Group #29599617 — subgroups

Subgroups — #29599617

Order 15616.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16839580	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839566	C₂	1	2	yes	maximal
3	#21446681	C₂	1	2	yes	maximal
4	#21443911	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758185	C₂²	1	1	yes	characteristic
7	C₂×#14519711	C₂²	1	2	yes	—
8	C₂×#14519701	—	2	2	no	—
9	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
10	#16839669	—	2	2	no	—
11	#16839580	—	2	2	no	—
12	#16839566	—	2	4	no	—
13	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₆₁	—	2	4	no	—
15	C₂×Q₃₂×C₆₁	—	2	2	no	—
16	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×#92418	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
20	C₂×#11758185	—	2	2	no	—
21	C₂×#11758185	—	2	1	no	—
22	#14519711	—	2	2	no	—
23	#14519711	—	2	2	no	—
24	#14519701	—	4	4	no	—
25	C₈×Dic₆₁	—	2	1	no	—
26	C₈×Dic₆₁	—	2	1	no	—
27	#14519684	—	4	2	no	—
28	#14519684	—	4	2	no	—
29	C₂²×Q₈×C₆₁	—	2	2	no	—
30	C₂×Q₁₆×C₆₁	—	2	2	no	—
31	C₂×Q₁₆×C₆₁	—	2	2	no	—
32	C₂×Q₁₆×C₆₁	—	2	2	no	—
33	C₂×Q₁₆×C₆₁	—	2	2	no	—
34	C₂×Q₁₆×C₆₁	—	2	2	no	—
35	C₂×Q₁₆×C₆₁	D₄	1	4	yes	—
36	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	Q₃₂×C₆₁	—	2	8	no	—
38	C₂×C₁₆×C₆₁	—	2	2	no	—
39	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
40	C₂×Q₈×C₆₁	—	2	4	no	—
41	C₂×Q₈×C₆₁	—	4	2	no	—
42	C₂×Q₈×C₆₁	—	4	2	no	—
43	C₂×Q₈×C₆₁	—	4	2	no	—
44	C₂×Q₈×C₆₁	—	4	2	no	—
45	C₂×Q₈×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	—	4	2	no	—
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₆₁	—	4	2	no	—
49	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
50	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
51	Q₁₆×C₆₁	—	2	4	no	—
52	Q₁₆×C₆₁	—	2	4	no	—
53	Q₁₆×C₆₁	—	4	2	no	—
54	Q₁₆×C₆₁	—	4	2	no	—
55	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
58	C₂×C₈×C₆₁	#41	1	2	yes	—
59	C₄⋊Dic₆₁	—	4	2	no	—
60	C₄⋊Dic₆₁	—	4	2	no	—
61	C₄×Dic₆₁	—	2	1	no	—
62	C₄×Dic₆₁	—	2	1	no	—
63	C₂×#92418	—	2	1	no	—
64	C₂×#92418	—	2	2	no	—
65	C₁₆×C₆₁	—	2	2	no	—
66	#11758185	—	4	1	no	—
67	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₆₁	—	4	4	no	—
69	Q₈×C₆₁	—	4	4	no	—
70	Q₈×C₆₁	—	8	2	no	—
71	Q₈×C₆₁	—	8	2	no	—
72	C₂×C₄×C₆₁	—	4	4	no	—
73	C₂×C₄×C₆₁	—	8	2	no	—
74	C₂×C₄×C₆₁	—	8	2	no	—
75	C₂×C₄×C₆₁	#121	1	2	yes	—
76	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
78	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×Dic₆₁	—	2	1	no	—
80	C₂×Dic₆₁	—	2	1	no	—
81	C₂×Dic₆₁	—	2	1	no	—
82	C₂×Dic₆₁	—	2	2	no	—
83	C₂×Dic₆₁	—	8	2	no	—
84	C₂×Dic₆₁	—	8	2	no	—
85	C₂×Dic₆₁	—	8	2	no	—
86	C₂×Dic₆₁	Q₃₂	1	2	yes	complemented
87	C₈×C₆₁	#120	1	1	yes	characteristic
88	C₈×C₆₁	#123	1	2	yes	—
89	C₈×C₆₁	#124	1	1	yes	characteristic
90	#92418	—	4	1	no	—
91	#13450	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	#459	1	2	yes	—
93	C₂²×C₆₁	#466	1	1	yes	characteristic
94	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
95	C₂²×C₆₁	#508	1	1	yes	characteristic
96	C₂²×C₆₁	#510	1	2	yes	—
97	C₄×C₆₁	—	8	4	no	—
98	C₄×C₆₁	#394	1	1	yes	characteristic
99	C₄×C₆₁	#463	1	2	yes	—
100	C₄×C₆₁	#469	1	1	yes	characteristic
