Group #29596372 — subgroups

Subgroups — #29596372

Order 15616.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	#21447574	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	#21443901	C₂	1	1	yes	maximal, characteristic
4	#21443900	C₂	1	2	yes	maximal
5	C₆₁×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
12	C₄²×Dic₆₁	C₄	1	1	yes	characteristic
13	#16839580	—	2	4	no	—
14	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
17	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₆₁	—	2	2	no	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
21	#14519717	—	2	2	no	—
22	#14519717	—	2	2	no	—
23	C₂×#11758185	—	2	2	no	—
24	C₄×#92418	—	2	2	no	—
25	C₈×Dic₆₁	—	2	2	no	—
26	C₈×Dic₆₁	—	2	2	no	—
27	C₈×Dic₆₁	C₈	1	2	yes	—
28	C₈×Dic₆₁	C₈	1	2	yes	—
29	C₈×Dic₆₁	D₄	1	2	yes	—
30	C₈×Dic₆₁	Q₈	1	2	yes	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₆₁	—	2	2	no	—
35	M₃₂×C₆₁	—	2	2	no	—
36	C₂×C₁₆×C₆₁	—	2	2	no	—
37	C₄×C₈×C₆₁	—	2	2	no	—
38	C₂²×C₄×C₆₁	—	2	1	no	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	—	2	1	no	—
41	C₂²×Dic₆₁	M₁₆	1	2	yes	—
42	C₂×C₈×C₆₁	—	2	1	no	—
43	C₂×C₈×C₆₁	—	2	1	no	—
44	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₆₁	#41	1	2	yes	—
48	C₂×C₈×C₆₁	#41	1	2	yes	—
49	C₄²×C₆₁	—	2	1	no	—
50	C₄²×C₆₁	M₁₆	1	2	yes	—
51	C₄×Dic₆₁	—	2	8	no	—
52	C₄×Dic₆₁	—	2	1	no	—
53	C₄×Dic₆₁	C₄⋊C₄	1	2	yes	—
54	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
55	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
56	C₄×Dic₆₁	M₁₆	1	2	yes	—
57	C₂×#92418	—	2	1	no	—
58	C₂×#92418	—	2	1	no	—
59	C₂×#92418	—	2	1	no	—
60	C₂×#92418	—	2	1	no	—
61	C₂×#92418	C₄⋊C₄	1	2	yes	—
62	C₂×#92418	C₂×C₈	1	2	yes	—
63	C₁₆×C₆₁	—	4	2	no	—
64	#11758185	—	4	2	no	—
65	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₆₁	—	2	4	no	—
75	C₂×Dic₆₁	—	2	2	no	—
76	C₂×Dic₆₁	—	2	4	no	—
77	C₂×Dic₆₁	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₆₁	C₄⋊C₈	1	2	yes	—
79	C₈×C₆₁	—	2	2	no	—
80	C₈×C₆₁	#124	1	2	yes	—
81	C₈×C₆₁	#125	1	2	yes	—
82	#92418	—	2	2	no	—
83	#92418	—	2	2	no	—
84	#12883	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	—	2	2	no	—
86	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₆₁	#432	1	1	yes	characteristic
88	C₂²×C₆₁	#432	1	1	yes	characteristic
89	C₄×C₆₁	—	2	4	no	—
90	C₄×C₆₁	#382	1	1	yes	characteristic
91	C₄×C₆₁	#423	1	1	yes	characteristic
92	C₄×C₆₁	#431	1	1	yes	characteristic
93	C₄×C₆₁	#431	1	1	yes	characteristic
94	Dic₆₁	—	2	4	no	—
95	Dic₆₁	—	2	2	no	—
96	Dic₆₁	#364	1	4	yes	—
97	#2150	—	61	2	no	—
98	C₄²×C₈	—	61	1	no	—
99	#1376	—	61	2	no	—
100	#1376	—	61	1	no	—
101	#1376	D₆₁	1	1	yes	characteristic
102	C₂×C₆₁	—	2	1	no	—
103	C₂×C₆₁	—	2	1	no	—
104	C₂×C₆₁	#1831	1	1	yes	characteristic
105	C₂×C₆₁	C₂×#364	1	2	yes	—
