Group #29571016 — subgroups

Subgroups — #29571016

Order 15616.

452 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 143 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	#21444481	C₂	1	1	yes	maximal, characteristic
2	#21444411	C₂	1	1	yes	maximal, characteristic
3	C₈×#11758194	C₂	1	2	yes	maximal, complemented
4	#21443708	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2931	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₄×#11758194	C₂²	1	2	yes	—
8	#16839511	C₄	1	2	yes	—
9	#16839506	C₄	1	2	yes	—
10	#16839502	C₄	1	4	yes	—
11	#16839484	—	2	2	no	—
12	C₈×#92418	—	2	2	no	—
13	C₂×#131×C₆₁	C₄	1	2	yes	—
14	C₆₁×#532	C₂²	1	1	yes	characteristic
15	C₆₁×#436	—	2	2	no	—
16	C₈×D₄×C₆₁	—	2	2	no	—
17	C₆₁×#423	C₂²	1	1	yes	characteristic
18	C₆₁×#408	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758194	—	2	2	no	—
21	C₄⋊#92418	C₂×C₄	1	4	yes	—
22	C₂²⋊#92418	—	2	2	no	—
23	#14519688	C₂×C₄	1	4	yes	—
24	C₄×#92418	—	2	2	no	—
25	C₄×#92418	—	2	2	no	—
26	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
27	#131×C₆₁	—	4	4	no	—
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₁	—	2	2	no	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	#127×C₆₁	C₂×C₄	1	1	yes	characteristic
35	#124×C₆₁	C₂×C₄	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	4	no	—
41	C₄×C₈×C₆₁	D₄	1	2	yes	—
42	C₄×C₈×C₆₁	D₄	1	2	yes	—
43	#41×C₆₁	—	4	2	no	—
44	#41×C₆₁	—	4	2	no	—
45	C₂×Q₈×C₆₁	C₄²	1	2	yes	—
46	C₂×D₄×C₆₁	—	2	2	no	—
47	C₂×D₄×C₆₁	—	2	2	no	—
48	C₂×D₄×C₆₁	C₄²	1	2	yes	—
49	C₂²×C₄×C₆₁	—	2	2	no	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	#41	1	2	yes	—
52	M₁₆×C₆₁	—	4	4	no	—
53	M₁₆×C₆₁	—	4	2	no	—
54	C₂×C₈×C₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	—	2	2	no	—
56	C₂×C₈×C₆₁	—	4	1	no	—
57	C₂×C₈×C₆₁	—	4	4	no	—
58	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	—	2	2	no	—
62	C₂²⋊C₄×C₆₁	—	2	4	no	—
63	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₆₁	C₂×D₄	1	2	yes	—
66	#11758194	—	2	8	no	—
67	C₂×#92418	—	2	4	no	—
68	C₂×#92418	—	2	4	no	—
69	C₂³×C₆₁	—	2	2	no	—
70	C₂³×C₆₁	#117	1	1	yes	characteristic
71	Q₈×C₆₁	—	4	2	no	—
72	D₄×C₆₁	—	4	2	no	—
73	D₄×C₆₁	—	4	4	no	—
74	D₄×C₆₁	—	4	4	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	—	2	4	no	—
78	C₂×C₄×C₆₁	—	2	2	no	—
79	C₂×C₄×C₆₁	—	4	2	no	—
80	C₂×C₄×C₆₁	—	4	2	no	—
81	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
85	C₈×C₆₁	—	2	1	no	—
86	C₈×C₆₁	—	2	2	no	—
87	C₈×C₆₁	—	4	2	no	—
88	C₈×C₆₁	#100	1	2	yes	—
89	#92418	—	2	8	no	—
90	#92418	—	4	4	no	—
91	#8241	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	—	2	1	no	—
93	C₂²×C₆₁	—	2	2	no	—
94	C₂²×C₆₁	—	4	2	no	—
95	C₂²×C₆₁	—	4	4	no	—
96	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	4	2	no	—
100	C₄×C₆₁	C₂×#100	1	1	yes	characteristic
101	C₄×C₆₁	#413	1	1	yes	characteristic
102	C₄×C₆₁	#443	1	4	yes	—
103	#2931	D₆₁	1	1	yes	characteristic
104	#1543	—	61	1	no	—
105	#1525	—	61	1	no	—
106	C₈×M₁₆	—	61	2	no	—
107	#1299	—	61	2	no	—
108	C₂×C₆₁	—	4	1	no	—
109	C₂×C₆₁	—	8	2	no	—
110	C₂×C₆₁	C₄×#100	1	1	yes	characteristic
111	C₂×C₆₁	#1764	1	2	yes	—
112	C₂×#131	Dic₆₁	1	2	yes	—
113	#532	C₂×D₆₁	1	1	yes	characteristic
114	#436	—	2	2	no	—
115	C₈×D₄	—	2	2	no	—
116	#431	—	61	2	no	—
117	#423	C₂×D₆₁	1	1	yes	characteristic
118	#408	Dic₆₁	1	2	yes	—
119	C₄×M₁₆	—	61	2	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	#333	—	61	2	no	—
122	#331	—	61	2	no	—
123	#329	—	61	4	no	—
124	#322	—	122	2	no	—
125	C₈²	—	122	2	no	—
126	C₆₁	#8241	1	1	yes	characteristic, complemented, Sylow
127	C₂×#41	C₂×Dic₆₁	1	2	yes	—
128	#131	—	4	4	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	122	2	no	—
131	C₂×M₁₆	C₂×Dic₆₁	1	2	yes	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
135	C₄⋊Q₈	C₄×D₆₁	1	1	yes	characteristic
136	#127	C₄×D₆₁	1	1	yes	characteristic
137	#124	C₄×D₆₁	1	2	yes	—
138	#121	—	2	2	no	—
139	C₄×D₄	—	2	2	no	—
140	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	61	4	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	C₂²⋊C₈	—	122	2	no	—
145	#97	—	61	4	no	—
146	C₄×C₈	—	122	2	no	—
147	C₄×C₈	—	122	2	no	—
148	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
149	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
150	#41	—	4	2	no	—
151	#41	—	4	2	no	—
152	C₂×Q₈	C₄×Dic₆₁	1	2	yes	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₄×Dic₆₁	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#11758222	1	2	yes	—
158	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
159	M₁₆	—	4	4	no	—
160	M₁₆	—	4	2	no	—
161	M₁₆	—	122	8	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	4	no	—
166	C₂×C₈	—	122	4	no	—
167	C₂×C₈	—	122	4	no	—
168	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
169	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
174	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	#14519736	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
190	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
191	C₂×C₄	C₄×C₂²⋊D₆₁	1	2	yes	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	122	8	no	—
196	C₈	—	244	4	no	—
197	C₈	#14519694	1	2	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	4	4	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₄×C₂²⋊Dic₆₁	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	1	no	—
205	C₄	—	4	2	no	—
206	C₄	C₂×#14519694	1	1	yes	characteristic
207	C₄	#16839746	1	1	yes	characteristic
208	C₄	#16839877	1	4	yes	—
209	C₂	—	4	1	no	—
210	C₂	—	8	2	no	—
211	C₂	C₄×#14519694	1	1	yes	characteristic
212	C₂	#21445654	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21444481

