Group #29512136 — subgroups

Subgroups — C₆₁×#8703 #29512136

Order 15616.

530 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 150 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×#382	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1817	C₂	1	2	yes	maximal
3	C₆₁×#1280	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊C₈×C₆₁	C₂²	1	4	yes	—
7	C₆₁×#382	—	2	2	no	—
8	C₆₁×#382	—	2	2	no	—
9	C₂×#95×C₆₁	C₄	1	2	yes	—
10	C₂×#95×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#336	—	2	2	no	—
12	C₆₁×#336	—	2	2	no	—
13	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₆₁	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₆₁	—	2	4	no	—
18	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
19	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₆₁	—	4	4	no	—
21	C₂×C₄²×C₆₁	—	2	4	no	—
22	C₂×C₄²×C₆₁	D₄	1	2	yes	—
23	C₂×C₄²×C₆₁	D₄	1	2	yes	—
24	C₂²⋊C₈×C₆₁	—	2	8	no	—
25	#95×C₆₁	—	2	4	no	—
26	#95×C₆₁	—	2	4	no	—
27	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	—	2	2	no	—
29	C₂²×C₄×C₆₁	—	2	2	no	—
30	C₂²×C₄×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	—	2	4	no	—
33	C₂²×C₄×C₆₁	—	2	4	no	—
34	C₂²×C₄×C₆₁	—	2	8	no	—
35	C₂²×C₄×C₆₁	—	4	2	no	—
36	C₂²×C₄×C₆₁	—	4	4	no	—
37	C₂²×C₄×C₆₁	—	4	4	no	—
38	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₆₁	#41	1	4	yes	—
43	C₂×C₈×C₆₁	—	4	8	no	—
44	C₂×C₈×C₆₁	—	4	4	no	—
45	C₄⋊C₄×C₆₁	—	4	4	no	—
46	C₄⋊C₄×C₆₁	—	4	4	no	—
47	C₄²×C₆₁	—	2	4	no	—
48	C₄²×C₆₁	—	2	4	no	—
49	C₂³×C₆₁	—	2	2	no	—
50	C₂³×C₆₁	—	2	1	no	—
51	C₂³×C₆₁	—	2	1	no	—
52	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂³×C₆₁	#117	1	2	yes	—
54	C₂³×C₆₁	#131	1	4	yes	—
55	C₂×C₄×C₆₁	—	2	8	no	—
56	C₂×C₄×C₆₁	—	2	1	no	—
57	C₂×C₄×C₆₁	—	2	1	no	—
58	C₂×C₄×C₆₁	—	2	4	no	—
59	C₂×C₄×C₆₁	—	2	1	no	—
60	C₂×C₄×C₆₁	—	2	1	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	4	no	—
63	C₂×C₄×C₆₁	—	4	4	no	—
64	C₂×C₄×C₆₁	—	4	4	no	—
65	C₂×C₄×C₆₁	—	4	4	no	—
66	C₂×C₄×C₆₁	—	4	4	no	—
67	C₂×C₄×C₆₁	—	4	4	no	—
68	C₂×C₄×C₆₁	—	8	8	no	—
69	C₂×C₄×C₆₁	#100	1	2	yes	—
70	C₂×C₄×C₆₁	#101	1	2	yes	—
71	C₂×C₄×C₆₁	#123	1	4	yes	—
72	C₈×C₆₁	—	8	4	no	—
73	#8703	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₆₁	—	2	2	no	—
75	C₂²×C₆₁	—	2	2	no	—
76	C₂²×C₆₁	—	2	4	no	—
77	C₂²×C₆₁	—	2	4	no	—
78	C₂²×C₆₁	#381	1	1	yes	characteristic
79	C₂²×C₆₁	#408	1	2	yes	—
80	C₂²×C₆₁	#410	1	2	yes	—
81	C₂²×C₆₁	C₂×#100	1	1	yes	characteristic
82	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