106	C₂×M₃₂	—	61	2	no	—
107	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
108	C₂×C₄×C₈	—	61	1	no	—
109	C₂×C₄×C₈	—	61	1	no	—
110	C₂×C₄×C₈	—	61	1	no	—
111	C₂×C₄×C₈	—	61	1	no	—
112	C₂×C₄×C₈	—	61	1	no	—
113	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
114	C₄³	—	61	1	no	—
115	C₄⋊C₁₆	—	122	4	no	—
116	C₆₁	#12883	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	61	1	no	—
118	C₂²×C₈	—	61	1	no	—
119	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
121	C₂×C₄²	—	61	1	no	—
122	C₂×C₄²	—	61	1	no	—
123	C₂×C₄²	—	122	2	no	—
124	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	122	2	no	—
128	M₃₂	—	122	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	122	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	61	2	no	—
133	C₄×C₈	—	61	2	no	—
134	C₄×C₈	—	61	2	no	—
135	C₄×C₈	—	61	2	no	—
136	C₄×C₈	—	122	2	no	—
137	C₄×C₈	—	122	2	no	—
138	C₄×C₈	—	122	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	61	2	no	—
141	C₂²×C₄	—	122	1	no	—
142	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	61	2	no	—
146	C₂×C₈	—	61	2	no	—
147	C₂×C₈	—	122	1	no	—
148	C₂×C₈	—	122	1	no	—
149	C₂×C₈	—	122	1	no	—
150	C₂×C₈	—	122	1	no	—
151	C₂×C₈	C₈×D₆₁	1	2	yes	—
152	C₂×C₈	#11758222	1	2	yes	—
153	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
154	C₂×C₈	#11758224	1	2	yes	—
155	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
156	C₄²	—	2	1	no	—
157	C₄²	—	61	2	no	—
158	C₄²	—	61	1	no	—
159	C₄²	—	61	1	no	—
160	C₄²	—	61	2	no	—
161	C₄²	—	122	8	no	—
162	C₄²	—	122	1	no	—
163	C₄²	#11758189	1	2	yes	—
164	C₁₆	—	4	2	no	—
165	C₁₆	—	244	2	no	—
166	C₂³	M₁₆×D₆₁	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	61	2	no	—
170	C₂×C₄	—	61	2	no	—
171	C₂×C₄	—	122	4	no	—
172	C₂×C₄	—	122	4	no	—
173	C₂×C₄	—	122	2	no	—
174	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
175	C₂×C₄	#14519733	1	1	yes	characteristic
176	C₂×C₄	#14519734	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
178	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
179	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	122	2	no	—
182	C₈	—	122	2	no	—
183	C₈	#14519763	1	2	yes	—
184	C₈	#14519767	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₄⋊C₈×D₆₁	1	1	yes	characteristic
187	C₂²	#16839816	1	1	yes	characteristic
188	C₂²	#16839817	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	61	4	no	—
191	C₄	—	122	4	no	—
192	C₄	—	122	2	no	—
193	C₄	#16839626	1	1	yes	characteristic
194	C₄	#16839784	1	1	yes	characteristic
195	C₄	#16839812	1	1	yes	characteristic
196	C₄	#16839813	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#21445934	1	1	yes	characteristic
200	C₂	#364×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21447574

