Group #29573003 — subgroups

Subgroups — #29573003

Order 15616.

388 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 101 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	#21454531	C₂	1	1	yes	maximal, characteristic
2	#21444580	C₂	1	2	yes	maximal, complemented
3	#21444454	C₂	1	1	yes	maximal, characteristic
4	#21443729	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1536	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519784	C₄	1	2	yes	—
7	#16839840	—	2	2	no	—
8	D₄×#92418	—	2	2	no	—
9	#16839819	C₂²	1	2	yes	—
10	#16839786	C₂²	1	1	yes	characteristic
11	#16839729	C₄	1	2	yes	—
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839514	—	2	2	no	—
14	#16839512	—	2	2	no	—
15	#16839502	—	2	2	no	—
16	C₆₁×#532	C₂²	1	1	yes	characteristic
17	C₆₁×#432	C₂²	1	2	yes	—
18	C₆₁×#329	—	2	2	no	—
19	#14519784	—	4	4	no	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	C₂×C₄	1	2	yes	—
22	C₂²×#92418	—	2	2	no	—
23	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
24	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
25	C₄⋊#92418	—	2	2	no	—
26	C₄⋊#92418	—	2	2	no	—
27	C₂²⋊#92418	—	2	4	no	—
28	C₂²⋊#92418	—	2	2	no	—
29	#14519688	—	2	2	no	—
30	C₄×#92418	D₄	1	2	yes	—
31	C₄×#92418	D₄	1	2	yes	—
32	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₆₁	Q₈	1	1	yes	characteristic
34	#127×C₆₁	Q₈	1	1	yes	characteristic
35	#124×C₆₁	D₄	1	2	yes	—
36	#121×C₆₁	—	2	2	no	—
37	C₄×D₄×C₆₁	—	2	2	no	—
38	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₁	—	2	2	no	—
40	C₂²⋊C₈×C₆₁	—	2	2	no	—
41	#97×C₆₁	—	2	2	no	—
42	#41×C₆₁	—	4	2	no	—
43	#41×C₆₁	—	4	2	no	—
44	C₂×Q₈×C₆₁	C₄⋊C₄	1	2	yes	—
45	C₂×D₄×C₆₁	—	2	2	no	—
46	C₂×D₄×C₆₁	—	2	2	no	—
47	C₂×D₄×C₆₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	#41	1	2	yes	—
51	C₂×C₈×C₆₁	—	4	2	no	—
52	C₄⋊C₄×C₆₁	—	2	2	no	—
53	C₂²⋊C₄×C₆₁	—	2	4	no	—
54	C₄²×C₆₁	C₂×D₄	1	2	yes	—
55	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	#11758194	—	4	4	no	—
58	#11758194	—	4	2	no	—
59	C₂×#92418	—	2	2	no	—
60	C₂×#92418	—	2	2	no	—
61	C₂×#92418	—	4	1	no	—
62	C₂×#92418	—	4	4	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂×#92418	C₂²⋊C₄	1	2	yes	—
65	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₆₁	—	2	2	no	—
68	C₂³×C₆₁	#117	1	1	yes	characteristic
69	Q₈×C₆₁	—	4	2	no	—
70	D₄×C₆₁	—	4	2	no	—
71	D₄×C₆₁	—	4	4	no	—
72	D₄×C₆₁	—	4	4	no	—
73	C₂×C₄×C₆₁	—	2	1	no	—
74	C₂×C₄×C₆₁	—	2	1	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	4	2	no	—
77	C₂×C₄×C₆₁	—	4	2	no	—
78	C₂×C₄×C₆₁	—	4	2	no	—
79	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#121	1	2	yes	—
82	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₆₁	—	8	2	no	—
84	#92418	—	2	1	no	—
85	#92418	—	2	2	no	—
86	#92418	—	4	2	no	—
87	#92418	—	8	2	no	—
88	#92418	#100	1	2	yes	—
89	#8630	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	2	1	no	—
91	C₂²×C₆₁	—	2	2	no	—
92	C₂²×C₆₁	—	4	2	no	—
