	no	—
105	C₄×C₈	C₂×Dic₃	1	2	yes	—
106	C₄×C₈	#67	1	2	yes	—
107	C₂³×C₃	—	4	4	no	—
108	C₂³×C₃	—	8	8	no	—
109	C₂³×C₃	—	8	4	no	—
110	C₂³×C₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₃×D₄	—	4	4	no	—
112	C₃×D₄	—	4	4	no	—
113	C₃×D₄	—	8	8	no	—
114	C₃×D₄	—	8	8	no	—
115	C₃×D₄	—	8	4	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	#102	1	2	yes	—
118	C₂×C₃×C₄	#102	1	2	yes	—
119	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₄	#121	1	2	yes	—
121	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
122	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
123	C₃×C₈	—	2	2	no	—
124	C₃×C₈	C₄×D₄	1	2	yes	—
125	C₃×C₈	#124	1	1	yes	characteristic
126	C₃×C₈	#127	1	1	yes	characteristic
127	C₃⋊C₈	—	4	4	no	—
128	C₂⁴	—	4	2	no	—
129	C₂×D₄	—	2	4	no	—
130	C₂×D₄	—	4	4	no	—
131	C₂×D₄	—	4	4	no	—
132	C₂×D₄	—	4	2	no	—
133	C₂×D₄	—	4	4	no	—
134	C₂×D₄	—	4	4	no	—
135	C₂×D₄	—	8	8	no	—
136	C₂²×C₄	#214	1	2	yes	—
137	C₂²×C₄	C₂×#67	1	1	yes	characteristic
138	D₈	—	2	8	no	—
139	D₈	—	4	8	no	—
140	D₈	—	4	2	no	—
141	D₈	—	4	2	no	—
142	M₁₆	—	6	4	no	—
143	M₁₆	—	12	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	12	2	no	—
146	C₂×C₈	#217	1	2	yes	—
147	C₂×C₈	D₃×D₄	1	2	yes	—
148	C₂×C₈	#237	1	2	yes	—
149	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
150	C₂×C₈	C₂×#67	1	1	yes	characteristic
151	C₄²	#213	1	1	yes	characteristic
152	C₄²	#213	1	1	yes	characteristic
153	C₄²	#217	1	2	yes	—
154	C₁₆	—	12	2	no	—
155	C₂²×C₃	—	2	2	no	—
156	C₂²×C₃	—	8	8	no	—
157	C₂²×C₃	—	8	4	no	—
158	C₂²×C₃	—	16	8	no	—
159	C₂²×C₃	C₂×#102	1	1	yes	characteristic
160	C₂²×C₃	#460	1	2	yes	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	#361	1	4	yes	complemented
163	C₃×C₄	#390	1	1	yes	characteristic
164	C₃×C₄	#418	1	1	yes	characteristic
165	C₃×C₄	#459	1	2	yes	—
166	C₂³	—	4	4	no	—
167	C₂³	—	8	8	no	—
168	C₂³	—	8	4	no	—
169	C₂³	C₂×#214	1	1	yes	characteristic
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	8	no	—
173	D₄	—	8	4	no	—
174	D₄	—	8	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	#832	1	2	yes	—
177	C₂×C₄	#832	1	2	yes	—
178	C₂×C₄	C₂×#213	1	1	yes	characteristic
179	C₂×C₄	#932	1	2	yes	—
180	C₂×C₄	#938	1	2	yes	—
181	C₂×C₄	C₂×#217	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	12	4	no	—
184	C₈	D₄×Dic₃	1	2	yes	—
185	C₈	#936	1	1	yes	characteristic
186	C₈	#940	1	1	yes	characteristic
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	—	16	4	no	—
189	C₂×C₃	#1982	1	1	yes	characteristic
190	C₂×C₃	C₂×#361	1	2	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	8	4	no	—
193	C₂²	—	8	8	no	—
194	C₂²	—	16	8	no	—
195	C₂²	#5075	1	2	yes	—
196	C₂²	C₂×#832	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	#4563	1	4	yes	complemented
199	C₄	#5072	1	2	yes	—
200	C₄	#5215	1	1	yes	characteristic
201	C₄	#5221	1	1	yes	characteristic
202	C₃	#13257	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	2	2	no	—
204	C₂	—	16	4	no	—
205	C₂	#71761	1	1	yes	characteristic
206	C₂	C₂×#4563	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#493

C₂

yes

maximal, characteristic

C₂×#4563

C₂

yes

maximal, complemented

C₄×#819

C₂

yes

maximal, characteristic, complemented

#69743

C₂

yes

maximal, characteristic, complemented

#13257

—

maximal, Sylow

C₂²×C₃×D₈

—

C₂²×C₃×D₈

C₂²

yes

—

C₂×C₃×#127

—

C₃×#493

—

C₃×#493

C₄

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#819

C₂²

yes

—

C₂×#812

C₂²

yes

—

C₄×#208

—

#4563

—

#4552

—

C₄⋊(C₃⋊C₁₆)