83	C₂²×C₆₁	#433	1	4	yes	—
84	C₄×C₆₁	—	2	4	no	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	2	2	no	—
87	C₄×C₆₁	—	4	2	no	—
88	C₄×C₆₁	—	8	4	no	—
89	C₂×#382	C₂×C₆₁	1	1	yes	characteristic
90	#1817	C₂×C₆₁	1	2	yes	—
91	#1280	C₂×C₆₁	1	4	yes	—
92	C₂×C₆₁	—	2	4	no	—
93	C₂×C₆₁	#1469	1	2	yes	—
94	C₂×C₆₁	#1470	1	2	yes	—
95	C₂×C₆₁	#1818	1	1	yes	characteristic
96	C₂×C₆₁	#1820	1	1	yes	characteristic
97	C₂×C₆₁	#1821	1	1	yes	characteristic
98	C₂²×C₄⋊C₄	C₄×C₆₁	1	2	yes	—
99	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
100	C₂×C₂²⋊C₈	C₂²×C₆₁	1	4	yes	—
101	#382	—	2	2	no	—
102	#382	—	2	2	no	—
103	C₂×#95	C₄×C₆₁	1	2	yes	—
104	C₂×#95	C₂²×C₆₁	1	2	yes	—
105	#336	—	2	2	no	—
106	#336	—	2	2	no	—
107	C₆₁	#8703	1	1	yes	characteristic, direct product factor, Sylow
108	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
109	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
110	C₂³×C₄	C₂×C₄×C₆₁	1	4	yes	—
111	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
112	C₂²×C₈	—	2	4	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	4	4	no	—
116	C₂×C₄²	—	2	4	no	—
117	C₂×C₄²	D₄×C₆₁	1	2	yes	—
118	C₂×C₄²	D₄×C₆₁	1	2	yes	—
119	C₂²⋊C₈	—	2	8	no	—
120	#95	—	2	4	no	—
121	#95	—	2	4	no	—
122	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	—	4	4	no	—
133	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
134	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
135	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
137	C₂²×C₄	#41×C₆₁	1	4	yes	—
138	C₂×C₈	—	4	8	no	—
139	C₂×C₈	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	1	no	—
146	C₂³	—	2	1	no	—
147	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
148	C₂³	#117×C₆₁	1	2	yes	—
149	C₂³	#131×C₆₁	1	4	yes	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	8	no	—
164	C₂×C₄	#100×C₆₁	1	2	yes	—
165	C₂×C₄	#101×C₆₁	1	2	yes	—
166	C₂×C₄	#123×C₆₁	1	4	yes	—
167	C₈	—	8	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₆₁×#381	1	1	yes	characteristic
173	C₂²	C₆₁×#408	1	2	yes	—
174	C₂²	C₆₁×#410	1	2	yes	—
175	C₂²	C₂×#100×C₆₁	1	1	yes	characteristic
176	C₂²	C₂×#101×C₆₁	1	1	yes	characteristic
177	C₂²	C₆₁×#433	1	4	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	8	4	no	—
183	C₂	—	2	4	no	—
184	C₂	C₆₁×#1469	1	2	yes	—
185	C₂	C₆₁×#1470	1	2	yes	—
186	C₂	C₆₁×#1818	1	1	yes	characteristic
187	C₂	C₆₁×#1820	1	1	yes	characteristic
188	C₂	C₆₁×#1821	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₆₁×#382

