Group #29579747 — subgroups

Subgroups — #29579747

Order 15616.

526 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 211 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×#16840160	C₂	1	1	yes	maximal, characteristic
2	#21446780	C₂	1	2	yes	maximal
3	#21446621	C₂	1	2	yes	maximal
4	#21445976	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	C₂²	1	2	yes	—
7	#16840160	—	2	2	no	—
8	#16840160	C₄	1	4	yes	complemented
9	C₂×#14519709	—	2	2	no	—
10	C₂×#14519709	—	2	2	no	—
11	C₂×#14519702	—	2	4	no	—
12	#16839632	C₂²	1	1	yes	characteristic
13	#16839632	C₂²	1	1	yes	characteristic
14	#16839525	—	2	4	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₈²×C₆₁	—	2	2	no	—
18	C₂²×Dic₁₂₂	—	2	2	no	—
19	#14519777	—	2	2	no	—
20	#14519777	—	2	2	no	—
21	#14519777	—	2	2	no	—
22	#14519777	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₆₁	—	2	2	no	—
24	C₂×C₄⋊Dic₆₁	—	2	2	no	—
25	C₂×C₄⋊Dic₆₁	—	2	2	no	—
26	C₂×C₄⋊Dic₆₁	—	2	2	no	—
27	C₂×C₄⋊Dic₆₁	—	2	1	no	—
28	C₂×C₄⋊Dic₆₁	—	2	1	no	—
29	#14519709	—	2	8	no	—
30	#14519709	—	4	4	no	—
31	#14519702	—	4	8	no	—
32	C₂²×C₈×C₆₁	D₄	1	4	yes	—
33	C₂²×C₈×C₆₁	D₄	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₆₁	—	2	2	no	—
36	C₄×C₈×C₆₁	—	2	2	no	—
37	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₆₁	D₄	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
41	C₂²×Dic₆₁	—	4	2	no	—
42	C₂²×Dic₆₁	—	4	1	no	—
43	C₂²×Dic₆₁	—	4	1	no	—
44	C₂×Dic₁₂₂	—	2	4	no	—
45	C₂×Dic₁₂₂	—	4	4	no	—
46	C₂×Dic₁₂₂	—	4	4	no	—
47	C₂×Dic₁₂₂	—	4	2	no	—
48	C₂×C₈×C₆₁	—	2	4	no	—
49	C₂×C₈×C₆₁	—	2	4	no	—
50	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₆₁	SD₁₆	1	8	yes	—
52	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
53	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
54	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	2	yes	—
57	C₄⋊Dic₆₁	—	2	4	no	—
58	C₄⋊Dic₆₁	—	2	4	no	—
59	C₄⋊Dic₆₁	—	4	2	no	—
60	C₄⋊Dic₆₁	—	4	2	no	—
61	C₄⋊Dic₆₁	—	4	4	no	—
62	C₄⋊Dic₆₁	—	4	2	no	—
63	C₄⋊Dic₆₁	—	4	2	no	—
64	C₄⋊Dic₆₁	—	4	4	no	—
65	C₂³×C₆₁	#127	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₆₁	#121	1	2	yes	—
69	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
72	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₆₁	#135	1	4	yes	—
74	C₂×Dic₆₁	—	4	4	no	—
75	C₂×Dic₆₁	—	8	2	no	—
76	C₂×Dic₆₁	—	8	2	no	—
77	C₂×Dic₆₁	—	8	1	no	—
78	C₂×Dic₆₁	—	8	2	no	—
79	C₂×Dic₆₁	—	8	1	no	—
80	C₂×Dic₆₁	—	8	2	no	—
81	Dic₁₂₂	—	4	8	no	—
82	Dic₁₂₂	—	8	4	no	—
83	C₈×C₆₁	—	2	8	no	—
84	C₈×C₆₁	#103	1	8	yes	complemented
85	#9819	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	#390	1	1	yes	characteristic
87	C₂²×C₆₁	#492	1	1	yes	characteristic
88	C₂²×C₆₁	#492	1	2	yes	—
89	C₂²×C₆₁	#494	1	1	yes	characteristic
90	C₂²×C₆₁	#495	1	2	yes	—
91	C₄×C₆₁	C₂×#103	1	2	yes	—
92	C₄×C₆₁	C₄×SD₁₆	1	4	yes	—
93	C₄×C₆₁	#465	1	4	yes	—
94	C₄×C₆₁	#487	1	2	yes	—
95	Dic₆₁	—	8	4	no	—
96	Dic₆₁	—	16	1	no	—
97	Dic₆₁	—	16	1	no	—
98	C₂×#498	—	61	1	no	—
99	#1990	—	61	2	no	—
100	#1956	—	61	2	no	—
101	#1844	—	61	1	no	—
102	C₂×C₈²	D₆₁	1	1	yes	characteristic
103	C₂×C₆₁	#1705	1	1	yes	characteristic