C₂

yes

maximal

C₄×C₈×Dic₆₁

C₂

yes

maximal, characteristic

#21443901

C₂

yes

maximal, characteristic

#21443900

C₂

yes

maximal

C₆₁×#1376

C₂

yes

maximal, characteristic

C₂×#14519717

C₂²

yes

—

C₂×C₄×#92418

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

C₄²×Dic₆₁

C₄

yes

characteristic

#16839580

—

C₂×M₃₂×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

—

—

—

C₈×Dic₆₁

—

C₈×Dic₆₁

—

C₈×Dic₆₁

C₈

yes

—

C₈×Dic₆₁

C₈

yes

—

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

M₃₂×C₆₁

—

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

M₁₆

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₄⋊C₄

yes

—

C₄×Dic₆₁

C₂×C₈

yes

characteristic

C₄×Dic₆₁

C₂×C₈

yes

characteristic

C₄×Dic₆₁

M₁₆

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#92418

C₂×C₈

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₄⋊C₈

yes

—

C₂×Dic₆₁

C₄⋊C₈

yes

—

C₈×C₆₁

—

C₈×C₆₁

#124

yes

—

C₈×C₆₁

#125

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₈

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#364

yes

—

#2150

—

C₄²×C₈

—

#1376

—

100

#1376

—

101

#1376

D₆₁

yes

characteristic

102

C₂×C₆₁

—

103

C₂×C₆₁

—

104

C₂×C₆₁

#1831

yes

characteristic

105

C₂×C₆₁

C₂×#364

yes

—

106

C₂×M₃₂

—

107

C₂×M₃₂

C₂×D₆₁

yes

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

114

C₄³

—

115

C₄⋊C₁₆

—

122

—

116

C₆₁

#12883

yes

characteristic, complemented, Sylow

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₆₁

yes

characteristic

120

C₂²×C₈

C₂²×D₆₁

yes

characteristic

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

122

—

124

C₂×C₄²

C₄×D₆₁

yes

characteristic

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

122

—

128

M₃₂

—

122

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

122

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

122

—

137

C₄×C₈

—

122

—

138

C₄×C₈

—

122

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

122

—

142

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

122

—

148

C₂×C₈

—

122

—

149

C₂×C₈

—

122

—

150

C₂×C₈

—

122

—

151

C₂×C₈

C₈×D₆₁

yes

—

152

C₂×C₈

#11758222

yes

—

153

C₂×C₈

D₄×D₆₁

yes

characteristic

154

C₂×C₈

#11758224

yes

—

155

C₂×C₈

Q₈×D₆₁

yes

characteristic

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

122

—

162

C₄²

—

122

—

163

C₄²

#11758189

yes

—

164

C₁₆

—

165

C₁₆

—

244

—

166

C₂³

M₁₆×D₆₁

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

122

—

172

C₂×C₄

—

122

—

173

C₂×C₄

—

122

—

174

C₂×C₄

C₄⋊C₄×D₆₁

yes

characteristic

175

C₂×C₄

#14519733

yes

characteristic

176

C₂×C₄

#14519734

yes

characteristic

177

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

178

C₂×C₄

M₁₆×D₆₁

yes

characteristic

179

C₂×C₄

C₂×#11758189

yes

characteristic

180

C₈

—

181

C₈

—

122

—

182

C₈

—

122

—

183

C₈

#14519763

yes

—

184

C₈

#14519767

yes

—

185

C₂²

—

186

C₂²

C₄⋊C₈×D₆₁

yes

characteristic

187

C₂²

#16839816

yes

characteristic

188

C₂²

#16839817

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

122

—

192

C₄

—

122

—

193

C₄

#16839626

yes

characteristic

194

C₄

#16839784

yes

characteristic

195

C₄

#16839812

yes

characteristic

196

C₄

#16839813

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#21445934

yes

characteristic

200

C₂

#364×D₆₁

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1376	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1376	D₆₁
hypocenter	C₆₁	#12883
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#1376

C₂

socle

C₂²×C₆₁

C₂×C₄⋊C₈

prime radical

C₂³×C₆₁

C₂×M₁₆

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
116. C₆₁	#12883	semidirect

H ≅ G/N

type

116. C₆₁

#12883

semidirect

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

subgroup	index
#21447574	2
C₄×C₈×Dic₆₁	2
#21443901	2
#21443900	2
C₆₁×#1376	2
#12883	61

subgroup

index

C₄×C₈×Dic₆₁

C₆₁×#1376

Subgroups — #29596372

Conjugacy classes of subgroups

Complements

Maximal subgroups