Group #29580163 — subgroups

Subgroups — #29580163

Order 15616.

438 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 159 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	#21454182	C₂	1	1	yes	maximal, characteristic
2	#21445672	C₂	1	1	yes	maximal, characteristic
3	#21443703	C₂	1	4	yes	maximal
4	C₆₁×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519769	—	2	1	no	—
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×#14519688	—	2	1	no	—
8	#16839660	—	2	2	no	—
9	#16839660	C₄	1	4	yes	—
10	#16839632	C₂²	1	2	yes	—
11	#16839628	—	2	1	no	—
12	#16839517	—	2	1	no	—
13	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	—	2	2	no	—
17	#14519769	—	4	2	no	—
18	#14519769	—	4	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
22	C₂×#11758196	—	2	4	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	Q₈	1	4	yes	—
25	#14519688	—	4	2	no	—
26	#14519684	—	2	4	no	—
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	D₄	1	2	yes	—
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₆₁	—	2	4	no	—
34	C₄⋊C₈×C₆₁	—	4	4	no	—
35	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	—	2	4	no	—
41	C₂×C₈×C₆₁	—	4	2	no	—
42	C₂×C₈×C₆₁	—	4	4	no	—
43	C₄²×C₆₁	—	2	2	no	—
44	C₄²×C₆₁	—	2	2	no	—
45	C₄²×C₆₁	—	2	2	no	—
46	C₄²×C₆₁	—	4	2	no	—
47	C₄²×C₆₁	D₈	1	2	yes	—
48	C₄²×C₆₁	SD₁₆	1	2	yes	—
49	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
50	C₄²×C₆₁	#41	1	2	yes	—
51	C₄⋊Dic₆₁	—	4	2	no	—
52	C₄⋊Dic₆₁	—	4	2	no	—
53	C₄⋊Dic₆₁	—	4	2	no	—
54	C₄⋊Dic₆₁	—	4	2	no	—
55	C₄⋊Dic₆₁	—	4	4	no	—
56	#11758196	—	4	8	no	—
57	C₂×#92418	—	2	4	no	—
58	C₂×#92418	—	4	2	no	—
59	C₂×#92418	—	4	2	no	—
60	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	#102	1	4	yes	—
68	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₆₁	#137	1	2	yes	—
76	C₂×C₄×C₆₁	#137	1	2	yes	—
77	C₂×Dic₆₁	—	4	4	no	—
78	C₂×Dic₆₁	—	4	4	no	—
79	C₂×Dic₆₁	—	4	4	no	—
80	C₈×C₆₁	—	8	2	no	—
81	#92418	—	4	4	no	—
82	#9904	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	#391	1	1	yes	characteristic
84	C₂²×C₆₁	#413	1	2	yes	—
85	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
86	C₂²×C₆₁	#417	1	2	yes	—
87	C₂²×C₆₁	#419	1	1	yes	characteristic
88	C₄×C₆₁	—	2	4	no	—
89	C₄×C₆₁	—	2	2	no	—
90	C₄×C₆₁	—	4	2	no	—
91	C₄×C₆₁	#474	1	2	yes	—
92	C₄×C₆₁	#475	1	2	yes	—
93	C₄×C₆₁	#490	1	1	yes	characteristic
94	C₄×C₆₁	#491	1	1	yes	characteristic
95	C₄×C₆₁	#501	1	2	yes	—
96	Dic₆₁	—	8	4	no	—
97	#2859	—	61	1	no	—
98	#1842	D₆₁	1	1	yes	characteristic
99	#1767	—	61	1	no	—
100	#1296	—	61	4	no	—
101	C₂×C₆₁	#1547	1	2	yes	—
102	C₂×C₆₁	#1559	1	2	yes	—
103	C₂×C₆₁	#1988	1	1	yes	characteristic
