Group #29516921 — subgroups

Subgroups — C₆₁×#13488 #29516921

Order 15616.

302 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 106 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×C₆₁×#362	C₂	1	2	yes	maximal
2	C₆₁×#1971	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1849	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1383	C₂	1	2	yes	maximal
5	C₆₁×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
8	C₂×#108×C₆₁	—	2	1	no	—
9	C₂×#108×C₆₁	C₂²	1	2	yes	—
10	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#423	—	2	1	no	—
12	C₂×#103×C₆₁	—	2	1	no	—
13	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#391	—	2	1	no	—
15	C₆₁×#362	—	2	4	no	—
16	C₂²×Q₈×C₆₁	—	2	1	no	—
17	C₂×Q₁₆×C₆₁	—	2	2	no	—
18	C₂×Q₁₆×C₆₁	—	2	2	no	—
19	C₂×Q₁₆×C₆₁	—	2	1	no	—
20	C₂×Q₁₆×C₆₁	D₄	1	2	yes	—
21	C₂×M₁₆×C₆₁	—	2	2	no	—
22	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
25	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
26	M₃₂×C₆₁	—	2	2	no	—
27	M₃₂×C₆₁	—	2	2	no	—
28	C₂×C₁₆×C₆₁	—	2	2	no	—
29	#108×C₆₁	—	2	2	no	—
30	#108×C₆₁	—	2	2	no	—
31	#108×C₆₁	—	4	2	no	—
32	#106×C₆₁	—	2	2	no	—
33	C₄⋊C₈×C₆₁	—	4	2	no	—
34	#103×C₆₁	—	4	2	no	—
35	#97×C₆₁	—	2	2	no	—
36	#95×C₆₁	—	4	2	no	—
37	C₂×Q₈×C₆₁	—	2	2	no	—
38	C₂×Q₈×C₆₁	—	4	1	no	—
39	C₂×Q₈×C₆₁	—	4	1	no	—
40	C₂×Q₈×C₆₁	—	4	1	no	—
41	C₂×Q₈×C₆₁	—	4	1	no	—
42	C₂×Q₈×C₆₁	—	4	1	no	—
43	C₂²×C₄×C₆₁	—	4	1	no	—
44	C₂²×C₄×C₆₁	—	4	1	no	—
45	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₆₁	—	2	4	no	—
48	Q₁₆×C₆₁	—	4	2	no	—
49	M₁₆×C₆₁	—	4	2	no	—
50	M₁₆×C₆₁	—	4	2	no	—
51	C₂×C₈×C₆₁	—	2	1	no	—
52	C₂×C₈×C₆₁	—	2	1	no	—
53	C₂×C₈×C₆₁	—	4	2	no	—
54	C₂×C₈×C₆₁	D₈	1	2	yes	—
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	#41	1	2	yes	—
57	C₂×C₈×C₆₁	#41	1	2	yes	—
58	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
59	C₄⋊C₄×C₆₁	—	4	1	no	—
60	C₄⋊C₄×C₆₁	—	4	1	no	—
61	C₄²×C₆₁	—	2	1	no	—
62	C₄²×C₆₁	—	2	1	no	—
63	C₁₆×C₆₁	—	4	2	no	—
64	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
65	Q₈×C₆₁	—	4	2	no	—
66	Q₈×C₆₁	—	4	2	no	—
67	Q₈×C₆₁	—	8	1	no	—
68	Q₈×C₆₁	—	8	1	no	—
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	—	8	1	no	—
73	C₂×C₄×C₆₁	—	8	1	no	—
74	C₂×C₄×C₆₁	—	8	2	no	—
75	C₂×C₄×C₆₁	—	8	1	no	—
76	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
82	C₈×C₆₁	—	2	2	no	—
83	C₈×C₆₁	—	8	2	no	—
84	C₈×C₆₁	#123	1	2	yes	—
85	C₈×C₆₁	#126	1	2	yes	—
86	#13488	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₆₁	—	2	2	no	—
88	C₂²×C₆₁	#461	1	1	yes	characteristic
89	C₂²×C₆₁	#481	1	1	yes	characteristic
90	C₂²×C₆₁	#486	1	1	yes	characteristic
91	C₄×C₆₁	—	4	2	no	—
92	C₄×C₆₁	—	8	2	no	—
93	C₄×C₆₁	—	16	1	no	—
94	C₄×C₆₁	#397	1	1	yes	characteristic
95	C₄×C₆₁	#459	1	1	yes	characteristic
