Group #29579093 — subgroups

Subgroups — #29579093

Order 15616.

538 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 167 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	#21454217	C₂	1	1	yes	maximal, characteristic
2	#21445672	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839510	C₂	1	4	yes	maximal
4	C₆₁×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519777	C₄	1	4	yes	—
6	C₂×#14519777	C₂²	1	2	yes	—
7	C₂×C₄⋊#92418	C₂²	1	2	yes	—
8	C₂×#14519688	—	2	1	no	—
9	#16839632	—	2	1	no	—
10	#16839632	—	2	1	no	—
11	#16839632	—	2	2	no	—
12	#16839517	—	2	1	no	—
13	#16839510	—	2	8	no	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#336	—	2	2	no	—
17	C₂²×Dic₁₂₂	C₂×C₄	1	4	yes	—
18	C₂²×#92418	—	2	2	no	—
19	#14519777	—	2	4	no	—
20	#14519777	—	2	4	no	—
21	#14519777	—	4	8	no	—
22	C₂×C₄⋊Dic₆₁	—	2	4	no	—
23	C₂×C₄⋊Dic₆₁	—	2	4	no	—
24	C₂×C₄⋊Dic₆₁	—	2	2	no	—
25	C₂×C₄⋊Dic₆₁	—	2	2	no	—
26	C₄⋊#92418	—	2	2	no	—
27	C₄⋊#92418	D₄	1	4	yes	—
28	#14519688	—	4	2	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₁	D₄	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	—	2	4	no	—
36	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	—	2	4	no	—
42	C₂×Dic₁₂₂	—	2	8	no	—
43	C₂×Dic₁₂₂	—	4	8	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	—	4	4	no	—
46	C₄²×C₆₁	—	2	4	no	—
47	C₄²×C₆₁	—	2	2	no	—
48	C₄²×C₆₁	—	2	4	no	—
49	C₄²×C₆₁	SD₁₆	1	2	yes	—
50	C₄²×C₆₁	Q₁₆	1	2	yes	—
51	C₄²×C₆₁	C₂×D₄	1	2	yes	—
52	C₄²×C₆₁	#41	1	2	yes	—
53	C₄⋊Dic₆₁	—	4	8	no	—
54	C₄⋊Dic₆₁	—	4	8	no	—
55	C₄⋊Dic₆₁	—	4	4	no	—
56	C₄⋊Dic₆₁	—	4	2	no	—
57	C₄⋊Dic₆₁	—	4	2	no	—
58	C₂×#92418	—	2	4	no	—
59	C₂×#92418	—	4	2	no	—
60	C₂×#92418	—	4	2	no	—
61	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#100	1	4	yes	—
68	C₂×C₄×C₆₁	#103	1	4	yes	—
69	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₆₁	#121	1	2	yes	—
71	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#137	1	2	yes	—
76	C₂×C₄×C₆₁	#137	1	4	yes	—
77	C₂×Dic₆₁	—	4	4	no	—
78	C₂×Dic₆₁	—	4	8	no	—
79	Dic₁₂₂	—	8	8	no	—
80	C₈×C₆₁	—	8	2	no	—
81	#92418	—	4	4	no	—
82	#9702	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	#390	1	1	yes	characteristic
84	C₂²×C₆₁	C₂×#100	1	2	yes	—
85	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
86	C₂²×C₆₁	#419	1	2	yes	—
87	C₂²×C₆₁	#419	1	1	yes	characteristic
88	C₄×C₆₁	—	2	4	no	—
89	C₄×C₆₁	—	2	4	no	—
90	C₄×C₆₁	—	2	2	no	—
91	C₄×C₆₁	#461	1	2	yes	—
92	C₄×C₆₁	#462	1	2	yes	—
93	C₄×C₆₁	#491	1	2	yes	—
94	C₄×C₆₁	#497	1	2	yes	—
95	Dic₆₁	—	8	4	no	—
96	#2868	—	61	1	no	—
97	#1831	D₆₁	1	1	yes	characteristic
98	#1767	—	61	1	no	—
99	C₂×#333	—	61	4	no	—
100	C₂×C₆₁	#1548	1	2	yes	—
101	C₂×C₆₁	#1558	1	2	yes	—
102	C₂×C₆₁	#1960	1	1	yes	characteristic
103	C₂×C₆₁	#1965	1	1	yes	characteristic