C₂

yes

maximal, characteristic

#21444411

C₂

yes

maximal, characteristic

C₈×#11758194

C₂

yes

maximal, complemented

#21443708

C₂

yes

maximal, complemented

C₆₁×#2931

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

—

C₄×#11758194

C₂²

yes

—

#16839511

C₄

yes

—

#16839506

C₄

yes

—

#16839502

C₄

yes

—

#16839484

—

C₈×#92418

—

C₂×#131×C₆₁

C₄

yes

—

C₆₁×#532

C₂²

yes

characteristic

C₆₁×#436

—

C₈×D₄×C₆₁

—

C₆₁×#423

C₂²

yes

characteristic

C₆₁×#408

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×#11758194

—

C₄⋊#92418

C₂×C₄

yes

—

C₂²⋊#92418

—

#14519688

C₂×C₄

yes

—

C₄×#92418

—

C₄×#92418

—

C₂×#41×C₆₁

C₂×C₄

yes

—

#131×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₆₁

C₂×C₄

yes

characteristic

#127×C₆₁

C₂×C₄

yes

characteristic

#124×C₆₁

C₂×C₄

yes

—

#121×C₆₁

—

C₄×D₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊C₈×C₆₁

—

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₄²

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

C₄²

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

—

#11758194

—

C₂×#92418

—

C₂×#92418

—

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×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#100

