Group #29585062 — subgroups

Subgroups — #29585062

Order 15616.

526 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 155 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	#21446626	C₂	1	2	yes	maximal
2	#21445979	C₂	1	1	yes	maximal, characteristic, complemented
3	#21445978	C₂	1	2	yes	maximal, complemented
4	#21443646	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1957	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519711	—	2	2	no	—
7	C₂×#14519710	—	2	4	no	—
8	C₂×#14519698	—	2	4	no	—
9	C₂×C₄×#92418	C₂²	1	2	yes	—
10	#16839632	C₂²	1	2	yes	—
11	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×#102×C₆₁	—	2	2	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#384	C₂²	1	1	yes	characteristic
15	C₂²×#92418	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₆₁	—	2	2	no	—
17	C₂×C₄⋊Dic₆₁	—	2	2	no	—
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	#14519711	—	4	4	no	—
20	#14519710	—	4	8	no	—
21	#14519698	—	4	8	no	—
22	C₄×#92418	—	2	2	no	—
23	C₄×#92418	—	2	2	no	—
24	C₂²×D₄×C₆₁	—	2	2	no	—
25	C₂²×D₄×C₆₁	—	2	1	no	—
26	C₂²×C₈×C₆₁	D₄	1	2	yes	—
27	#127×C₆₁	—	2	1	no	—
28	#127×C₆₁	—	2	2	no	—
29	#127×C₆₁	Q₈	1	2	yes	—
30	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#102×C₆₁	—	4	4	no	—
34	C₄×C₈×C₆₁	—	2	2	no	—
35	C₂⁴×C₆₁	—	4	1	no	—
36	C₂×D₄×C₆₁	—	2	4	no	—
37	C₂×D₄×C₆₁	—	2	2	no	—
38	C₂×D₄×C₆₁	—	4	1	no	—
39	C₂×D₄×C₆₁	—	4	2	no	—
40	C₂×D₄×C₆₁	—	4	2	no	—
41	C₂×D₄×C₆₁	—	4	4	no	—
42	C₂×D₄×C₆₁	—	4	4	no	—
43	C₂×D₄×C₆₁	—	4	2	no	—
44	C₂²×C₄×C₆₁	—	4	1	no	—
45	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₆₁	—	4	2	no	—
48	C₂×C₈×C₆₁	—	2	2	no	—
49	C₂×C₈×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	D₈	1	4	yes	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄⋊C₄×C₆₁	—	4	4	no	—
53	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₆₁	#41	1	1	yes	characteristic
55	C₄²×C₆₁	#41	1	2	yes	—
56	C₄⋊Dic₆₁	—	4	4	no	—
57	C₄⋊Dic₆₁	—	4	4	no	—
58	C₄⋊Dic₆₁	—	4	4	no	—
59	C₂×#92418	—	2	4	no	—
60	C₂×#92418	—	2	4	no	—
61	C₂×#92418	D₈	1	8	yes	—
62	C₂³×C₆₁	—	4	2	no	—
63	C₂³×C₆₁	—	8	2	no	—
64	C₂³×C₆₁	—	8	1	no	—
65	C₂³×C₆₁	—	8	2	no	—
66	C₂³×C₆₁	—	8	1	no	—
67	C₂³×C₆₁	#127	1	1	yes	characteristic
68	D₄×C₆₁	—	4	8	no	—
69	D₄×C₆₁	—	4	4	no	—
70	D₄×C₆₁	—	8	2	no	—
71	D₄×C₆₁	—	8	4	no	—
72	C₂×C₄×C₆₁	—	8	2	no	—
73	C₂×C₄×C₆₁	—	8	1	no	—
74	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#123	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
79	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₆₁	#137	1	4	yes	—
81	C₂×C₄×C₆₁	#137	1	2	yes	—
82	C₂×Dic₆₁	—	8	2	no	—
83	C₂×Dic₆₁	—	8	2	no	—
84	C₂×Dic₆₁	—	8	2	no	—
85	C₈×C₆₁	—	2	4	no	—
86	#92418	—	2	8	no	—
87	#10673	—	61	1	no	maximal, Sylow
88	C₂²×C₆₁	—	8	4	no	—
89	C₂²×C₆₁	—	8	2	no	—
90	C₂²×C₆₁	—	16	1	no	—
91	C₂²×C₆₁	—	16	1	no	—
92	C₂²×C₆₁	—	16	2	no	—
93	C₂²×C₆₁	#399	1	1	yes	characteristic
94	C₂²×C₆₁	#493	1	2	yes	—
95	C₂²×C₆₁	#493	1	1	yes	characteristic
96	C₂²×C₆₁	#497	1	2	yes	—
97	C₂²×C₆₁	#497	1	1	yes	characteristic
98	C₄×C₆₁	—	16	1	no	—
99	C₄×C₆₁	#474	1	4	yes	—
100	C₄×C₆₁	#474	1	2	yes	—
101	C₄×C₆₁	#480	1	2	yes	—
102	C₄×C₆₁	#480	1	4	yes	—
103	Dic₆₁	—	16	2	no	—
