Group #29598175 — subgroups

Subgroups — #29598175

Order 15616.

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

normal subgroups only

This group has 450 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	#21454532	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839547	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14519783	C₂²	1	2	yes	—
6	#16840312	C₄	1	1	yes	characteristic
7	C₂²×#11758185	C₂²	1	2	yes	—
8	#16839841	—	2	2	no	—
9	C₈×D₂₄₄	—	2	2	no	—
10	#16839786	C₄	1	1	yes	characteristic
11	#16839728	C₄	1	2	yes	—
12	#16839547	—	2	8	no	—
13	C₄×#11758185	—	2	2	no	—
14	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×C₄⋊C₁₆	—	2	2	no	—
17	C₂×#11758222	C₂×C₄	1	2	yes	—
18	#14519783	—	2	4	no	—
19	#14519783	—	2	4	no	—
20	C₂×#11758194	C₂×C₄	1	2	yes	—
21	C₂×#11758189	—	2	2	no	—
22	C₂×C₈×D₆₁	—	2	2	no	—
23	#14519777	C₈	1	1	yes	characteristic
24	#14519774	C₈	1	1	yes	characteristic
25	#14519765	C₈	1	2	yes	—
26	#14519751	—	2	2	no	—
27	C₄×D₂₄₄	—	2	2	no	—
28	C₂×#11758185	—	2	2	no	—
29	C₂×#11758185	—	2	2	no	—
30	C₂×#11758185	D₄	1	4	yes	complemented
31	C₄⋊#92418	—	2	2	no	—
32	#14519689	—	2	4	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₆₁	—	2	4	no	—
37	C₂×C₁₆×C₆₁	—	2	2	no	—
38	C₄×C₈×C₆₁	D₄	1	2	yes	—
39	C₄×C₈×C₆₁	D₄	1	2	yes	—
40	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₆₁	—	2	2	no	—
43	#11758222	—	2	4	no	—
44	#11758222	—	2	4	no	—
45	C₂×D₂₄₄	—	2	2	no	—
46	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
47	C₂×C₄×D₆₁	—	2	2	no	—
48	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	#41	1	2	yes	—
55	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
58	#11758198	—	2	4	no	—
59	C₄⋊Dic₆₁	—	2	2	no	—
60	#11758194	—	2	4	no	—
61	C₂×#92418	—	2	2	no	—
62	#11758189	—	4	4	no	—
63	C₈×D₆₁	—	4	2	no	—
64	C₈×D₆₁	—	4	2	no	—
65	C₁₆×C₆₁	—	4	2	no	—
66	#11758185	—	2	4	no	—
67	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×D₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	—	2	4	no	—
70	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#131	1	2	yes	—
76	C₂²⋊D₆₁	—	4	4	no	—
77	C₂×Dic₆₁	—	2	2	no	—
78	D₂₄₄	—	2	4	no	—
79	D₂₄₄	—	4	4	no	—
80	C₄×D₆₁	—	4	2	no	—
81	C₄×D₆₁	—	4	2	no	—
82	Dic₁₂₂	—	2	4	no	—
83	C₈×C₆₁	—	2	2	no	—
84	C₈×C₆₁	#121	1	2	yes	—
85	C₈×C₆₁	#124	1	1	yes	characteristic
86	C₈×C₆₁	#127	1	1	yes	characteristic
87	#92418	—	4	2	no	—
88	#13191	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	—	2	2	no	—
90	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₆₁	#436	1	2	yes	—
92	C₂×D₆₁	—	4	4	no	—
93	C₂×D₆₁	—	4	2	no	—
94	C₄×C₆₁	—	2	2	no	—
95	C₄×C₆₁	#350	1	4	yes	complemented
96	C₄×C₆₁	#390	1	1	yes	characteristic
97	C₄×C₆₁	#408	1	1	yes	characteristic
98	C₄×C₆₁	C₈×D₄	1	2	yes	—
99	Dic₆₁	—	4	2	no	—
100	#2931	—	61	1	no	—
101	#2150	D₆₁	1	1	yes	characteristic, complemented
102	C₂×#350	—	61	4	no	—
103	C₂×C₄×C₁₆	—	61	1	no	—
104	C₂×C₆₁	—	2	2	no	—
105	C₂×C₆₁	#1954	1	1	yes	characteristic
106	C₂×C₆₁	C₂×#350	1	2	yes	—
107	D₆₁	—	8	2	no	—
108	C₂×#131	—	61	2	no	—
