Group #29572780 — subgroups

Subgroups — #29572780

Order 15616.

456 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 151 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	#21445672	C₂	1	2	yes	maximal
2	C₂×#16839504	C₂	1	2	yes	maximal
3	#21443700	C₂	1	2	yes	maximal
4	C₄×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	—	2	2	no	—
6	C₂×#14519688	C₂²	1	4	yes	—
7	#16839517	—	2	2	no	—
8	#16839504	—	2	4	no	—
9	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#389	C₄	1	2	yes	—
11	C₆₁×#387	—	2	2	no	—
12	C₆₁×#382	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
14	C₄×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	—	2	4	no	—
17	C₄⋊#92418	—	2	8	no	—
18	#14519688	—	2	4	no	—
19	#14519688	—	2	4	no	—
20	#125×C₆₁	—	2	2	no	—
21	#125×C₆₁	D₄	1	2	yes	—
22	#125×C₆₁	Q₈	1	2	yes	—
23	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₆₁	—	2	2	no	—
28	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	D₄	1	2	yes	—
31	C₂×C₄²×C₆₁	D₄	1	2	yes	—
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#95×C₆₁	—	2	2	no	—
34	#95×C₆₁	—	2	2	no	—
35	#95×C₆₁	—	2	2	no	—
36	C₂²×C₄×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	—	2	2	no	—
38	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₄⋊C₄×C₆₁	—	2	4	no	—
45	C₄⋊C₄×C₆₁	—	2	4	no	—
46	C₄⋊C₄×C₆₁	—	4	4	no	—
47	C₄⋊C₄×C₆₁	—	4	2	no	—
48	C₄⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄²×C₆₁	—	2	4	no	—
51	C₄²×C₆₁	—	4	1	no	—
52	C₄²×C₆₁	—	4	2	no	—
53	C₄²×C₆₁	—	4	4	no	—
54	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	2	yes	—
57	C₂×#92418	—	2	8	no	—
58	C₂×#92418	—	4	4	no	—
59	C₂×#92418	—	4	4	no	—
60	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	1	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	4	no	—
66	C₂×C₄×C₆₁	—	2	1	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	—	4	2	no	—
74	C₂×C₄×C₆₁	—	4	2	no	—
75	C₂×C₄×C₆₁	—	4	2	no	—
76	C₂×C₄×C₆₁	#101	1	2	yes	—
77	C₂×C₄×C₆₁	#104	1	4	yes	—
78	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#121	1	2	yes	—
81	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
82	#92418	—	4	8	no	—
83	#8580	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	#380	1	1	yes	characteristic
85	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
86	C₂²×C₆₁	#413	1	1	yes	characteristic
87	C₂²×C₆₁	C₂×#104	1	2	yes	—
88	C₂²×C₆₁	#421	1	2	yes	—
89	C₄×C₆₁	—	2	1	no	—
90	C₄×C₆₁	—	2	1	no	—
91	C₄×C₆₁	—	2	2	no	—
92	C₄×C₆₁	—	2	2	no	—
93	C₄×C₆₁	—	4	1	no	—
94	C₄×C₆₁	—	4	4	no	—
95	C₄×C₆₁	—	4	2	no	—
96	C₄×C₆₁	—	8	2	no	—
97	C₄×C₆₁	#330	1	4	yes	complemented
98	C₄×#125	D₆₁	1	1	yes	characteristic
99	#1767	—	61	2	no	—
100	C₂×#330	—	61	2	no	—
101	#1295	—	61	2	no	—
102	C₂×C₆₁	C₂×#330	1	1	yes	characteristic
103	C₂×C₆₁	#1529	1	2	yes	—
104	C₂×C₆₁	#1537	1	1	yes	characteristic
105	C₂×C₆₁	#1801	1	1	yes	characteristic
106	C₂×C₆₁	#1806	1	2	yes	—
107	C₂×#125	C₂×D₆₁	1	1	yes	characteristic
108	C₂×C₄⋊C₈	—	122	2	no	—
109	C₂×#97	—	61	4	no	—
110	#389	Dic₆₁	1	2	yes	—
111	#387	—	2	2	no	—
112	#382	C₂×D₆₁	1	1	yes	characteristic
113	C₄×C₄⋊C₄	—	2	2	no	—
114	C₄×C₄⋊C₄	Dic₆₁	1	2	yes	—
115	C₄³	C₂×D₆₁	1	1	yes	characteristic
116	#336	—	122	2	no	—
117	#330	—	122	4	no	—
118	C₆₁	#8580	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	122	4	no	—
120	#125	—	2	2	no	—
121	#125	Dic₁₂₂	1	2	yes	—
122	#125	D₂₄₄	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
126	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
129	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
132	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
133	C₄⋊C₈	—	122	8	no	—
134	#97	—	122	4	no	—
135	#97	—	122	4	no	—
136	#95	—	2	2	no	—
137	#95	—	2	2	no	—
138	#95	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
143	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
146	C₂×C₈	—	122	8	no	—
147	C₂×C₈	—	244	4	no	—
148	C₂×C₈	—	244	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	C₄⋊Dic₆₁	1	4	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	4	2	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
161	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
162	C₄²	#11758222	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	#14519696	1	2	yes	—
180	C₂×C₄	#14519704	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
182	C₂×C₄	#14519736	1	1	yes	characteristic
183	C₂×C₄	#14519751	1	2	yes	—
184	C₂×C₄	#14519756	1	2	yes	—
185	C₈	—	244	8	no	—
186	C₂²	#16839622	1	1	yes	characteristic
187	C₂²	C₂×#14519696	1	1	yes	characteristic
188	C₂²	#16839746	1	1	yes	characteristic
189	C₂²	C₂×#14519704	1	2	yes	—
190	C₂²	#16839779	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	1	no	—
197	C₄	—	4	4	no	—
198	C₄	—	8	2	no	—
199	C₄	#16839504	1	4	yes	complemented
200	C₂	C₂×#16839504	1	1	yes	characteristic
201	C₂	#21444426	1	2	yes	—
202	C₂	#21444457	1	1	yes	characteristic
203	C₂	#21445815	1	1	yes	characteristic
204	C₂	#21445839	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445672