93	C₂²×C₆₁	—	4	4	no	—
94	C₂²×C₆₁	#380	1	1	yes	characteristic
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	2	no	—
97	C₄×C₆₁	—	2	2	no	—
98	C₄×C₆₁	—	4	2	no	—
99	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
100	C₄×C₆₁	#413	1	1	yes	characteristic
101	#2931	—	61	1	no	—
102	#1568	—	61	2	no	—
103	#1536	—	61	1	no	—
104	#1536	D₆₁	1	1	yes	characteristic
105	#1307	—	61	2	no	—
106	C₂×C₆₁	—	4	1	no	—
107	C₂×C₆₁	—	8	2	no	—
108	C₂×C₆₁	#1800	1	1	yes	characteristic
109	C₂×C₆₁	#1814	1	2	yes	—
110	C₂×#131	—	61	2	no	—
111	#532	C₂×D₆₁	1	1	yes	characteristic
112	#436	—	122	2	no	—
113	C₈×D₄	—	122	2	no	—
114	#432	—	61	2	no	—
115	#432	C₂×D₆₁	1	2	yes	—
116	#423	—	61	1	no	—
117	#408	—	61	2	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	#335	—	122	2	no	—
120	#334	—	122	2	no	—
121	#329	—	2	2	no	—
122	#329	—	122	2	no	—
123	C₆₁	#8630	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	C₄×D₆₁	1	2	yes	—
125	#131	—	244	4	no	—
126	C₂×M₁₆	—	61	2	no	—
127	C₂×M₁₆	—	122	2	no	—
128	C₂²×C₈	—	61	1	no	—
129	C₂²×C₈	—	61	1	no	—
130	C₂²×C₈	—	122	2	no	—
131	C₄⋊Q₈	Dic₁₂₂	1	1	yes	characteristic
132	#127	Dic₁₂₂	1	1	yes	characteristic
133	#124	C₂²⋊D₆₁	1	2	yes	—
134	#121	—	2	2	no	—
135	C₄×D₄	—	2	2	no	—
136	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	122	2	no	—
139	C₄⋊C₈	—	122	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	122	4	no	—
142	C₂²⋊C₈	—	122	2	no	—
143	#97	—	2	2	no	—
144	#97	—	122	2	no	—
145	C₄×C₈	—	61	2	no	—
146	C₄×C₈	—	61	2	no	—
147	#41	—	4	2	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	#11758196	1	2	yes	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	#11758196	1	2	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
155	C₂²×C₄	#11758224	1	2	yes	—
156	M₁₆	—	244	4	no	—
157	M₁₆	—	244	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	61	1	no	—
160	C₂×C₈	—	61	2	no	—
161	C₂×C₈	—	61	1	no	—
162	C₂×C₈	—	122	2	no	—
163	C₂×C₈	—	122	2	no	—
164	C₂×C₈	—	244	4	no	—
165	C₂×C₈	—	244	1	no	—
166	C₂×C₈	—	244	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
170	C₄²	D₄×D₆₁	1	2	yes	—
171	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
172	C₂³	—	2	2	no	—
173	C₂³	#14519733	1	1	yes	characteristic
174	Q₈	—	4	2	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
185	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
186	C₂×C₄	#14519749	1	2	yes	—
187	C₂×C₄	#14519754	1	2	yes	—
188	C₈	—	8	2	no	—
189	C₈	—	61	2	no	—
190	C₈	—	122	1	no	—
191	C₈	—	122	2	no	—
192	C₈	—	488	2	no	—
193	C₈	—	244	2	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	#16839620	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	#100×D₆₁	1	1	yes	characteristic
204	C₄	#16839743	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	—	8	2	no	—
207	C₂	#21445809	1	1	yes	characteristic
208	C₂	#21445871	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21454531