109	#532	—	61	1	no	—
110	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
111	C₂²×C₁₆	—	61	2	no	—
112	#436	—	122	2	no	—
113	C₈×D₄	—	122	2	no	—
114	#423	—	61	1	no	—
115	#408	—	61	2	no	—
116	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	2	no	—
118	#350	—	122	8	no	—
119	C₄×C₁₆	—	122	2	no	—
120	C₆₁	#13191	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	61	2	no	—
122	#131	—	122	4	no	—
123	#131	—	122	4	no	—
124	C₂×M₁₆	—	61	2	no	—
125	C₂×M₁₆	—	122	2	no	—
126	C₂²×C₈	—	122	2	no	—
127	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
129	C₄⋊Q₈	—	61	1	no	—
130	#127	—	61	1	no	—
131	#124	—	61	2	no	—
132	#121	—	122	2	no	—
133	C₄×D₄	—	122	2	no	—
134	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
135	M₃₂	—	2	4	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	61	4	no	—
138	C₂×C₁₆	—	122	2	no	—
139	C₂×C₁₆	—	122	2	no	—
140	C₄⋊C₈	—	122	2	no	—
141	C₂²⋊C₈	—	122	4	no	—
142	C₄×C₈	D₂₄₄	1	2	yes	—
143	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
144	#41	—	122	4	no	—
145	#41	—	122	4	no	—
146	C₂×Q₈	—	61	2	no	—
147	C₂×D₄	—	61	2	no	—
148	C₂×D₄	—	122	2	no	—
149	C₂×D₄	—	122	2	no	—
150	C₂²×C₄	—	122	2	no	—
151	C₂²×C₄	#11758189	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
153	M₁₆	—	122	4	no	—
154	M₁₆	—	244	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	122	2	no	—
157	C₂×C₈	—	244	2	no	—
158	C₂×C₈	—	244	2	no	—
159	C₂×C₈	#11758198	1	2	yes	—
160	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
161	C₂×C₈	D₄×D₆₁	1	2	yes	—
162	C₂×C₈	#11758226	1	2	yes	—
163	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
164	C₄⋊C₄	—	122	2	no	—
165	C₂²⋊C₄	—	122	4	no	—
166	C₄²	C₈×D₆₁	1	1	yes	characteristic
167	C₄²	C₈×D₆₁	1	1	yes	characteristic
168	C₄²	#11758198	1	2	yes	—
169	C₁₆	—	4	2	no	—
170	C₁₆	—	122	4	no	—
171	C₂³	—	122	2	no	—
172	C₂³	C₂×#11758189	1	1	yes	characteristic
173	Q₈	—	122	4	no	—
174	D₄	—	122	4	no	—
175	D₄	—	244	4	no	—
176	D₄	—	244	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	122	2	no	—
179	C₂×C₄	—	244	2	no	—
180	C₂×C₄	—	244	2	no	—
181	C₂×C₄	#14519689	1	2	yes	—
182	C₂×C₄	#14519689	1	2	yes	—
183	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
184	C₂×C₄	#14519739	1	2	yes	—
185	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
186	C₂×C₄	#14519785	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	244	2	no	—
189	C₈	#14519752	1	2	yes	—
190	C₈	#14519766	1	1	yes	characteristic
191	C₈	#14519773	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	244	4	no	—
194	C₂²	—	244	2	no	—
195	C₂²	C₂×#14519689	1	1	yes	characteristic
196	C₂²	#16839839	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	244	2	no	—
199	C₄	#16839547	1	4	yes	complemented
200	C₄	#16839663	1	1	yes	characteristic
201	C₄	#16839730	1	1	yes	characteristic
202	C₄	#16839827	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	488	2	no	—
205	C₂	#21446612	1	1	yes	characteristic
206	C₂	C₂×#16839547	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454532