104	C₂×C₆₁	#1981	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	—	61	4	no	—
106	C₂×C₄⋊Q₈	—	61	2	no	—
107	C₂×C₄⋊C₈	—	61	2	no	—
108	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
109	C₂×#97	—	122	1	no	—
110	#384	—	122	1	no	—
111	#384	—	122	1	no	—
112	#384	—	122	2	no	—
113	C₄³	C₂×D₆₁	1	1	yes	characteristic
114	#336	—	2	2	no	—
115	#336	—	122	1	no	—
116	#333	—	122	8	no	—
117	C₆₁	#9702	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	61	4	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	122	2	no	—
121	C₄⋊Q₈	—	122	4	no	—
122	C₄⋊Q₈	—	122	4	no	—
123	C₄⋊Q₈	—	244	8	no	—
124	C₂×C₄⋊C₄	—	122	4	no	—
125	C₂×C₄⋊C₄	—	122	4	no	—
126	C₂×C₄⋊C₄	—	122	2	no	—
127	C₂×C₄⋊C₄	—	122	2	no	—
128	C₂×C₄²	C₄×D₆₁	1	2	yes	—
129	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
130	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
131	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
132	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	61	4	no	—
135	C₄⋊C₈	—	122	2	no	—
136	#97	—	244	2	no	—
137	C₂×Q₈	—	122	8	no	—
138	C₂×Q₈	—	244	8	no	—
139	C₂²×C₄	—	122	4	no	—
140	C₂²×C₄	#11758198	1	2	yes	—
141	C₂²×C₄	#11758198	1	2	yes	—
142	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	122	4	no	—
148	C₂×C₈	—	244	2	no	—
149	C₂×C₈	—	244	2	no	—
150	C₄⋊C₄	—	244	8	no	—
151	C₄⋊C₄	—	244	8	no	—
152	C₄⋊C₄	—	244	4	no	—
153	C₄⋊C₄	—	244	2	no	—
154	C₄⋊C₄	—	244	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	#11758190	1	2	yes	—
159	C₄²	Dic₂₄₄	1	2	yes	—
160	C₄²	D₄×D₆₁	1	2	yes	—
161	C₄²	#11758226	1	2	yes	—
162	C₂³	C₂×#11758198	1	1	yes	characteristic
163	Q₈	—	488	8	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	8	no	—
170	C₂×C₄	—	244	4	no	—
171	C₂×C₄	#14519693	1	4	yes	—
172	C₂×C₄	#14519702	1	4	yes	—
173	C₂×C₄	#14519739	1	2	yes	—
174	C₂×C₄	#14519752	1	2	yes	—
175	C₂×C₄	#14519766	1	1	yes	characteristic
176	C₂×C₄	#14519773	1	1	yes	characteristic
177	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
178	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
179	C₂×C₄	#14519807	1	4	yes	—
180	C₂×C₄	#14519810	1	2	yes	—
181	C₈	—	8	2	no	—
182	C₈	—	244	4	no	—
183	C₂²	#16839663	1	1	yes	characteristic
184	C₂²	C₂×#14519693	1	2	yes	—
185	C₂²	C₂×#14519702	1	1	yes	characteristic
186	C₂²	#16839770	1	2	yes	—
187	C₂²	#16839772	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	488	4	no	—
192	C₄	#16839968	1	2	yes	—
193	C₄	#16839973	1	2	yes	—
194	C₄	#16840132	1	2	yes	—
195	C₄	#16840157	1	2	yes	—
196	C₂	#21444499	1	2	yes	—
197	C₂	#21444544	1	2	yes	—
198	C₂	#21446638	1	1	yes	characteristic
199	C₂	#21446658	1	1	yes	characteristic
200	C₂	#21446741	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21454217

