Group #29579626 — subgroups

Subgroups — #29579626

Order 15616.

474 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 139 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	#21446809	C₂	1	2	yes	maximal
2	#21444578	C₂	1	1	yes	maximal, characteristic
3	#21444500	C₂	1	2	yes	maximal
4	C₈×#11758194	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#494	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	1	yes	characteristic
7	#16839786	C₂²	1	2	yes	—
8	C₄×#11758194	C₂²	1	1	yes	characteristic
9	#16839531	—	2	4	no	—
10	#16839515	—	2	2	no	—
11	#16839511	—	2	4	no	—
12	#16839484	—	2	1	no	—
13	C₈×#92418	—	2	1	no	—
14	C₂²×Q₁₆×C₆₁	C₄	1	4	yes	—
15	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#494	—	2	1	no	—
17	C₆₁×#494	—	2	1	no	—
18	C₆₁×#494	—	2	2	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	D₄	1	2	yes	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	D₄	1	2	yes	—
25	C₂²⋊#92418	—	2	1	no	—
26	#14519688	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	—	2	1	no	—
29	C₂²×Q₈×C₆₁	C₂×C₄	1	4	yes	—
30	C₂×Q₁₆×C₆₁	—	2	4	no	—
31	C₂×Q₁₆×C₆₁	—	2	4	no	—
32	C₂×Q₁₆×C₆₁	—	4	8	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₄⋊Q₈×C₆₁	—	2	2	no	—
35	C₄⋊Q₈×C₆₁	—	2	2	no	—
36	C₄⋊Q₈×C₆₁	—	2	2	no	—
37	C₄⋊Q₈×C₆₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
40	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	2	yes	—
44	C₂×Q₈×C₆₁	—	2	4	no	—
45	C₂×Q₈×C₆₁	—	2	4	no	—
46	C₂×Q₈×C₆₁	—	4	8	no	—
47	C₂²×C₄×C₆₁	—	2	4	no	—
48	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
51	Q₁₆×C₆₁	—	4	8	no	—
52	Q₁₆×C₆₁	—	4	8	no	—
53	C₂×C₈×C₆₁	—	2	4	no	—
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
56	C₄⋊C₄×C₆₁	—	4	2	no	—
57	C₄⋊C₄×C₆₁	—	4	2	no	—
58	C₄⋊C₄×C₆₁	—	4	2	no	—
59	C₄⋊C₄×C₆₁	—	4	2	no	—
60	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₆₁	#41	1	1	yes	characteristic
64	#11758194	—	2	2	no	—
65	#11758194	—	4	4	no	—
66	#11758194	Q₁₆	1	4	yes	complemented
67	C₂×#92418	—	2	2	no	—
68	C₂×#92418	—	2	2	no	—
69	C₂×#92418	—	4	2	no	—
70	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
71	Q₈×C₆₁	—	4	8	no	—
72	Q₈×C₆₁	—	8	4	no	—
73	C₂×C₄×C₆₁	—	2	2	no	—
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	4	4	no	—
76	C₂×C₄×C₆₁	—	8	4	no	—
77	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₆₁	#135	1	2	yes	—
85	C₈×C₆₁	—	2	2	no	—
86	C₈×C₆₁	#100	1	4	yes	—
87	#92418	—	2	4	no	—
88	#92418	—	4	2	no	—
89	#92418	—	8	2	no	—
90	#9798	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	—	2	1	no	—
92	C₂²×C₆₁	—	2	1	no	—
93	C₂²×C₆₁	#390	1	1	yes	characteristic
94	C₂²×C₆₁	C₄×Q₁₆	1	2	yes	—
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	8	4	no	—
98	C₄×C₆₁	C₂×#100	1	2	yes	—
99	C₄×C₆₁	#467	1	2	yes	—
100	C₄×C₆₁	#473	1	2	yes	—
101	C₄×C₆₁	#494	1	1	yes	characteristic
102	C₄×C₆₁	#495	1	1	yes	characteristic
103	C₂×#494	D₆₁	1	1	yes	characteristic