C₂

yes

maximal, characteristic

C₂×#16839547

C₂

yes

maximal, complemented

C₂×C₄×#11758185

C₂

yes

maximal, characteristic, complemented

C₆₁×#2150

C₂

yes

maximal, characteristic, complemented

C₂×#14519783

C₂²

yes

—

#16840312

C₄

yes

characteristic

C₂²×#11758185

C₂²

yes

—

#16839841

—

C₈×D₂₄₄

—

#16839786

C₄

yes

characteristic

#16839728

C₄

yes

—

#16839547

—

C₄×#11758185

—

C₂×M₃₂×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₂×#11758222

C₂×C₄

yes

—

#14519783

—

#14519783

—

C₂×#11758194

C₂×C₄

yes

—

C₂×#11758189

—

C₂×C₈×D₆₁

—

#14519777

C₈

yes

characteristic

#14519774

C₈

yes

characteristic

#14519765

C₈

yes

—

#14519751

—

C₄×D₂₄₄

—

C₂×#11758185

—

C₂×#11758185

—

C₂×#11758185

D₄

yes

complemented

C₄⋊#92418

—

#14519689

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₆₁

—

#11758222

—

#11758222

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂×C₈

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

C₂×C₈

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

#11758194

—

C₂×#92418

—

#11758189

—

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

#131

yes

—

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

#121

yes

—

C₈×C₆₁

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₁

#436

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₈×D₄

yes

—

Dic₆₁

—

100

#2931

—

101

#2150

D₆₁

yes

characteristic, complemented

102

C₂×#350

—

103

C₂×C₄×C₁₆

—

104

C₂×C₆₁

—

105

C₂×C₆₁

#1954

yes

characteristic

106

C₂×C₆₁

C₂×#350

yes

—

107

D₆₁

—

108

C₂×#131

—

109

#532

—

110

C₂×M₃₂

C₂×D₆₁

yes

—

111

C₂²×C₁₆

—

112

#436

—

122

—

113

C₈×D₄

—

122

—

114

#423

—

115

#408

—

116

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

117

C₄⋊C₁₆

—

118

#350

—

122

—

119

C₄×C₁₆

—

122

—

120

C₆₁

#13191

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

#131

—

122

—

123

#131

—

122

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

122

—

126

C₂²×C₈

—

122

—

127

C₂²×C₈

C₄×D₆₁

yes

characteristic

128

C₂²×C₈

C₂²×D₆₁

yes

characteristic

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

122

—

133

C₄×D₄

—

122

—

134

C₂×C₄²

C₄×D₆₁

yes

characteristic

135

M₃₂

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

122

—

139

C₂×C₁₆

—

122

—

140

C₄⋊C₈

—

122

—

141

C₂²⋊C₈

—

122

—

142

C₄×C₈

D₂₄₄

yes

—

143

C₄×C₈

C₂²⋊D₆₁

yes

—

144

#41

—

122

—

145

#41

—

122

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

122

—

149

C₂×D₄

—

122

—

150

C₂²×C₄

—

122

—

151

C₂²×C₄

#11758189

yes

—

152

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

153

M₁₆

—

122

—

154

M₁₆

—

244

—

155

C₂×C₈

—

156

C₂×C₈

—

122

—

157

C₂×C₈

—

244

—

158

C₂×C₈

—

244

—

159

C₂×C₈

#11758198

yes

—

160

C₂×C₈

C₂×D₂₄₄

yes

characteristic

161

C₂×C₈

D₄×D₆₁

yes

—

162

C₂×C₈

#11758226

yes

—

163

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

164

C₄⋊C₄

—

122

—

165

C₂²⋊C₄

—

122

—

166

C₄²

C₈×D₆₁

yes

characteristic

167

C₄²

C₈×D₆₁

yes

characteristic

168

C₄²

#11758198

yes

—

169

C₁₆

—

170

C₁₆

—

122

—

171

C₂³

—

122

—

172

C₂³

C₂×#11758189

yes

characteristic

173

Q₈

—

122

—

174

D₄

—

122

—

175

D₄

—

244

—

176

D₄

—

244

—

177

C₂×C₄

—

178

C₂×C₄

—

122

—

179

C₂×C₄

—

244

—

180

C₂×C₄

—

244

—

181

C₂×C₄

#14519689

yes

—

182

C₂×C₄

#14519689

yes

—

183

C₂×C₄

C₂×#11758198

yes

characteristic

184

C₂×C₄

#14519739

yes

—

185

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

186

C₂×C₄

#14519785

yes

—

187

C₈

—

188

C₈

—

244

—

189

C₈

#14519752

yes

—

190

C₈

#14519766

yes

characteristic

191

C₈

#14519773

yes

characteristic

192

C₂²

—

193

C₂²

—

244

—

194

C₂²

—

244

—

195

C₂²

C₂×#14519689

yes

characteristic

196

C₂²

#16839839

yes

—

197

C₄

—

198

C₄

—

244

—

199

C₄

#16839547

yes

complemented

200

C₄

#16839663

yes

characteristic

201

C₄

#16839730

yes

characteristic

202

C₄

#16839827

yes

—

203

C₂

—

204

C₂

—

488

—

205

C₂

#21446612

yes

characteristic

206

C₂

C₂×#16839547

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2150	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	#16840312	C₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2150	D₆₁
hypocenter	C₆₁	#13191
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#2150

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₈

prime radical

#16840312

C₄

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
2. C₂×#16839547	C₂	semidirect
3. C₂×C₄×#11758185	C₂	semidirect
4. C₆₁×#2150	C₂	semidirect
30. C₂×#11758185	D₄	semidirect
95. C₄×C₆₁	#350	semidirect
101. #2150	D₆₁	semidirect
120. C₆₁	#13191	semidirect
199. C₄	#16839547	semidirect

H ≅ G/N

type

2. C₂×#16839547

C₂

semidirect

3. C₂×C₄×#11758185

C₂

semidirect

4. C₆₁×#2150

C₂

semidirect

30. C₂×#11758185

D₄

semidirect

95. C₄×C₆₁

#350

semidirect

101. #2150

D₆₁

semidirect

120. C₆₁

#13191

semidirect

199. C₄

#16839547

semidirect

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

subgroup	index
#21454532	2
C₂×#16839547	2
C₂×C₄×#11758185	2
C₆₁×#2150	2
#13191	61

subgroup

index

C₆₁×#2150

Subgroups — #29598175

Conjugacy classes of subgroups

Complements

Maximal subgroups