104	C₂×C₆₁	#1992	1	1	yes	characteristic
105	C₂×C₆₁	#1998	1	1	yes	characteristic
106	C₂×#125	—	122	1	no	—
107	C₂×C₄⋊C₈	—	2	2	no	—
108	C₂×C₄⋊C₈	—	61	2	no	—
109	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
110	C₂×#97	—	122	1	no	—
111	#389	—	61	4	no	—
112	#389	—	122	2	no	—
113	#384	—	61	2	no	—
114	#382	—	122	1	no	—
115	C₄³	C₂×D₆₁	1	1	yes	characteristic
116	#336	—	122	1	no	—
117	C₆₁	#9904	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	122	2	no	—
120	#125	—	244	2	no	—
121	#125	—	244	2	no	—
122	C₂×C₄⋊C₄	—	61	4	no	—
123	C₂×C₄⋊C₄	—	122	4	no	—
124	C₂×C₄⋊C₄	—	122	2	no	—
125	C₂×C₄⋊C₄	—	122	2	no	—
126	C₂×C₄²	C₄×D₆₁	1	2	yes	—
127	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
128	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
130	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	—	4	4	no	—
133	C₄⋊C₈	—	61	4	no	—
134	C₄⋊C₈	—	122	2	no	—
135	#97	—	244	2	no	—
136	#95	—	122	4	no	—
137	C₂²×C₄	—	122	4	no	—
138	C₂²×C₄	#11758198	1	2	yes	—
139	C₂²×C₄	#11758198	1	2	yes	—
140	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	4	no	—
145	C₂×C₈	—	122	4	no	—
146	C₂×C₈	—	244	2	no	—
147	C₂×C₈	—	244	2	no	—
148	C₄⋊C₄	—	244	4	no	—
149	C₄⋊C₄	—	244	2	no	—
150	C₄⋊C₄	—	244	8	no	—
151	C₄⋊C₄	—	244	2	no	—
152	C₄⋊C₄	—	244	2	no	—
153	C₄⋊C₄	—	244	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	#11758190	1	2	yes	—
159	C₄²	D₄₈₈	1	2	yes	—
160	C₄²	#11758224	1	2	yes	—
161	C₄²	Q₈×D₆₁	1	2	yes	—
162	C₂³	C₂×#11758198	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	244	4	no	—
170	C₂×C₄	—	244	4	no	—
171	C₂×C₄	—	244	4	no	—
172	C₂×C₄	#14519699	1	4	yes	—
173	C₂×C₄	#14519742	1	2	yes	—
174	C₂×C₄	#14519757	1	2	yes	—
175	C₂×C₄	#14519764	1	1	yes	characteristic
176	C₂×C₄	#14519775	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
178	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
179	C₂×C₄	#14519803	1	2	yes	—
180	C₂×C₄	#14519808	1	2	yes	—
181	C₂×C₄	#14519810	1	2	yes	—
182	C₈	—	8	2	no	—
183	C₈	—	244	4	no	—
184	C₂²	#16839667	1	1	yes	characteristic
185	C₂²	#16839744	1	2	yes	—
186	C₂²	C₂×#14519699	1	1	yes	characteristic
187	C₂²	#16839761	1	2	yes	—
188	C₂²	#16839772	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	488	4	no	—
193	C₄	#16840043	1	2	yes	—
194	C₄	#16840048	1	2	yes	—
195	C₄	#16840126	1	1	yes	characteristic
196	C₄	#16840130	1	1	yes	characteristic
197	C₄	#16840168	1	2	yes	—
198	C₂	#21444494	1	2	yes	—
199	C₂	#21444547	1	2	yes	—
200	C₂	#21446770	1	1	yes	characteristic
201	C₂	#21446787	1	1	yes	characteristic
202	C₂	#21446811	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21454182