C₂

yes

maximal, characteristic

#21445672

C₂

yes

maximal, characteristic

C₂×#16839510

C₂

yes

maximal

C₆₁×#1831

C₂

yes

maximal, characteristic

C₂×#14519777

C₄

yes

—

C₂×#14519777

C₂²

yes

—

C₂×C₄⋊#92418

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₆₁×#336

—

C₂²×Dic₁₂₂

C₂×C₄

yes

—

C₂²×#92418

—

#14519777

—

#14519777

—

#14519777

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₄⋊#92418

—

C₄⋊#92418

D₄

yes

—

#14519688

—

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

—

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×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

Q₁₆

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

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

#100

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₁

#137

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#103

yes

characteristic

C₂²×C₆₁

#419

yes

—

C₂²×C₆₁

#419

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#461

yes

—

C₄×C₆₁

#462

yes

—

C₄×C₆₁

#491

yes

—

C₄×C₆₁

#497

yes

—

Dic₆₁

—

#2868

—

#1831

D₆₁

yes

characteristic

#1767

—

C₂×#333

—

100

C₂×C₆₁

#1548

yes

—

101

C₂×C₆₁

#1558

yes

—

102

C₂×C₆₁

#1960

yes

characteristic

103

C₂×C₆₁

#1965

yes

characteristic

104

C₂×C₆₁

#1981

yes

characteristic

105

C₂×C₄⋊Q₈

—

106

C₂×C₄⋊Q₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

109

C₂×#97

—

122

—

110

#384

—

122

—

111

#384

—

122

—

112

#384

—

122

—

113

C₄³

C₂×D₆₁

yes

characteristic

114

#336

—

115

#336

—

122

—

116

#333

—

122

—

117

C₆₁

#9702

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

122

—

121

C₄⋊Q₈

—

122

—

122

C₄⋊Q₈

—

122

—

123

C₄⋊Q₈

—

244

—

124

C₂×C₄⋊C₄

—

122

—

125

C₂×C₄⋊C₄

—

122

—

126

C₂×C₄⋊C₄

—

122

—

127

C₂×C₄⋊C₄

—

122

—

128

C₂×C₄²

C₄×D₆₁

yes

—

129

C₂×C₄²

D₂₄₄

yes

characteristic

130

C₂×C₄²

D₂₄₄

yes

characteristic

131

C₂×C₄²

C₂²⋊D₆₁

yes

—

132

C₂×C₄²

C₂²×D₆₁

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

122

—

136

#97

—

244

—

137

C₂×Q₈

—

122

—

138

C₂×Q₈

—

244

—

139

C₂²×C₄

—

122

—

140

C₂²×C₄

#11758198

yes

—

141

C₂²×C₄

#11758198

yes

—

142

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

143

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

144

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

122

—

148

C₂×C₈

—

244

—

149

C₂×C₈

—

244

—

150

C₄⋊C₄

—

244

—

151

C₄⋊C₄

—

244

—

152

C₄⋊C₄

—

244

—

153

C₄⋊C₄

—

244

—

154

C₄⋊C₄

—

244

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#11758190

yes

—

159

C₄²

Dic₂₄₄

yes

—

160

C₄²

D₄×D₆₁

yes

—

161

C₄²

#11758226

yes

—

162

C₂³

C₂×#11758198

yes

characteristic

163

Q₈

—

488

—

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

#14519693

yes

—

172

C₂×C₄

#14519702

yes

—

173

C₂×C₄

#14519739

yes

—

174

C₂×C₄

#14519752

yes

—

175

C₂×C₄

#14519766

yes

characteristic

176

C₂×C₄

#14519773

yes

characteristic

177

C₂×C₄

C₂×#11758190

yes

characteristic

178

C₂×C₄

C₂×Dic₂₄₄

yes

characteristic

179

C₂×C₄

#14519807

yes

—

180

C₂×C₄

#14519810

yes

—

181

C₈

—

182

C₈

—

244

—

183

C₂²

#16839663

yes

characteristic

184

C₂²

C₂×#14519693

yes

—

185

C₂²

C₂×#14519702

yes

characteristic

186

C₂²

#16839770

yes

—

187

C₂²

#16839772

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

488

—

192

C₄

#16839968

yes

—

193

C₄

#16839973

yes

—

194

C₄

#16840132

yes

—

195

C₄

#16840157

yes

—

196

C₂

#21444499

yes

—

197

C₂

#21444544

yes

—

198

C₂

#21446638

yes

characteristic

199

C₂

#21446658

yes

characteristic

200

C₂

#21446741

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11758198
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1831	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1831	D₆₁
hypocenter	C₆₁	#9702
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×#11758198

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1831

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 ◁ #1831

N	H ≅ G/N	type
117. C₆₁	#9702	semidirect

H ≅ G/N

type

117. C₆₁

#9702

semidirect

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

subgroup	index
#21454217	2
#21445672	2
C₂×#16839510	2
C₆₁×#1831	2
#9702	61

subgroup

index

C₆₁×#1831

Subgroups — #29579093

Conjugacy classes of subgroups

Complements

Maximal subgroups