C₂

yes

maximal, characteristic

#21444580

C₂

yes

maximal, complemented

#21444454

C₂

yes

maximal, characteristic

#21443729

C₂

yes

maximal, complemented

C₆₁×#1536

C₂

yes

maximal, characteristic

C₂×#14519784

C₄

yes

—

#16839840

—

D₄×#92418

—

#16839819

C₂²

yes

—

#16839786

C₂²

yes

characteristic

#16839729

C₄

yes

—

C₂×C₄×#92418

C₂²

yes

—

—

—

C₆₁×#532

C₂²

yes

characteristic

C₆₁×#432

C₂²

yes

—

C₆₁×#329

—

#14519784

—

C₂×#11758194

—

C₂×#11758194

C₂×C₄

yes

—

C₂²×#92418

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄×#92418

D₄

yes

—

C₂×#41×C₆₁

C₂×C₄

yes

—

C₄⋊Q₈×C₆₁

Q₈

yes

characteristic

#127×C₆₁

Q₈

yes

characteristic

#124×C₆₁

D₄

yes

—

#121×C₆₁

—

C₄×D₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

#97×C₆₁

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₄⋊C₄

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

—

—

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂³×C₆₁

—

C₂³×C₆₁

#117

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

—

—

—

—

—

yes

—

#8630

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#380

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#100

yes

characteristic

100

C₄×C₆₁

#413

yes

characteristic

101

#2931

—

102

#1568

—

103

#1536

—

104

#1536

D₆₁

yes

characteristic

105

#1307

—

106

C₂×C₆₁

—

107

C₂×C₆₁

—

108

C₂×C₆₁

#1800

yes

characteristic

109

C₂×C₆₁

#1814

yes

—

110

C₂×#131

—

111

#532

C₂×D₆₁

yes

characteristic

112

#436

—

122

—

113

C₈×D₄

—

122

—

114

#432

—

115

#432

C₂×D₆₁

yes

—

116

#423

—

117

#408

—

118

C₂×C₄×C₈

—

119

#335

—

122

—

120

#334

—

122

—

121

#329

—

122

#329

—

122

—

123

C₆₁

#8630

yes

characteristic, complemented, Sylow

124

C₂×#41

C₄×D₆₁

yes

—

125

#131

—

244

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

122

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

—

122

—

131

C₄⋊Q₈

Dic₁₂₂

yes

characteristic

132

#127

Dic₁₂₂

yes

characteristic

133

#124

C₂²⋊D₆₁

yes

—

134

#121

—

135

C₄×D₄

—

136

C₂×C₄²

C₂²×D₆₁

yes

characteristic

137

C₄⋊C₈

—

138

C₄⋊C₈

—

122

—

139

C₄⋊C₈

—

122

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

122

—

142

C₂²⋊C₈

—

122

—

143

#97

—

144

#97

—

122

—

145

C₄×C₈

—

146

C₄×C₈

—

147

#41

—

148

#41

—

149

C₂×Q₈

#11758196

yes

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

#11758196

yes

—

153

C₂²×C₄

—

154

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

155

C₂²×C₄

#11758224

yes

—

156

M₁₆

—

244

—

157

M₁₆

—

244

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

122

—

163

C₂×C₈

—

122

—

164

C₂×C₈

—

244

—

165

C₂×C₈

—

244

—

166

C₂×C₈

—

244

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

C₂×Dic₁₂₂

yes

characteristic

170

C₄²

D₄×D₆₁

yes

—

171

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

172

C₂³

—

173

C₂³

#14519733

yes

characteristic

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂²⋊C₄×D₆₁

yes

characteristic

185

C₂×C₄

C₂×#11758196

yes

characteristic

186

C₂×C₄

#14519749

yes

—

187

C₂×C₄

#14519754

yes

—

188

C₈

—

189

C₈

—

190

C₈

—

122

—

191

C₈

—

122

—

192

C₈

—

488

—

193

C₈

—

244

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#16839620

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#100×D₆₁

yes

characteristic

204

C₄

#16839743

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#21445809

yes

characteristic

208

C₂

#21445871

yes

—

	subgroup	G/H
center	C₂²	#16839620
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1536	C₂
socle	C₂²×C₆₁	#380
prime radical	C₆₁×#532	C₂²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1536	D₆₁
hypocenter	C₆₁	#8630
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

subgroup

G/H

center

C₂²

#16839620

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1536

C₂

socle

C₂²×C₆₁

#380

prime radical

C₆₁×#532

C₂²

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

N	H ≅ G/N	type
2. #21444580	C₂	semidirect
4. #21443729	C₂	semidirect
123. C₆₁	#8630	semidirect

H ≅ G/N

type

2. #21444580

C₂

semidirect

4. #21443729

C₂

semidirect

123. C₆₁

#8630

semidirect

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

subgroup	index
#21454531	2
#21444580	2
#21444454	2
#21443729	2
C₆₁×#1536	2
#8630	61

subgroup

index

C₆₁×#1536

Subgroups — #29573003

Conjugacy classes of subgroups

Complements

Maximal subgroups