104	#1997	—	61	2	no	—
105	#1567	—	61	1	no	—
106	#1548	—	61	2	no	—
107	C₈×M₁₆	—	61	1	no	—
108	C₂×C₆₁	—	2	2	no	—
109	C₂×C₆₁	#1965	1	1	yes	characteristic
110	C₂×C₆₁	#1969	1	1	yes	characteristic
111	C₂×C₆₁	#1979	1	1	yes	characteristic
112	C₂²×Q₁₆	Dic₆₁	1	4	yes	—
113	C₂×C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
114	#494	—	2	1	no	—
115	#494	—	2	1	no	—
116	#494	—	2	2	no	—
117	#431	—	61	1	no	—
118	#423	—	61	2	no	—
119	C₄×M₁₆	—	61	1	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	#343	—	122	4	no	—
122	#335	—	122	2	no	—
123	#333	—	122	4	no	—
124	#322	—	122	1	no	—
125	C₈²	—	122	1	no	—
126	C₆₁	#9798	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	C₂×Dic₆₁	1	4	yes	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	—	2	4	no	—
130	C₂×Q₁₆	—	4	8	no	—
131	C₂×M₁₆	—	61	2	no	—
132	C₂×M₁₆	—	122	2	no	—
133	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
141	C₄⋊C₈	—	61	2	no	—
142	C₄⋊C₈	—	122	2	no	—
143	C₄⋊C₈	—	122	2	no	—
144	C₂²⋊C₈	—	122	1	no	—
145	#97	—	122	1	no	—
146	C₄×C₈	—	122	1	no	—
147	C₄×C₈	—	122	1	no	—
148	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	4	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
156	C₂²×C₄	#11758224	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
158	Q₁₆	—	4	8	no	—
159	Q₁₆	—	4	8	no	—
160	M₁₆	—	61	4	no	—
161	M₁₆	—	122	2	no	—
162	M₁₆	—	244	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	122	2	no	—
165	C₂×C₈	—	122	2	no	—
166	C₂×C₈	—	244	2	no	—
167	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
168	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	D₄×D₆₁	1	1	yes	characteristic
174	C₄²	#11758224	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
176	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
177	C₂³	D₄×Dic₆₁	1	1	yes	characteristic
178	Q₈	—	4	8	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	4	no	—
184	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
185	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
186	C₂×C₄	#14519748	1	1	yes	characteristic
187	C₂×C₄	#14519748	1	1	yes	characteristic
188	C₂×C₄	#14519764	1	1	yes	characteristic
189	C₂×C₄	#14519773	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
191	C₂×C₄	#14519800	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	122	4	no	—
194	C₈	—	488	2	no	—
195	C₈	—	244	2	no	—
196	C₈	#14519694	1	4	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#16839665	1	1	yes	characteristic
200	C₂²	Q₁₆×Dic₆₁	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	4	no	—
204	C₄	C₂×#14519694	1	2	yes	—
205	C₄	#16840000	1	2	yes	—
206	C₄	#16840034	1	2	yes	—
207	C₄	#16840140	1	1	yes	characteristic
208	C₄	#16840147	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	#21446660	1	1	yes	characteristic
211	C₂	#21446681	1	1	yes	characteristic
212	C₂	#21446735	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21446809