C₂

yes

maximal

C₂×#16839504

C₂

yes

maximal

#21443700

C₂

yes

maximal

C₄×#125×C₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

—

C₂×#14519688

C₂²

yes

—

#16839517

—

#16839504

—

C₂×#125×C₆₁

C₂²

yes

characteristic

C₆₁×#389

C₄

yes

—

C₆₁×#387

—

C₆₁×#382

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₆₁

—

C₄×C₄⋊C₄×C₆₁

C₄

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

C₄⋊#92418

—

#14519688

—

#14519688

—

#125×C₆₁

—

#125×C₆₁

D₄

yes

—

#125×C₆₁

Q₈

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#95×C₆₁

—

#95×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

#41

yes

—

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

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#101

yes

—

C₂×C₄×C₆₁

#104

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#421

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#330

yes

complemented

C₄×#125

D₆₁

yes

characteristic

#1767

—

100

C₂×#330

—

101

#1295

—

102

C₂×C₆₁

C₂×#330

yes

characteristic

103

C₂×C₆₁

#1529

yes

—

104

C₂×C₆₁

#1537

yes

characteristic

105

C₂×C₆₁

#1801

yes

characteristic

106

C₂×C₆₁

#1806

yes

—

107

C₂×#125

C₂×D₆₁

yes

characteristic

108

C₂×C₄⋊C₈

—

122

—

109

C₂×#97

—

110

#389

Dic₆₁

yes

—

111

#387

—

112

#382

C₂×D₆₁

yes

characteristic

113

C₄×C₄⋊C₄

—

114

C₄×C₄⋊C₄

Dic₆₁

yes

—

115

C₄³

C₂×D₆₁

yes

characteristic

116

#336

—

122

—

117

#330

—

122

—

118

C₆₁

#8580

yes

characteristic, complemented, Sylow

119

C₂²×C₈

—

122

—

120

#125

—

121

#125

Dic₁₂₂

yes

—

122

#125

D₂₄₄

yes

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

126

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

129

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

130

C₂×C₄²

C₂²⋊D₆₁

yes

—

131

C₂×C₄²

C₂²⋊D₆₁

yes

—

132

C₂×C₄²

C₂²×D₆₁

yes

characteristic

133

C₄⋊C₈

—

122

—

134

#97

—

122

—

135

#97

—

122

—

136

#95

—

137

#95

—

138

#95

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

142

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

143

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

144

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

—

146

C₂×C₈

—

122

—

147

C₂×C₈

—

244

—

148

C₂×C₈

—

244

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

C₄⋊Dic₆₁

yes

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×Dic₁₂₂

yes

characteristic

161

C₄²

C₂×D₂₄₄

yes

characteristic

162

C₄²

#11758222

yes

—

163

C₂³

C₂×C₂²⋊Dic₆₁

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#14519696

yes

—

180

C₂×C₄

#14519704

yes

—

181

C₂×C₄

C₂×C₄⋊Dic₆₁

yes

characteristic

182

C₂×C₄

#14519736

yes

characteristic

183

C₂×C₄

#14519751

yes

—

184

C₂×C₄

#14519756

yes

—

185

C₈

—

244

—

186

C₂²

#16839622

yes

characteristic

187

C₂²

C₂×#14519696

yes

characteristic

188

C₂²

#16839746

yes

characteristic

189

C₂²

C₂×#14519704

yes

—

190

C₂²

#16839779

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

#16839504

yes

complemented

200

C₂

C₂×#16839504

yes

characteristic

201

C₂

#21444426

yes

—

202

C₂

#21444457

yes

characteristic

203

C₂

#21445815

yes

characteristic

204

C₂

#21445839

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#125×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#125	D₆₁
hypocenter	C₆₁	#8580
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₁

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₄×#125×C₆₁

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 ◁ C₄×#125

N	H ≅ G/N	type
97. C₄×C₆₁	#330	semidirect
118. C₆₁	#8580	semidirect
199. C₄	#16839504	semidirect

H ≅ G/N

type

97. C₄×C₆₁

#330

semidirect

118. C₆₁

#8580

semidirect

199. C₄

#16839504

semidirect

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

subgroup	index
#21445672	2
C₂×#16839504	2
#21443700	2
C₄×#125×C₆₁	2
#8580	61

subgroup

index

C₄×#125×C₆₁

Subgroups — #29572780

Conjugacy classes of subgroups

Complements

Maximal subgroups