96	C₄×C₆₁	#480	1	1	yes	characteristic
97	C₄×C₆₁	#489	1	1	yes	characteristic
98	C₂×#362	C₂×C₆₁	1	2	yes	—
99	#1971	C₂×C₆₁	1	1	yes	characteristic
100	#1849	C₂×C₆₁	1	1	yes	characteristic
101	#1383	C₂×C₆₁	1	2	yes	—
102	#1377	C₂×C₆₁	1	1	yes	characteristic
103	C₂×C₆₁	—	2	1	no	—
104	C₂×C₆₁	—	2	1	no	—
105	C₂×C₆₁	#2071	1	1	yes	characteristic
106	C₂×C₆₁	#2239	1	2	yes	—
107	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
108	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
109	C₂×#108	—	2	1	no	—
110	C₂×#108	C₂²×C₆₁	1	2	yes	—
111	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
112	#423	—	2	1	no	—
113	C₂×#103	—	2	1	no	—
114	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
115	#391	—	2	1	no	—
116	#362	—	2	4	no	—
117	C₆₁	#13488	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×Q₈	—	2	1	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	D₄×C₆₁	1	2	yes	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	4	2	no	—
134	#106	—	2	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	#103	—	4	2	no	—
137	#97	—	2	2	no	—
138	#95	—	4	2	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	4	1	no	—
141	C₂×Q₈	—	4	1	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	—	4	1	no	—
147	C₂²×C₄	SD₁₆×C₆₁	1	2	yes	—
148	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
149	Q₁₆	—	2	4	no	—
150	Q₁₆	—	4	2	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	D₈×C₆₁	1	2	yes	—
157	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
158	C₂×C₈	#41×C₆₁	1	2	yes	—
159	C₂×C₈	#41×C₆₁	1	2	yes	—
160	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	4	2	no	—
166	C₂³	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
167	Q₈	—	4	2	no	—
168	Q₈	—	4	2	no	—
169	Q₈	—	8	1	no	—
170	Q₈	—	8	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	1	no	—
178	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
179	C₂×C₄	#123×C₆₁	1	1	yes	characteristic
180	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
181	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
182	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
183	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	8	2	no	—
186	C₈	#123×C₆₁	1	2	yes	—
187	C₈	#126×C₆₁	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	C₆₁×#461	1	1	yes	characteristic
190	C₂²	C₆₁×#481	1	1	yes	characteristic
191	C₂²	C₆₁×#486	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	8	2	no	—
194	C₄	—	16	1	no	—
195	C₄	C₆₁×#397	1	1	yes	characteristic
196	C₄	C₆₁×#459	1	1	yes	characteristic
197	C₄	C₆₁×#480	1	1	yes	characteristic
198	C₄	C₆₁×#489	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	C₆₁×#2071	1	1	yes	characteristic
202	C₂	C₆₁×#2239	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₆₁×#362