104	#1957	—	61	2	no	—
105	#1957	D₆₁	1	1	yes	characteristic
106	#1845	—	61	1	no	—
107	#1845	—	61	2	no	—
108	#1273	—	61	1	no	—
109	C₂×C₆₁	—	16	2	no	—
110	C₂×C₆₁	#1712	1	1	yes	characteristic
111	C₂×C₆₁	#1722	1	1	yes	characteristic
112	C₂×C₆₁	#1722	1	2	yes	—
113	C₂×C₆₁	#2093	1	2	yes	—
114	C₂×C₆₁	#2093	1	1	yes	characteristic
115	C₂×#127	C₂×D₆₁	1	1	yes	characteristic
116	C₂×#107	—	122	4	no	—
117	C₂×#107	—	122	2	no	—
118	C₂×#102	—	2	2	no	—
119	C₂×#102	—	122	4	no	—
120	C₂×C₄×C₈	—	61	2	no	—
121	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
122	#384	—	61	2	no	—
123	#384	C₂×D₆₁	1	1	yes	characteristic
124	C₆₁	#10673	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₄	—	2	2	no	—
126	C₂²×D₄	—	2	1	no	—
127	C₂²×C₈	—	61	4	no	—
128	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
129	#127	—	2	1	no	—
130	#127	—	2	2	no	—
131	#127	Dic₁₂₂	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	122	2	no	—
135	C₂×C₄⋊C₄	—	122	2	no	—
136	C₂×C₄⋊C₄	—	122	2	no	—
137	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
138	#107	—	244	8	no	—
139	#107	—	244	4	no	—
140	#102	—	4	4	no	—
141	#102	—	244	8	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	122	2	no	—
144	C₄×C₈	—	122	2	no	—
145	C₂⁴	—	4	1	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	4	2	no	—
152	C₂×D₄	—	4	4	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	—	244	2	no	—
156	C₂²×C₄	D₄×D₆₁	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	61	8	no	—
161	C₂×C₈	—	122	4	no	—
162	C₂×C₈	—	122	4	no	—
163	C₂×C₈	#11758199	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	244	4	no	—
167	C₄⋊C₄	—	244	4	no	—
168	C₄⋊C₄	—	244	4	no	—
169	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
170	C₄²	#11758222	1	1	yes	characteristic
171	C₄²	#11758224	1	2	yes	—
172	C₂³	—	4	2	no	—
173	C₂³	—	8	2	no	—
174	C₂³	—	8	1	no	—
175	C₂³	—	8	2	no	—
176	C₂³	—	8	1	no	—
177	C₂³	#14519773	1	1	yes	characteristic
178	D₄	—	4	8	no	—
179	D₄	—	4	4	no	—
180	D₄	—	8	2	no	—
181	D₄	—	8	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	—	488	2	no	—
185	C₂×C₄	—	488	2	no	—
186	C₂×C₄	—	488	2	no	—
187	C₂×C₄	#14519754	1	2	yes	—
188	C₂×C₄	#14519756	1	1	yes	characteristic
189	C₂×C₄	#14519758	1	1	yes	characteristic
190	C₂×C₄	#14519759	1	2	yes	—
191	C₂×C₄	D₈×D₆₁	1	4	yes	—
192	C₂×C₄	C₂×#11758199	1	2	yes	—
193	C₂×C₄	#14519806	1	4	yes	—
194	C₂×C₄	#14519807	1	2	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	122	8	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	2	no	—
199	C₂²	—	16	1	no	—
200	C₂²	—	16	1	no	—
201	C₂²	—	16	2	no	—
202	C₂²	#16839698	1	1	yes	characteristic
203	C₂²	#16840137	1	2	yes	—
204	C₂²	#16840138	1	1	yes	characteristic
205	C₂²	#16840153	1	2	yes	—
206	C₂²	#16840157	1	1	yes	characteristic
207	C₄	—	16	1	no	—
208	C₄	—	976	2	no	—
209	C₄	#16840040	1	4	yes	—
210	C₄	#16840042	1	2	yes	—
211	C₄	#16840073	1	2	yes	—
212	C₄	#16840074	1	4	yes	—
213	C₂	—	16	2	no	—
214	C₂	#21445433	1	1	yes	characteristic
215	C₂	#21445488	1	1	yes	characteristic
216	C₂	#21445492	1	2	yes	—
217	C₂	#21447345	1	2	yes	—
218	C₂	#21447347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21446626

