Group #29579835 — subgroups

Subgroups — #29579835

Order 15616.

494 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#16840167	C₂	1	1	yes	maximal, characteristic
2	#21446779	C₂	1	2	yes	maximal
3	#21446620	C₂	1	2	yes	maximal
4	C₂×#16839484	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	C₂²	1	2	yes	—
7	#16840167	—	2	2	no	—
8	#16840167	C₄	1	4	yes	complemented
9	C₂×#14519711	—	2	2	no	—
10	C₂×#14519700	—	2	4	no	—
11	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839524	—	2	4	no	—
14	#16839484	—	2	2	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#384	C₂²	1	2	yes	—
17	C₆₁×#340	—	2	2	no	—
18	C₂²×Dic₁₂₂	—	2	2	no	—
19	C₂²×#92418	D₄	1	2	yes	—
20	C₂²×#92418	D₄	1	2	yes	—
21	#14519777	—	2	2	no	—
22	#14519777	—	2	2	no	—
23	#14519777	—	2	2	no	—
24	#14519777	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₁	—	2	2	no	—
26	C₂×C₄⋊Dic₆₁	—	2	2	no	—
27	#14519711	—	2	8	no	—
28	#14519700	—	4	8	no	—
29	C₄×#92418	—	2	1	no	—
30	C₄×#92418	—	2	1	no	—
31	C₄×#92418	—	2	1	no	—
32	C₄×#92418	—	2	1	no	—
33	C₂²×C₈×C₆₁	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₁	D₄	1	2	yes	—
40	C₂²×C₄×C₆₁	—	4	2	no	—
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₆₁	—	4	2	no	—
45	C₂×Dic₁₂₂	—	2	4	no	—
46	C₂×Dic₁₂₂	—	4	4	no	—
47	C₂×Dic₁₂₂	—	4	4	no	—
48	C₂×Dic₁₂₂	—	4	2	no	—
49	C₂×C₈×C₆₁	—	2	4	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
51	C₄⋊C₄×C₆₁	—	4	4	no	—
52	C₄⋊C₄×C₆₁	—	4	4	no	—
53	C₄⋊C₄×C₆₁	—	4	4	no	—
54	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	1	yes	characteristic
57	C₄²×C₆₁	#41	1	1	yes	characteristic
58	C₄⋊Dic₆₁	—	2	4	no	—
59	C₄⋊Dic₆₁	—	2	4	no	—
60	C₄⋊Dic₆₁	—	4	2	no	—
61	C₄⋊Dic₆₁	—	4	2	no	—
62	C₂×#92418	—	2	2	no	—
63	C₂×#92418	—	2	2	no	—
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	—	2	4	no	—
66	C₂×#92418	Q₁₆	1	4	yes	—
67	C₂³×C₆₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₆₁	—	8	2	no	—
69	C₂×C₄×C₆₁	—	8	2	no	—
70	C₂×C₄×C₆₁	—	8	2	no	—
71	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
76	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₆₁	#135	1	2	yes	—
79	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
80	C₂×C₄×C₆₁	#137	1	4	yes	—
81	C₂×Dic₆₁	—	4	4	no	—
82	C₂×Dic₆₁	—	8	2	no	—
83	C₂×Dic₆₁	—	8	2	no	—
84	Dic₁₂₂	—	4	8	no	—
85	Dic₁₂₂	—	8	4	no	—
86	C₈×C₆₁	—	2	4	no	—
87	#92418	—	2	4	no	—
88	#92418	—	4	2	no	—
89	#9834	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	#390	1	1	yes	characteristic
91	C₂²×C₆₁	#494	1	1	yes	characteristic
92	C₂²×C₆₁	#495	1	1	yes	characteristic
93	C₂²×C₆₁	#496	1	2	yes	—
94	C₂²×C₆₁	#497	1	2	yes	—
95	C₄×C₆₁	—	16	2	no	—
96	C₄×C₆₁	#419	1	2	yes	—
97	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
98	C₄×C₆₁	#442	1	2	yes	—
99	C₄×C₆₁	#467	1	2	yes	—
100	C₄×C₆₁	#469	1	2	yes	—
101	C₄×C₆₁	#491	1	2	yes	—
102	Dic₆₁	—	8	4	no	—
103	C₂×#500	—	61	1	no	—
104	#1990	—	61	2	no	—
105	#1956	—	61	2	no	—
106	#1834	D₆₁	1	1	yes	characteristic
107	C₂×#322	—	61	1	no	—
108	C₂×C₆₁	#1719	1	2	yes	—
109	C₂×C₆₁	#1723	1	2	yes	—
110	C₂×C₆₁	#1960	1	1	yes	characteristic