104	C₂×C₆₁	#1707	1	2	yes	—
105	C₂×C₆₁	#1708	1	1	yes	characteristic
106	C₂×C₆₁	#1956	1	1	yes	characteristic
107	C₂×C₆₁	#1968	1	2	yes	—
108	C₂×C₄⋊Q₈	—	61	2	no	—
109	#498	—	61	4	no	—
110	#498	—	122	2	no	—
111	C₂×#106	—	122	2	no	—
112	C₂×#106	—	122	2	no	—
113	C₂×#103	—	122	4	no	—
114	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
116	#384	—	61	1	no	—
117	#384	—	61	1	no	—
118	#340	—	122	4	no	—
119	C₈²	—	2	2	no	—
120	C₆₁	#9819	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	122	2	no	—
122	C₂²×C₈	D₂₄₄	1	4	yes	—
123	C₂²×C₈	D₂₄₄	1	2	yes	—
124	C₄⋊Q₈	—	61	4	no	—
125	C₄⋊Q₈	—	122	2	no	—
126	C₄⋊Q₈	—	122	2	no	—
127	C₄⋊Q₈	—	122	2	no	—
128	C₂×C₄⋊C₄	—	122	2	no	—
129	C₂×C₄⋊C₄	—	122	2	no	—
130	C₂×C₄⋊C₄	—	122	2	no	—
131	C₂×C₄⋊C₄	—	122	2	no	—
132	C₂×C₄⋊C₄	—	122	1	no	—
133	C₂×C₄⋊C₄	—	122	1	no	—
134	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
135	#106	—	122	8	no	—
136	#106	—	244	4	no	—
137	#103	—	244	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₄×D₆₁	1	2	yes	—
141	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
142	C₂×Q₈	—	122	4	no	—
143	C₂×Q₈	—	244	4	no	—
144	C₂×Q₈	—	244	4	no	—
145	C₂×Q₈	—	244	2	no	—
146	C₂²×C₄	—	244	1	no	—
147	C₂²×C₄	—	244	1	no	—
148	C₂²×C₄	—	244	2	no	—
149	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₂₄₄	1	2	yes	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	#11758190	1	8	yes	—
154	C₂×C₈	#11758190	1	4	yes	—
155	C₂×C₈	Dic₂₄₄	1	4	yes	—
156	C₂×C₈	#11758198	1	4	yes	—
157	C₄⋊C₄	—	122	4	no	—
158	C₄⋊C₄	—	122	4	no	—
159	C₄⋊C₄	—	244	4	no	—
160	C₄⋊C₄	—	244	2	no	—
161	C₄⋊C₄	—	244	2	no	—
162	C₄⋊C₄	—	244	4	no	—
163	C₄⋊C₄	—	244	2	no	—
164	C₄⋊C₄	—	244	2	no	—
165	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
166	C₄²	#11758222	1	2	yes	—
167	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
168	C₂³	#14519774	1	1	yes	characteristic
169	Q₈	—	488	4	no	—
170	Q₈	—	244	8	no	—
171	C₂×C₄	—	488	2	no	—
172	C₂×C₄	—	488	1	no	—
173	C₂×C₄	—	488	1	no	—
174	C₂×C₄	—	488	2	no	—
175	C₂×C₄	—	488	2	no	—
176	C₂×C₄	—	488	2	no	—
177	C₂×C₄	—	244	4	no	—
178	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
179	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
180	C₂×C₄	#14519751	1	2	yes	—
181	C₂×C₄	#14519765	1	1	yes	characteristic
182	C₂×C₄	C₂×#11758190	1	4	yes	—
183	C₂×C₄	C₂×#11758190	1	2	yes	—
184	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
185	C₂×C₄	#14519799	1	4	yes	—
186	C₈	—	2	8	no	—
187	C₈	#14519702	1	8	yes	complemented
188	C₂²	#16839664	1	1	yes	characteristic
189	C₂²	#16840135	1	2	yes	—
190	C₂²	#16840135	1	1	yes	characteristic
191	C₂²	#16840142	1	1	yes	characteristic
192	C₂²	#16840146	1	2	yes	—
193	C₄	—	976	1	no	—
194	C₄	—	976	1	no	—
195	C₄	—	488	4	no	—
196	C₄	C₂×#14519702	1	2	yes	—
197	C₄	C₄×#11758190	1	4	yes	—
198	C₄	#16839991	1	4	yes	—
199	C₄	#16840113	1	2	yes	—
200	C₂	#21445406	1	1	yes	characteristic
201	C₂	#21445416	1	2	yes	—
202	C₂	#21445420	1	1	yes	characteristic
203	C₂	#21446621	1	1	yes	characteristic
204	C₂	#21446673	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840160