101	Dic₆₁	—	2	2	no	—
102	Dic₆₁	—	4	1	no	—
103	Dic₆₁	—	16	2	no	—
104	C₂²×Q₃₂	D₆₁	1	1	yes	characteristic
105	C₂×C₄⋊C₁₆	—	61	1	no	—
106	C₂×#358	—	61	2	no	—
107	#1969	—	61	2	no	—
108	#1380	—	61	1	no	—
109	C₂×C₆₁	#2054	1	1	yes	characteristic
110	C₂×C₆₁	#2209	1	2	yes	—
111	C₂×C₆₁	#2211	1	2	yes	—
112	C₂×C₆₁	#2216	1	1	yes	characteristic
113	C₂×C₆₁	#2219	1	1	yes	characteristic
114	C₂²×Q₁₆	C₂×D₆₁	1	2	yes	—
115	C₂×Q₃₂	—	2	4	no	—
116	C₂×Q₃₂	—	2	2	no	—
117	C₂²×C₁₆	—	61	1	no	—
118	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
119	C₂×#107	—	61	2	no	—
120	C₂×#103	—	122	2	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	#391	—	122	2	no	—
123	C₄⋊C₁₆	—	122	2	no	—
124	#358	—	122	4	no	—
125	C₆₁	#13450	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	2	2	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	C₂²⋊D₆₁	1	4	yes	—
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	122	2	no	—
136	C₂×C₄²	—	61	1	no	—
137	Q₃₂	—	2	8	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	122	2	no	—
140	C₂×C₁₆	—	122	1	no	—
141	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
142	#107	—	122	2	no	—
143	#107	—	122	2	no	—
144	#103	—	244	4	no	—
145	C₄×C₈	—	122	1	no	—
146	C₄×C₈	—	122	1	no	—
147	#95	—	244	2	no	—
148	#95	—	244	2	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	61	1	no	—
157	C₂²×C₄	—	61	1	no	—
158	C₂²×C₄	—	244	2	no	—
159	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
160	Q₁₆	—	2	4	no	—
161	Q₁₆	—	2	4	no	—
162	Q₁₆	—	4	2	no	—
163	Q₁₆	—	4	2	no	—
164	C₂×C₈	—	122	1	no	—
165	C₂×C₈	—	122	2	no	—
166	C₂×C₈	#11758224	1	1	yes	characteristic
167	C₂×C₈	#11758224	1	2	yes	—
168	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
169	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
170	C₄⋊C₄	—	244	2	no	—
171	C₄⋊C₄	—	244	2	no	—
172	C₄²	—	122	1	no	—
173	C₄²	—	122	1	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	244	1	no	—
176	C₂³	D₈×D₆₁	1	1	yes	characteristic
177	Q₈	—	4	4	no	—
178	Q₈	—	4	4	no	—
179	Q₈	—	8	2	no	—
180	Q₈	—	8	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	61	2	no	—
185	C₂×C₄	—	122	1	no	—
186	C₂×C₄	—	122	1	no	—
187	C₂×C₄	—	122	2	no	—
188	C₂×C₄	—	122	1	no	—
189	C₂×C₄	—	488	2	no	—
190	C₂×C₄	—	488	2	no	—
191	C₂×C₄	—	488	2	no	—
192	C₂×C₄	#14519748	1	1	yes	characteristic
193	C₂×C₄	#14519749	1	2	yes	—
194	C₂×C₄	#14519796	1	1	yes	characteristic
195	C₂×C₄	#14519801	1	2	yes	—
196	C₈	—	244	1	no	—
197	C₈	#14519746	1	1	yes	characteristic
198	C₈	#14519758	1	2	yes	—
199	C₈	#14519764	1	1	yes	characteristic
200	C₂²	#16839955	1	2	yes	—
201	C₂²	#16839995	1	1	yes	characteristic
202	C₂²	Q₃₂×D₆₁	1	1	yes	characteristic
203	C₂²	#16840195	1	1	yes	characteristic
204	C₂²	#16840204	1	2	yes	—
205	C₄	—	8	4	no	—
206	C₄	—	122	2	no	—
207	C₄	—	976	2	no	—
208	C₄	—	244	1	no	—
209	C₄	#16839676	1	1	yes	characteristic
210	C₄	#16839975	1	2	yes	—
211	C₄	#16840012	1	1	yes	characteristic
212	C₂	#21447127	1	1	yes	characteristic
213	C₂	#21447867	1	2	yes	—
214	C₂	#21447883	1	2	yes	—
215	C₂	#21447907	1	1	yes	characteristic
216	C₂	#21447924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839580