111	C₂×C₆₁	#1969	1	1	yes	characteristic
112	C₂×C₆₁	#1972	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	61	2	no	—
114	#500	—	61	4	no	—
115	#500	—	122	2	no	—
116	C₂×#107	—	122	2	no	—
117	C₂×#103	—	122	4	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	C₂×C₄×C₈	—	61	1	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	#384	C₂×D₆₁	1	2	yes	—
122	#340	—	2	2	no	—
123	#340	—	122	4	no	—
124	#322	—	122	2	no	—
125	C₆₁	#9834	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	122	2	no	—
127	C₂²×C₈	—	61	2	no	—
128	C₂²×C₈	—	61	2	no	—
129	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
130	C₄⋊Q₈	—	61	4	no	—
131	C₄⋊Q₈	—	122	2	no	—
132	C₄⋊Q₈	—	122	2	no	—
133	C₄⋊Q₈	—	122	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	122	2	no	—
138	C₂×C₄⋊C₄	—	122	2	no	—
139	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
140	#107	—	122	8	no	—
141	#103	—	244	8	no	—
142	C₄×C₈	—	122	1	no	—
143	C₄×C₈	—	122	1	no	—
144	C₄×C₈	—	122	1	no	—
145	C₄×C₈	—	122	1	no	—
146	C₄×C₈	C₄×D₆₁	1	2	yes	—
147	C₄×C₈	D₂₄₄	1	2	yes	—
148	C₂×Q₈	—	122	4	no	—
149	C₂×Q₈	—	244	4	no	—
150	C₂×Q₈	—	244	4	no	—
151	C₂×Q₈	—	244	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	244	2	no	—
154	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
155	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	61	4	no	—
159	C₂×C₈	—	122	2	no	—
160	C₂×C₈	—	122	2	no	—
161	C₂×C₈	—	122	2	no	—
162	C₂×C₈	—	122	4	no	—
163	C₂×C₈	#11758198	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	122	4	no	—
168	C₄⋊C₄	—	122	4	no	—
169	C₄⋊C₄	—	244	2	no	—
170	C₄⋊C₄	—	244	2	no	—
171	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
172	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
173	C₄²	#11758226	1	1	yes	characteristic
174	C₄²	#11758226	1	1	yes	characteristic
175	C₂³	#14519773	1	1	yes	characteristic
176	Q₈	—	488	4	no	—
177	Q₈	—	244	8	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	488	2	no	—
182	C₂×C₄	—	488	2	no	—
183	C₂×C₄	—	244	4	no	—
184	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
185	C₂×C₄	#14519739	1	1	yes	characteristic
186	C₂×C₄	#14519739	1	1	yes	characteristic
187	C₂×C₄	#14519752	1	1	yes	characteristic
188	C₂×C₄	#14519752	1	1	yes	characteristic
189	C₂×C₄	#14519766	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
191	C₂×C₄	#14519796	1	2	yes	—
192	C₂×C₄	#14519804	1	4	yes	—
193	C₂×C₄	#14519807	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	122	4	no	—
196	C₈	—	244	2	no	—
197	C₂²	#16839663	1	1	yes	characteristic
198	C₂²	#16840140	1	1	yes	characteristic
199	C₂²	#16840145	1	1	yes	characteristic
200	C₂²	#16840148	1	2	yes	—
201	C₂²	#16840157	1	2	yes	—
202	C₄	—	16	2	no	—
203	C₄	—	488	4	no	—
204	C₄	#16839770	1	2	yes	—
205	C₄	#16839856	1	2	yes	—
206	C₄	#16839873	1	2	yes	—
207	C₄	#16840004	1	2	yes	—
208	C₄	#16840016	1	2	yes	—
209	C₄	#16840132	1	2	yes	—
210	C₂	#21445471	1	2	yes	—
211	C₂	#21445499	1	2	yes	—
212	C₂	#21446638	1	1	yes	characteristic
213	C₂	#21446678	1	1	yes	characteristic
214	C₂	#21446693	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840167