yes

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

100

C₄×C₆₁

C₂×#100

yes

characteristic

101

C₄×C₆₁

#413

yes

characteristic

102

C₄×C₆₁

#443

yes

—

103

#2931

D₆₁

yes

characteristic

104

#1543

—

105

#1525

—

106

C₈×M₁₆

—

107

#1299

—

108

C₂×C₆₁

—

109

C₂×C₆₁

—

110

C₂×C₆₁

C₄×#100

yes

characteristic

111

C₂×C₆₁

#1764

yes

—

112

C₂×#131

Dic₆₁

yes

—

113

#532

C₂×D₆₁

yes

characteristic

114

#436

—

115

C₈×D₄

—

116

#431

—

117

#423

C₂×D₆₁

yes

characteristic

118

#408

Dic₆₁

yes

—

119

C₄×M₁₆

—

120

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

121

#333

—

122

#331

—

123

#329

—

124

#322

—

122

—

125

C₈²

—

122

—

126

C₆₁

#8241

yes

characteristic, complemented, Sylow

127

C₂×#41

C₂×Dic₆₁

yes

—

128

#131

—

129

C₂×M₁₆

—

130

C₂×M₁₆

—

122

—

131

C₂×M₁₆

C₂×Dic₆₁

yes

—

132

C₂²×C₈

—

133

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

134

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

135

C₄⋊Q₈

C₄×D₆₁

yes

characteristic

136

#127

C₄×D₆₁

yes

characteristic

137

#124

C₄×D₆₁

yes

—

138

#121

—

139

C₄×D₄

—

140

C₂×C₄²

C₂²×D₆₁

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₂²⋊C₈

—

144

C₂²⋊C₈

—

122

—

145

#97

—

146

C₄×C₈

—

122

—

147

C₄×C₈

—

122

—

148

C₄×C₈

C₂²⋊D₆₁

yes

—

149

C₄×C₈

C₂²⋊D₆₁

yes

—

150

#41

—

151

#41

—

152

C₂×Q₈

C₄×Dic₆₁

yes

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

C₄×Dic₆₁

yes

—

156

C₂²×C₄

—

157

C₂²×C₄

#11758222

yes

—

158

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

122

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

122

—

167

C₂×C₈

—

122

—

168

C₂×C₈

C₂²⋊Dic₆₁

yes

—

169

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

170

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

171

C₄⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

C₂×C₄×D₆₁

yes

characteristic

174

C₄²

C₂×C₄×D₆₁

yes

characteristic

175

C₄²

C₂×C₂²⋊D₆₁

yes

—

176

C₂³

—

177

C₂³

#14519736

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

D₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

C₂×C₄×Dic₆₁

yes

characteristic

189

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

190

C₂×C₄

C₄×C₂²⋊D₆₁

yes

—

191

C₂×C₄

C₄×C₂²⋊D₆₁

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

122

—

196

C₈

—

244

—

197

C₈

#14519694

yes

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₄×C₂²⋊Dic₆₁

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

C₂×#14519694

yes

characteristic

207

C₄

#16839746

yes

characteristic

208

C₄

#16839877

yes

—

209

C₂

—

210

C₂

—

211

C₂

C₄×#14519694

yes

characteristic

212

C₂

#21445654

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2931	C₂
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₆₁×#532	C₂²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#2931	D₆₁
hypocenter	C₆₁	#8241
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₆₁

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#2931

C₂

socle

C₂²×C₆₁

C₄×C₂²⋊C₄

prime radical

C₆₁×#532

C₂²

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

N	H ≅ G/N	type
3. C₈×#11758194	C₂	semidirect
4. #21443708	C₂	semidirect
126. C₆₁	#8241	semidirect

H ≅ G/N

type

3. C₈×#11758194

C₂

semidirect

4. #21443708

C₂

semidirect

126. C₆₁

#8241

semidirect

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

subgroup	index
#21444481	2
#21444411	2
C₈×#11758194	2
#21443708	2
C₆₁×#2931	2
#8241	61

subgroup

index

C₆₁×#2931

Subgroups — #29571016

Conjugacy classes of subgroups

Complements

Maximal subgroups