C₂

yes

maximal, characteristic

C₂×#16839566

C₂

yes

maximal

#21446681

C₂

yes

maximal

#21443911

C₂

yes

maximal, characteristic

C₂²×Q₃₂×C₆₁

C₂

yes

maximal, characteristic

C₂²×#11758185

C₂²

yes

characteristic

C₂×#14519711

C₂²

yes

—

C₂×#14519701

—

C₂×C₈×Dic₆₁

C₂²

yes

—

—

—

C₂²×Q₁₆×C₆₁

C₂²

yes

—

C₂×Q₃₂×C₆₁

—

C₂×Q₃₂×C₆₁

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂²×#92418

D₄

yes

characteristic

C₂×C₄⋊Dic₆₁

—

C₂×C₄×Dic₆₁

D₄

yes

—

—

—

—

—

C₈×Dic₆₁

—

C₈×Dic₆₁

—

#14519684

—

#14519684

—

C₂²×Q₈×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

D₈

yes

characteristic

C₂²×Dic₆₁

D₈

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

C₂×C₈×C₆₁

yes

C₂×C₈×C₆₁

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂×D₈

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

Q₃₂

yes

C₈×C₆₁

yes

C₈×C₆₁

yes

—

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

—

C₂²×C₆₁

#466

yes

characteristic

C₂²×C₆₁

C₂×Q₃₂

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

#394

yes

characteristic

C₄×C₆₁

#463

yes

—

100

C₄×C₆₁

#469

yes

characteristic

101

Dic₆₁

—

102

Dic₆₁

—

103

Dic₆₁

—

104

C₂²×Q₃₂

D₆₁

yes

characteristic

105

C₂×C₄⋊C₁₆

—

106

C₂×#358

—

107

#1969

—

108

#1380

—

109

C₂×C₆₁

#2054

yes

characteristic

110

C₂×C₆₁

#2209

yes

—

111

C₂×C₆₁

#2211

yes

—

112

C₂×C₆₁

#2216

yes

characteristic

113

C₂×C₆₁

#2219

yes

characteristic

114

C₂²×Q₁₆

C₂×D₆₁

yes

—

115

C₂×Q₃₂

—

116

C₂×Q₃₂

—

117

C₂²×C₁₆

—

118

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

119

C₂×#107

—

120

C₂×#103

—

122

—

121

C₂×C₄×C₈

—

122

#391

—

122

—

123

C₄⋊C₁₆

—

122

—

124

#358

—

122

—

125

C₆₁

#13450

yes

characteristic, complemented, Sylow

126

C₂²×Q₈

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

C₂²⋊D₆₁

yes

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂²×D₆₁

yes

characteristic

135

C₂×C₄⋊C₄

—

122

—

136

C₂×C₄²

—

137

Q₃₂

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

122

—

140

C₂×C₁₆

—

122

—

141

C₂×C₁₆

C₂²⋊D₆₁

yes

—

142

#107

—

122

—

143

#107

—

122

—

144

#103

—

244

—

145

C₄×C₈

—

122

—

146

C₄×C₈

—

122

—

147

#95

—

244

—

148

#95

—

244

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

244

—

159

C₂²×C₄

D₄×D₆₁