C₂

yes

maximal

C₆₁×#1971

C₂

yes

maximal, characteristic

C₆₁×#1849

C₂

yes

maximal, characteristic

C₆₁×#1383

C₂

yes

maximal

C₆₁×#1377

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₆₁

C₂²

yes

characteristic

C₂×M₃₂×C₆₁

C₂²

yes

—

C₂×#108×C₆₁

—

C₂×#108×C₆₁

C₂²

yes

—

C₂×#106×C₆₁

C₂²

yes

characteristic

C₆₁×#423

—

C₂×#103×C₆₁

—

C₂×#97×C₆₁

C₂²

yes

characteristic

C₆₁×#391

—

C₆₁×#362

—

C₂²×Q₈×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

D₄

yes

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

M₃₂×C₆₁

—

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

#108×C₆₁

—

#108×C₆₁

—

#108×C₆₁

—

#106×C₆₁

—

C₄⋊C₈×C₆₁

—

#103×C₆₁

—

#97×C₆₁

—

#95×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

SD₁₆

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₁₆×C₆₁

—

C₂³×C₆₁

C₂×SD₁₆

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×D₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#123

yes

—

C₈×C₆₁

#126

yes

—

#13488

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₂×C₆₁

yes

—

#1971

C₂×C₆₁

yes

characteristic

100

#1849

C₂×C₆₁

yes

characteristic

101

#1383

C₂×C₆₁

yes

—

102

#1377

C₂×C₆₁

yes

characteristic

103

C₂×C₆₁

—

104

C₂×C₆₁

—

105

C₂×C₆₁

#2071

yes

characteristic

106

C₂×C₆₁

#2239

yes

—

107

C₂²×Q₁₆

C₂²×C₆₁

yes

characteristic

108

C₂×M₃₂

C₂²×C₆₁

yes

—

109

C₂×#108

—

110

C₂×#108

C₂²×C₆₁

yes

—

111

C₂×#106

C₂²×C₆₁

yes

characteristic

112

#423

—

113

C₂×#103

—

114

C₂×#97

C₂²×C₆₁

yes

characteristic

115

#391

—

116

#362

—

117

C₆₁

#13488

yes

characteristic, direct product factor, Sylow

118

C₂²×Q₈

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

D₄×C₆₁

yes

—

123

C₂×M₁₆

—

124

C₂²×C₈

D₄×C₆₁

yes

characteristic

125

C₂²×C₈

C₂³×C₆₁

yes

characteristic

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

D₄×C₆₁

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

133

#108

—

134

#106

—

135

C₄⋊C₈

—

136

#103

—

137

#97

—

138

#95

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

SD₁₆×C₆₁

yes

—

148

C₂²×C₄

C₂×D₄×C₆₁

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

D₈×C₆₁

yes

—

157

C₂×C₈

C₂×D₄×C₆₁

yes

characteristic

158

C₂×C₈

#41×C₆₁

yes

—

159

C₂×C₈

#41×C₆₁

yes

—

160

C₂×C₈

#41×C₆₁

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

166

C₂³

C₂×SD₁₆×C₆₁

yes

characteristic

167

Q₈

—

168

Q₈

—

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

#121×C₆₁

yes

characteristic

179

C₂×C₄

#123×C₆₁

yes

characteristic

180

C₂×C₄

#124×C₆₁

yes

characteristic

181

C₂×C₄

C₂×D₈×C₆₁

yes

characteristic

182

C₂×C₄

AGL(1,ℤ/8)×C₆₁

yes

characteristic

183

C₂×C₄

#137×C₆₁

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

#123×C₆₁

yes

—

187

C₈

#126×C₆₁

yes

—

188

C₂²

—

189

C₂²

C₆₁×#461

yes

characteristic

190

C₂²

C₆₁×#481

yes

characteristic

191

C₂²

C₆₁×#486

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

C₆₁×#397

yes

characteristic

196

C₄

C₆₁×#459

yes

characteristic

197

C₄

C₆₁×#480

yes

characteristic

198

C₄

C₆₁×#489

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

C₆₁×#2071

yes

characteristic

202

C₂

C₆₁×#2239

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₁	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂³×C₆₁	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#97×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

#481

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₆₁

prime radical

C₂³×C₆₁

C₂×SD₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×#97×C₆₁ ◁ G

N	H ≅ G/N	type
86. #13488	C₆₁	direct (N × H)
117. C₆₁	#13488	direct (N × H)

H ≅ G/N

type

86. #13488

C₆₁

direct (N × H)

117. C₆₁

#13488

direct (N × H)

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

subgroup	index
C₂×C₆₁×#362	2
C₆₁×#1971	2
C₆₁×#1849	2
C₆₁×#1383	2
C₆₁×#1377	2
#13488	61

subgroup

index

C₂×C₆₁×#362

C₆₁×#1971

C₆₁×#1849

C₆₁×#1383

C₆₁×#1377

#13488

Subgroups — C₆₁×#13488 #29516921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#13488 #29516921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13488 #29516921