C₂

yes

maximal, characteristic

#21445672

C₂

yes

maximal, characteristic

#21443703

C₂

yes

maximal

C₆₁×#1842

C₂

yes

maximal, characteristic

C₂×#14519769

—

C₂×C₄⋊#92418

C₂²

yes

—

C₂×#14519688

—

#16839660

—

#16839660

C₄

yes

—

#16839632

C₂²

yes

—

#16839628

—

#16839517

—

C₂×C₄⋊C₈×C₆₁

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

#14519769

—

#14519769

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×#11758196

—

C₄⋊#92418

—

C₄⋊#92418

Q₈

yes

—

#14519688

—

#14519684

—

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₈

yes

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂×Q₈

yes

—

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

—

—

—

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#102

yes

—

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#102

yes

characteristic

C₂²×C₆₁

#417

yes

—

C₂²×C₆₁

#419

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#474

yes

—

C₄×C₆₁

#475

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

#2859

—

#1842

D₆₁

yes

characteristic

#1767

—

100

#1296

—

101

C₂×C₆₁

#1547

yes

—

102

C₂×C₆₁

#1559

yes

—

103

C₂×C₆₁

#1988

yes

characteristic

104

C₂×C₆₁

#1992

yes

characteristic

105

C₂×C₆₁

#1998

yes

characteristic

106

C₂×#125

—

122

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

110

C₂×#97

—

122

—

111

#389

—

112

#389

—

122

—

113

#384

—

114

#382

—

122

—

115

C₄³

C₂×D₆₁

yes

characteristic

116

#336

—

122

—

117

C₆₁

#9904

yes

characteristic, complemented, Sylow

118

C₂²×C₈

—

119

C₂²×C₈

—

122

—

120

#125

—

244

—

121

#125

—

244

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

122

—

124

C₂×C₄⋊C₄

—

122

—

125

C₂×C₄⋊C₄

—

122

—

126

C₂×C₄²

C₄×D₆₁

yes

—

127

C₂×C₄²

D₂₄₄

yes

characteristic

128

C₂×C₄²

D₂₄₄

yes

characteristic

129

C₂×C₄²

C₂²⋊D₆₁

yes

—

130

C₂×C₄²

C₂²×D₆₁

yes

characteristic

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

122

—

135

#97

—

244

—

136

#95

—

122

—

137

C₂²×C₄

—

122

—

138

C₂²×C₄

#11758198

yes

—

139

C₂²×C₄

#11758198

yes

—

140

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

141

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

142

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

122

—

146

C₂×C₈

—

244

—

147

C₂×C₈

—

244

—

148

C₄⋊C₄

—

244

—

149

C₄⋊C₄

—

244

—

150

C₄⋊C₄

—

244

—

151

C₄⋊C₄

—

244

—

152

C₄⋊C₄

—

244

—

153

C₄⋊C₄

—

244

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#11758190

yes

—

159

C₄²

D₄₈₈

yes

—

160

C₄²

#11758224

yes

—

161

C₄²

Q₈×D₆₁

yes

—

162

C₂³

C₂×#11758198

yes

characteristic

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

244

—

170

C₂×C₄

—

244

—

171

C₂×C₄

—

244

—

172

C₂×C₄

#14519699

yes

—

173

C₂×C₄

#14519742

yes

—

174

C₂×C₄

#14519757

yes

—

175

C₂×C₄

#14519764

yes

characteristic

176

C₂×C₄

#14519775

yes

characteristic

177

C₂×C₄

C₂×D₄₈₈

yes

characteristic

178

C₂×C₄

C₂×#11758190

yes

characteristic

179

C₂×C₄

#14519803

yes

—

180

C₂×C₄

#14519808

yes

—

181

C₂×C₄

#14519810

yes

—

182

C₈

—

183

C₈

—

244

—

184

C₂²

#16839667

yes

characteristic

185

C₂²

#16839744

yes

—

186

C₂²

C₂×#14519699

yes

characteristic

187

C₂²

#16839761

yes

—

188

C₂²

#16839772

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

488

—

193

C₄

#16840043

yes

—

194

C₄

#16840048

yes

—

195

C₄

#16840126

yes

characteristic

196

C₄

#16840130

yes

characteristic

197

C₄

#16840168

yes

—

198

C₂

#21444494

yes

—

199

C₂

#21444547

yes

—

200

C₂

#21446770

yes

characteristic

201

C₂

#21446787

yes

characteristic

202

C₂

#21446811

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11758198
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1842	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1842	D₆₁
hypocenter	C₆₁	#9904
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₂×#11758198

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1842

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
117. C₆₁	#9904	semidirect

H ≅ G/N

type

117. C₆₁

#9904

semidirect

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

subgroup	index
#21454182	2
#21445672	2
#21443703	2
C₆₁×#1842	2
#9904	61

subgroup

index

C₆₁×#1842

Subgroups — #29580163

Conjugacy classes of subgroups

Complements

Maximal subgroups