C₂

yes

maximal, characteristic

C₆₁×#1817

C₂

yes

maximal

C₆₁×#1280

C₂

yes

maximal

C₂²×C₄⋊C₄×C₆₁

C₄

yes

—

C₂²×C₄²×C₆₁

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₆₁

C₂²

yes

—

C₆₁×#382

—

C₆₁×#382

—

C₂×#95×C₆₁

C₄

yes

—

C₂×#95×C₆₁

C₂²

yes

—

C₆₁×#336

—

C₆₁×#336

—

C₂³×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂³×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂³×C₄×C₆₁

C₂×C₄

yes

—

C₂³×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂²⋊C₈×C₆₁

—

#95×C₆₁

—

#95×C₆₁

—

C₂⁴×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

#41

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₆₁

#117

yes

—

C₂³×C₆₁

#131

yes

—

C₂×C₄×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

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#100

yes

—

C₂×C₄×C₆₁

#101

yes

—

C₂×C₄×C₆₁

#123

yes

—

C₈×C₆₁

—

#8703

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#381

yes

characteristic

C₂²×C₆₁

#408

yes

—

C₂²×C₆₁

#410

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₂×#382

C₂×C₆₁

yes

characteristic

#1817

C₂×C₆₁

yes

—

#1280

C₂×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

#1469

yes

—

C₂×C₆₁

#1470

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×C₄⋊C₄

C₄×C₆₁

yes

—

C₂²×C₄²

C₂²×C₆₁

yes

characteristic

100

C₂×C₂²⋊C₈

C₂²×C₆₁

yes

—

101

#382

—

102

#382

—

103

C₂×#95

C₄×C₆₁

yes

—

104

C₂×#95

C₂²×C₆₁

yes

—

105

#336

—

106

#336

—

107

C₆₁

#8703

yes

characteristic, direct product factor, Sylow

108

C₂³×C₄

C₂×C₄×C₆₁

yes

characteristic

109

C₂³×C₄

C₂×C₄×C₆₁

yes

characteristic

110

C₂³×C₄

C₂×C₄×C₆₁

yes

—

111

C₂³×C₄

C₂³×C₆₁

yes

characteristic

112

C₂²×C₈

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

—

117

C₂×C₄²

D₄×C₆₁

yes

—

118

C₂×C₄²

D₄×C₆₁

yes

—

119

C₂²⋊C₈

—

120

#95

—

121

#95

—

122

C₂⁴

C₂²×C₄×C₆₁

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊C₄×C₆₁

yes

—

134

C₂²×C₄

C₂²⋊C₄×C₆₁

yes

characteristic

135

C₂²×C₄

C₂²⋊C₄×C₆₁

yes

characteristic

136

C₂²×C₄

C₂×D₄×C₆₁

yes

—

137

C₂²×C₄

#41×C₆₁

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

148

C₂³

#117×C₆₁

yes

—

149

C₂³

#131×C₆₁

yes

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#100×C₆₁

yes

—

165

C₂×C₄

#101×C₆₁

yes

—

166

C₂×C₄

#123×C₆₁

yes

—

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₆₁×#381

yes

characteristic

173

C₂²

C₆₁×#408

yes

—

174

C₂²

C₆₁×#410

yes

—

175

C₂²

C₂×#100×C₆₁

yes

characteristic

176

C₂²

C₂×#101×C₆₁

yes

characteristic

177

C₂²

C₆₁×#433

yes

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₂

—

184

C₂

C₆₁×#1469

yes

—

185

C₂

C₆₁×#1470

yes

—

186

C₂

C₆₁×#1818

yes

characteristic

187

C₂

C₆₁×#1820

yes

characteristic

188

C₂

C₆₁×#1821

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₁

C₂×C₂²⋊C₄

commutator

C₂⁴

C₂²×C₄×C₆₁

Frattini, cosocle

C₂³×C₄

C₂³×C₆₁

prime radical

C₂⁴×C₆₁

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
73. #8703	C₆₁	direct (N × H)
107. C₆₁	#8703	direct (N × H)

H ≅ G/N

type

73. #8703

C₆₁

direct (N × H)

107. C₆₁

#8703

direct (N × H)

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

subgroup	index
C₂×C₆₁×#382	2
C₆₁×#1817	2
C₆₁×#1280	2
#8703	61

subgroup

index

C₂×C₆₁×#382

C₆₁×#1817

C₆₁×#1280

#8703

Subgroups — C₆₁×#8703 #29512136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#8703 #29512136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8703 #29512136