C₂

yes

maximal

#21445979

C₂

yes

maximal, characteristic, complemented

#21445978

C₂

yes

maximal, complemented

#21443646

C₂

yes

maximal, characteristic, complemented

C₆₁×#1957

C₂

yes

maximal, characteristic

C₂×#14519711

—

C₂×#14519710

—

C₂×#14519698

—

C₂×C₄×#92418

C₂²

yes

—

#16839632

C₂²

yes

—

C₂×#127×C₆₁

C₂²

yes

characteristic

C₂×#102×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#384

C₂²

yes

characteristic

C₂²×#92418

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

—

C₂²×D₄×C₆₁

—

C₂²×D₄×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

#127×C₆₁

—

#127×C₆₁

—

#127×C₆₁

Q₈

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#102×C₆₁

—

C₄×C₈×C₆₁

—

C₂⁴×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×Q₈

yes

C₄²×C₆₁

yes

C₄²×C₆₁

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

D₈

yes

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

#127

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

#92418

—

#10673

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#399

yes

characteristic

C₂²×C₆₁

#493

yes

—

C₂²×C₆₁

#493

yes

characteristic

C₂²×C₆₁

#497

yes

—

C₂²×C₆₁

#497

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#474

yes

—

100

C₄×C₆₁

#474

yes

—

101

C₄×C₆₁

#480

yes

—

102

C₄×C₆₁

#480

yes

—

103

Dic₆₁

—

104

#1957

—

105

#1957

D₆₁

yes

characteristic

106

#1845

—

107

#1845

—

108

#1273

—

109

C₂×C₆₁

—

110

C₂×C₆₁

#1712

yes

characteristic

111

C₂×C₆₁

#1722

yes

characteristic

112

C₂×C₆₁

#1722

yes

—

113

C₂×C₆₁

#2093

yes

—

114

C₂×C₆₁

#2093

yes

characteristic

115

C₂×#127

C₂×D₆₁

yes

characteristic

116

C₂×#107

—

122

—

117

C₂×#107

—

122

—

118

C₂×#102

—

119

C₂×#102

—

122

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

122

#384

—

123

#384

C₂×D₆₁

yes

characteristic

124

C₆₁

#10673

yes

characteristic, complemented, Sylow

125

C₂²×D₄

—

126

C₂²×D₄

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₆₁

yes

—

129

#127

—

130

#127

—

131

#127

Dic₁₂₂

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

122

—

135

C₂×C₄⋊C₄

—

122

—

136

C₂×C₄⋊C₄

—

122

—

137

C₂×C₄²

C₂²×D₆₁

yes

characteristic

138

#107

—

244

—

139

#107

—

244

—

140

#102

—

141

#102

—

244

—

142

C₄×C₈

—

143

C₄×C₈

—

122

—

144

C₄×C₈

—

122

—

145

C₂⁴

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

244

—

156

C₂²×C₄

D₄×D₆₁

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

122

—

162

C₂×C₈

—

122

—

163

C₂×C₈

#11758199

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

244

—

167

C₄⋊C₄

—

244

—

168

C₄⋊C₄

—

244

—

169

C₄²

C₂×Dic₁₂₂

yes

characteristic

170

C₄²

#11758222

yes

characteristic

171

C₄²

#11758224

yes

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

177

C₂³

#14519773

yes

characteristic

178

D₄

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

488

—

185

C₂×C₄

—

488

—

186

C₂×C₄

—

488

—

187

C₂×C₄

#14519754

yes

—

188

C₂×C₄

#14519756

yes

characteristic

189

C₂×C₄

#14519758

yes

characteristic

190

C₂×C₄

#14519759

yes

—

191

C₂×C₄

D₈×D₆₁

yes

—

192

C₂×C₄

C₂×#11758199

yes

—

193

C₂×C₄

#14519806

yes

—

194

C₂×C₄

#14519807

yes

—

195

C₈

—

196

C₈

—

122

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

#16839698

yes

characteristic

203

C₂²

#16840137

yes

—

204

C₂²

#16840138

yes

characteristic

205

C₂²

#16840153

yes

—

206

C₂²

#16840157

yes

characteristic

207

C₄

—

208

C₄

—

976

—

209

C₄

#16840040

yes

—

210

C₄

#16840042

yes

—

211

C₄

#16840073

yes

—

212

C₄

#16840074

yes

—

213

C₂

—

214

C₂

#21445433

yes

characteristic

215

C₂

#21445488

yes

characteristic

216

C₂

#21445492

yes

—

217

C₂

#21447345

yes

—

218

C₂

#21447347

yes

characteristic

	subgroup	G/H
center	C₂³	#14519773
commutator, cosocle	C₂×C₄²×C₆₁	C₂³
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1957	C₂
socle	C₂³×C₆₁	#127
prime radical	C₂×#127×C₆₁	C₂²
hypercenter	#1957	D₆₁
hypocenter	C₆₁	#10673
LCS	G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

subgroup

G/H

center

C₂³

#14519773

commutator, cosocle

C₂×C₄²×C₆₁

C₂³

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1957

C₂

socle

C₂³×C₆₁

#127

prime radical

C₂×#127×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂×C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

N	H ≅ G/N	type
2. #21445979	C₂	semidirect
3. #21445978	C₂	semidirect
4. #21443646	C₂	semidirect
124. C₆₁	#10673	semidirect

H ≅ G/N

type

2. #21445979

C₂

semidirect

3. #21445978

C₂

semidirect

4. #21443646

C₂

semidirect

124. C₆₁

#10673

semidirect

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

subgroup	index
#21446626	2
#21445979	2
#21445978	2
#21443646	2
C₆₁×#1957	2
#10673	61

subgroup

index

C₆₁×#1957

Subgroups — #29585062

Conjugacy classes of subgroups

Complements

Maximal subgroups