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

163

Q₁₆

—

164

C₂×C₈

—

122

—

165

C₂×C₈

—

122

—

166

C₂×C₈

#11758224

yes

characteristic

167

C₂×C₈

#11758224

yes

—

168

C₂×C₈

C₂×C₂²⋊D₆₁

yes

—

169

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

170

C₄⋊C₄

—

244

—

171

C₄⋊C₄

—

244

—

172

C₄²

—

122

—

173

C₄²

—

122

—

174

C₁₆

—

175

C₁₆

—

244

—

176

C₂³

D₈×D₆₁

yes

characteristic

177

Q₈

—

178

Q₈

—

179

Q₈

—

180

Q₈

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

122

—

186

C₂×C₄

—

122

—

187

C₂×C₄

—

122

—

188

C₂×C₄

—

122

—

189

C₂×C₄

—

488

—

190

C₂×C₄

—

488

—

191

C₂×C₄

—

488

—

192

C₂×C₄

#14519748

yes

characteristic

193

C₂×C₄

#14519749

yes

—

194

C₂×C₄

#14519796

yes

characteristic

195

C₂×C₄

#14519801

yes

—

196

C₈

—

244

—

197

C₈

#14519746

yes

characteristic

198

C₈

#14519758

yes

—

199

C₈

#14519764

yes

characteristic

200

C₂²

#16839955

yes

—

201

C₂²

#16839995

yes

characteristic

202

C₂²

Q₃₂×D₆₁

yes

characteristic

203

C₂²

#16840195

yes

characteristic

204

C₂²

#16840204

yes

—

205

C₄

—

206

C₄

—

122

—

207

C₄

—

976

—

208

C₄

—

244

—

209

C₄

#16839676

yes

characteristic

210

C₄

#16839975

yes

—

211

C₄

#16840012

yes

characteristic

212

C₂

#21447127

yes

characteristic

213

C₂

#21447867

yes

—

214

C₂

#21447883

yes

—

215

C₂

#21447907

yes

characteristic

216

C₂

#21447924

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₁
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂²×Q₃₂×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₂²×Q₃₂	D₆₁
hypocenter	C₆₁	#13450
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂³

D₈×D₆₁

commutator, cosocle

C₂²×C₈×C₆₁

C₂³

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₂²×Q₃₂×C₆₁

C₂

socle, prime radical

C₂³×C₆₁

C₂×D₈

hypercenter

C₂²×Q₃₂

D₆₁

hypocenter

C₆₁

#13450

LCS

G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
86. C₂×Dic₆₁	Q₃₂	semidirect
125. C₆₁	#13450	semidirect

H ≅ G/N

type

86. C₂×Dic₆₁

Q₃₂

semidirect

125. C₆₁

#13450

semidirect

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

subgroup	index
C₂×#16839580	2
C₂×#16839566	2
#21446681	2
#21443911	2
C₂²×Q₃₂×C₆₁	2
#13450	61

subgroup

index

C₂²×Q₃₂×C₆₁

Subgroups — #29599617

Conjugacy classes of subgroups

Complements

Maximal subgroups