C₂

yes

maximal, characteristic

#21446779

C₂

yes

maximal

#21446620

C₂

yes

maximal

C₂×#16839484

C₂

yes

maximal, characteristic

C₆₁×#1834

C₂

yes

maximal, characteristic

C₂×#14519777

C₂²

yes

—

#16840167

—

#16840167

C₄

yes

complemented

C₂×#14519711

—

C₂×#14519700

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₂×C₄×#92418

C₂²

yes

characteristic

#16839524

—

#16839484

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#384

C₂²

yes

—

C₆₁×#340

—

C₂²×Dic₁₂₂

—

C₂²×#92418

D₄

yes

—

C₂²×#92418

D₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

—

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

C₄²×C₆₁

yes

C₄²×C₆₁

yes

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

—

—

—

—

Q₁₆

yes

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#497

yes

—

C₄×C₆₁

—

C₄×C₆₁

#419

yes

—

C₄×C₆₁

C₄×Q₁₆

yes

—

C₄×C₆₁

#442

yes

—

C₄×C₆₁

#467

yes

—

100

C₄×C₆₁

#469

yes

—

101

C₄×C₆₁

#491

yes

—

102

Dic₆₁

—

103

C₂×#500

—

104

#1990

—

105

#1956

—

106

#1834

D₆₁

yes

characteristic

107

C₂×#322

—

108

C₂×C₆₁

#1719

yes

—

109

C₂×C₆₁

#1723

yes

—

110

C₂×C₆₁

#1960

yes

characteristic

111

C₂×C₆₁

#1969

yes

characteristic

112

C₂×C₆₁

#1972

yes

characteristic

113

C₂×C₄⋊Q₈

—

114

#500

—

115

#500

—

122

—

116

C₂×#107

—

122

—

117

C₂×#103

—

122

—

118

C₂×C₄×C₈

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

121

#384

C₂×D₆₁

yes

—

122

#340

—

123

#340

—

122

—

124

#322

—

122

—

125

C₆₁

#9834

yes

characteristic, complemented, Sylow

126

C₂²×Q₈

—

122

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

C₂²⋊D₆₁

yes

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

122

—

132

C₄⋊Q₈

—

122

—

133

C₄⋊Q₈

—

122

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

122

—

138

C₂×C₄⋊C₄

—

122

—

139

C₂×C₄²

C₂²×D₆₁

yes

characteristic

140

#107

—

122

—

141

#103

—

244

—

142

C₄×C₈

—

122

—

143

C₄×C₈

—

122

—

144

C₄×C₈

—

122

—

145

C₄×C₈

—

122

—

146

C₄×C₈

C₄×D₆₁

yes

—

147

C₄×C₈

D₂₄₄

yes

—

148

C₂×Q₈

—

122

—

149

C₂×Q₈

—

244

—

150

C₂×Q₈

—

244

—

151

C₂×Q₈

—

244

—

152

C₂²×C₄

—

153

C₂²×C₄

—

244

—

154

C₂²×C₄

D₄×D₆₁

yes

characteristic

155

C₂²×C₄

D₄×D₆₁

yes

characteristic

156

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

122

—

160

C₂×C₈

—

122

—

161

C₂×C₈

—

122

—

162

C₂×C₈

—

122

—

163

C₂×C₈

#11758198

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

122

—

168

C₄⋊C₄

—

122

—

169

C₄⋊C₄

—

244

—

170

C₄⋊C₄

—

244

—

171

C₄²

C₂×C₄×D₆₁

yes

characteristic

172

C₄²

C₂×D₂₄₄

yes

characteristic

173

C₄²

#11758226

yes

characteristic

174

C₄²

#11758226

yes

characteristic

175

C₂³

#14519773

yes

characteristic

176

Q₈

—

488

—

177

Q₈

—

244

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

488

—

182

C₂×C₄

—

488

—

183

C₂×C₄

—

244

—

184

C₂×C₄

C₂×#11758198

yes

characteristic

185

C₂×C₄

#14519739

yes

characteristic

186

C₂×C₄

#14519739

yes

characteristic

187

C₂×C₄

#14519752

yes

characteristic

188

C₂×C₄

#14519752

yes

characteristic

189

C₂×C₄

#14519766

yes

characteristic

190

C₂×C₄

Q₁₆×D₆₁

yes

—

191

C₂×C₄

#14519796

yes

—

192

C₂×C₄

#14519804

yes

—

193

C₂×C₄

#14519807

yes

—

194

C₈

—

195

C₈

—

122

—

196

C₈

—

244

—

197

C₂²

#16839663

yes

characteristic

198

C₂²

#16840140

yes

characteristic

199

C₂²

#16840145

yes

characteristic

200

C₂²

#16840148

yes

—

201

C₂²

#16840157

yes

—

202

C₄

—

203

C₄

—

488

—

204

C₄

#16839770

yes

—

205

C₄

#16839856

yes

—

206

C₄

#16839873

yes

—

207

C₄

#16840004

yes

—

208

C₄

#16840016

yes

—

209

C₄

#16840132

yes

—

210

C₂

#21445471

yes

—

211

C₂

#21445499

yes

—

212

C₂

#21446638

yes

characteristic

213

C₂

#21446678

yes

characteristic

214

C₂

#21446693

yes

characteristic

	subgroup	G/H
center	C₂³	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1834	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1834	D₆₁
hypocenter	C₆₁	#9834
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#14519773

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1834

C₂

socle, prime radical

C₂³×C₆₁

#127

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
8. #16840167	C₄	semidirect
125. C₆₁	#9834	semidirect

H ≅ G/N

type

8. #16840167

C₄

semidirect

125. C₆₁

#9834

semidirect

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

subgroup	index
C₂×#16840167	2
#21446779	2
#21446620	2
C₂×#16839484	2
C₆₁×#1834	2
#9834	61

subgroup

index

C₆₁×#1834

Subgroups — #29579835

Conjugacy classes of subgroups

Complements

Maximal subgroups