C₂

yes

maximal

#21444578

C₂

yes

maximal, characteristic

#21444500

C₂

yes

maximal

C₈×#11758194

C₂

yes

maximal, characteristic

C₂×C₆₁×#494

C₂

yes

maximal, characteristic

#16839815

C₂²

yes

characteristic

#16839786

C₂²

yes

—

C₄×#11758194

C₂²

yes

—

—

—

—

—

C₂²×Q₁₆×C₆₁

C₄

yes

—

C₂×C₄⋊Q₈×C₆₁

C₂²

yes

—

C₆₁×#494

—

C₆₁×#494

—

C₆₁×#494

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×#11758194

—

C₂×#11758194

D₄

yes

—

C₄⋊#92418

—

C₄⋊#92418

—

C₄⋊#92418

D₄

yes

—

—

—

—

—

C₂²×Q₈×C₆₁

C₂×C₄

yes

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

#41

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

C₄²×C₆₁

yes

—

—

Q₁₆

yes

—

—

—

C₂³×C₆₁

C₄×D₄

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₈×C₆₁

—

C₈×C₆₁

#100

yes

—

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#390

yes

characteristic

C₂²×C₆₁

C₄×Q₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#100

yes

—

C₄×C₆₁

#467

yes

—

100

C₄×C₆₁

#473

yes

—

101

C₄×C₆₁

#494

yes

characteristic

102

C₄×C₆₁

#495

yes

characteristic

103

C₂×#494

D₆₁

yes

characteristic

104

#1997

—

105

#1567

—

106

#1548

—

107

C₈×M₁₆

—

108

C₂×C₆₁

—

109

C₂×C₆₁

#1965

yes

characteristic

110

C₂×C₆₁

#1969

yes

characteristic

111

C₂×C₆₁

#1979

yes

characteristic

112

C₂²×Q₁₆

Dic₆₁

yes

—

113

C₂×C₄⋊Q₈

C₂×D₆₁

yes

—

114

#494

—

115

#494

—

116

#494

—

117

#431

—

118

#423

—

119

C₄×M₁₆

—

120

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

121

#343

—

122

—

122

#335

—

122

—

123

#333

—

122

—

124

#322

—

122

—

125

C₈²

—

122

—

126

C₆₁

#9798

yes

characteristic, complemented, Sylow

127

C₂²×Q₈

C₂×Dic₆₁

yes

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×M₁₆

—

132

C₂×M₁₆

—

122

—

133

C₂²×C₈

C₂×Dic₆₁

yes

—

134

C₄⋊Q₈

—

135

C₄⋊Q₈

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄²

C₂²×D₆₁

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

122

—

143

C₄⋊C₈

—

122

—

144

C₂²⋊C₈

—

122

—

145

#97

—

122

—

146

C₄×C₈

—

122

—

147

C₄×C₈

—

122

—

148

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

149

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

150

C₄×C₈

C₂²⋊D₆₁

yes

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂²×C₄

—

155

C₂²×C₄

D₄×D₆₁

yes

characteristic

156

C₂²×C₄

#11758224

yes

characteristic

157

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

158

Q₁₆

—

159

Q₁₆

—

160

M₁₆

—

161

M₁₆

—

122

—

162

M₁₆

—

244

—

163

C₂×C₈

—

164

C₂×C₈

—

122

—

165

C₂×C₈

—

122

—

166

C₂×C₈

—

244

—

167

C₂×C₈

C₂²⋊Dic₆₁

yes

—

168

C₂×C₈

C₂²⋊Dic₆₁

yes

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄²

D₄×D₆₁

yes

characteristic

174

C₄²

#11758224

yes

characteristic

175

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

176

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

177

C₂³

D₄×Dic₆₁

yes

characteristic

178

Q₈

—

179

Q₈

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

185

C₂×C₄

D₄×Dic₆₁

yes

characteristic

186

C₂×C₄

#14519748

yes

characteristic

187

C₂×C₄

#14519748

yes

characteristic

188

C₂×C₄

#14519764

yes

characteristic

189

C₂×C₄

#14519773

yes

characteristic

190

C₂×C₄

Q₁₆×D₆₁

yes

—

191

C₂×C₄

#14519800

yes

—

192

C₈

—

193

C₈

—

122

—

194

C₈

—

488

—

195

C₈

—

244

—

196

C₈

#14519694

yes

—

197

C₂²

—

198

C₂²

—

199

C₂²

#16839665

yes

characteristic

200

C₂²

Q₁₆×Dic₆₁

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

C₂×#14519694

yes

—

205

C₄

#16840000

yes

—

206

C₄

#16840034

yes

—

207

C₄

#16840140

yes

characteristic

208

C₄

#16840147

yes

characteristic

209

C₂

—

210

C₂

#21446660

yes

characteristic

211

C₂

#21446681

yes

characteristic

212

C₂

#21446735

yes

characteristic

	subgroup	G/H
center	C₂²	#16839665
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#494	C₂
socle	C₂²×C₆₁	#390
prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#494	D₆₁
hypocenter	C₆₁	#9798
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#16839665

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₆₁×#494

C₂

socle

C₂²×C₆₁

#390

prime radical

C₂³×C₆₁

C₄×D₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
66. #11758194	Q₁₆	semidirect
126. C₆₁	#9798	semidirect

H ≅ G/N

type

66. #11758194

Q₁₆

semidirect

126. C₆₁

#9798

semidirect

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

subgroup	index
#21446809	2
#21444578	2
#21444500	2
C₈×#11758194	2
C₂×C₆₁×#494	2
#9798	61

subgroup

index

C₂×C₆₁×#494

Subgroups — #29579626

Conjugacy classes of subgroups

Complements

Maximal subgroups