—

C₂×#493

D₃

yes

—

—

—

C₂²×C₃×D₄

—

C₂²×C₃×D₄

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₂×C₃×D₈

C₂×C₄

yes

—

C₂²×C₃×C₈

D₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

—

—

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×#208

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#819

—

#819

D₄

yes

complemented

#812

—

C₂×C₃⋊C₁₆

—

#803

—

C₄×C₃⋊C₈

—

C₂²×D₈

—

C₂²×D₈

C₂×D₃

yes

—

C₂×#127

—

C₂×M₃₂

—

#493

—

#493

Dic₃

yes

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

C₄⋊C₁₆

—

#361

—

#341

—

C₂⁴×C₃

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂²×C₃×C₄

SD₁₆

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

#41

yes

—

C₃×C₄²

C₂²⋊C₄

yes

—

C₃×C₄²

D₈

yes

characteristic

C₃×C₄²

D₈

yes

characteristic

#208

—

#208

—

C₂×C₃⋊C₈

—

C₃⋊C₁₆

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×Dic₃

yes

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

—

—

—

C₂×C₄²

#67

yes

characteristic

M₃₂

—

100

C₂×C₁₆

—

101

#108

—

102

#108

—

103

#97

—

104

C₄×C₈

—

105

C₄×C₈

C₂×Dic₃

yes

—

106

C₄×C₈

#67

yes

—

107

C₂³×C₃

—

108

C₂³×C₃

—

109

C₂³×C₃

—

110

C₂³×C₃

C₂×SD₁₆

yes

characteristic

111

C₃×D₄

—

112

C₃×D₄

—

113

C₃×D₄

—

114

C₃×D₄

—

115

C₃×D₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

#102

yes

—

118

C₂×C₃×C₄

#102

yes

—

119

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

120

C₂×C₃×C₄

#121

yes

—

121

C₂×C₃×C₄

C₂×D₈

yes

characteristic

122

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

123

C₃×C₈

—

124

C₃×C₈

C₄×D₄

yes

—

125

C₃×C₈

#124

yes

characteristic

126

C₃×C₈

#127

yes

characteristic

127

C₃⋊C₈

—

128

C₂⁴

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

#214

yes

—

137

C₂²×C₄

C₂×#67

yes

characteristic

138

D₈

—

139

D₈

—

140

D₈

—

141

D₈

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

#217

yes

—

147

C₂×C₈

D₃×D₄

yes

—

148

C₂×C₈

#237

yes

—

149

C₂×C₈

C₂²×Dic₃

yes

characteristic

150

C₂×C₈

C₂×#67

yes

characteristic

151

C₄²

#213

yes

characteristic

152

C₄²

#213

yes

characteristic

153

C₄²

#217

yes

—

154

C₁₆

—

155

C₂²×C₃

—

156

C₂²×C₃

—

157

C₂²×C₃

—

158

C₂²×C₃

—

159

C₂²×C₃

C₂×#102

yes

characteristic

160

C₂²×C₃

#460

yes

—

161

C₃×C₄

—

162

C₃×C₄

#361

yes

complemented

163

C₃×C₄

#390

yes

characteristic

164

C₃×C₄

#418

yes

characteristic

165

C₃×C₄

#459

yes

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

C₂×#214

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

#832

yes

—

177

C₂×C₄

#832

yes

—

178

C₂×C₄

C₂×#213

yes

characteristic

179

C₂×C₄

#932

yes

—

180

C₂×C₄

#938

yes

—

181

C₂×C₄

C₂×#217

yes

characteristic

182

C₈

—

183

C₈

—

184

C₈

D₄×Dic₃

yes

—

185

C₈

#936

yes

characteristic

186

C₈

#940

yes

characteristic

187

C₂×C₃

—

188

C₂×C₃

—

189

C₂×C₃

#1982

yes

characteristic

190

C₂×C₃

C₂×#361

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

#5075

yes

—

196

C₂²

C₂×#832

yes

characteristic

197

C₄

—

198

C₄

#4563

yes

complemented

199

C₄

#5072

yes

—

200

C₄

#5215

yes

characteristic

201

C₄

#5221

yes

characteristic

202

C₃

#13257

yes

characteristic, complemented, Sylow

203

C₂

—

204

C₂

—

205

C₂

#71761

yes

characteristic

206

C₂

C₂×#4563

yes

—

	subgroup	G/H
center	C₂²	C₂×#832
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×#493	C₂
socle	C₂²×C₃	C₂×#102
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#493	D₃
hypocenter	C₃	#13257
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂²

C₂×#832

commutator

C₂×C₃×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₃

Fitting, prime radical

C₂×C₃×#493

C₂

socle

C₂²×C₃

C₂×#102

cosocle

C₂²×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
2. C₂×#4563	C₂	semidirect
3. C₄×#819	C₂	semidirect
4. #69743	C₂	semidirect
40. #819	D₄	semidirect
162. C₃×C₄	#361	semidirect
198. C₄	#4563	semidirect
202. C₃	#13257	semidirect

H ≅ G/N

type

2. C₂×#4563

C₂

semidirect

3. C₄×#819

C₂

semidirect

4. #69743

C₂

semidirect

40. #819

D₄

semidirect

162. C₃×C₄

#361

semidirect

198. C₄

#4563

semidirect

202. C₃

#13257

semidirect

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

subgroup	index
C₂×C₃×#493	2
C₂×#4563	2
C₄×#819	2
#69743	2
#13257	3

subgroup

index

C₂×C₃×#493

Subgroups — #10713676

Conjugacy classes of subgroups

Complements

Maximal subgroups