C₂

yes

maximal, characteristic

#21446780

C₂

yes

maximal

#21446621

C₂

yes

maximal

#21445976

C₂

yes

maximal, characteristic

C₂×C₈²×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519777

C₂²

yes

—

#16840160

—

#16840160

C₄

yes

—

—

—

C₂²

yes

characteristic

#16839632

C₂²

yes

characteristic

#16839525

—

C₂×C₄×C₈×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₈²×C₆₁

—

C₂²×Dic₁₂₂

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519709

—

#14519709

—

#14519702

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

Q₁₆

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

yes

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#494

yes

characteristic

C₂²×C₆₁

#495

yes

—

C₄×C₆₁

C₂×#103

yes

—

C₄×C₆₁

C₄×SD₁₆

yes

—

C₄×C₆₁

#465

yes

—

C₄×C₆₁

#487

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

C₂×#498

—

#1990

—

100

#1956

—

101

#1844

—

102

C₂×C₈²

D₆₁

yes

characteristic

103

C₂×C₆₁

#1705

yes

characteristic

104

C₂×C₆₁

#1707

yes

—

105

C₂×C₆₁

#1708

yes

characteristic

106

C₂×C₆₁

#1956

yes

characteristic

107

C₂×C₆₁

#1968

yes

—

108

C₂×C₄⋊Q₈

—

109

#498

—

110

#498

—

122

—

111

C₂×#106

—

122

—

112

C₂×#106

—

122

—

113

C₂×#103

—

122

—

114

C₂×C₄×C₈

C₂×D₆₁

yes

—

115

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

116

#384

—

117

#384

—

118

#340

—

122

—

119

C₈²

—

120

C₆₁

#9819

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

—

122

C₂²×C₈

D₂₄₄

yes

—

123

C₂²×C₈

D₂₄₄

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

122

—

126

C₄⋊Q₈

—

122

—

127

C₄⋊Q₈

—

122

—

128

C₂×C₄⋊C₄

—

122

—

129

C₂×C₄⋊C₄

—

122

—

130

C₂×C₄⋊C₄

—

122

—

131

C₂×C₄⋊C₄

—

122

—

132

C₂×C₄⋊C₄

—

122

—

133

C₂×C₄⋊C₄

—

122

—

134

C₂×C₄²

C₂²×D₆₁

yes

characteristic

135

#106

—

122

—

136

#106

—

244

—

137

#103

—

244

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₄×D₆₁

yes

—

141

C₄×C₈

C₂²⋊D₆₁

yes

—

142

C₂×Q₈

—

122

—

143

C₂×Q₈

—

244

—

144

C₂×Q₈

—

244

—

145

C₂×Q₈

—

244

—

146

C₂²×C₄

—

244

—

147

C₂²×C₄

—

244

—

148

C₂²×C₄

—

244

—

149

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

150

C₂²×C₄

C₂×D₂₄₄

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

#11758190

yes

—

154

C₂×C₈

#11758190

yes

—

155

C₂×C₈

Dic₂₄₄

yes

—

156

C₂×C₈

#11758198

yes

—

157

C₄⋊C₄

—

122

—

158

C₄⋊C₄

—

122

—

159

C₄⋊C₄

—

244

—

160

C₄⋊C₄

—

244

—

161

C₄⋊C₄

—

244

—

162

C₄⋊C₄

—

244

—

163

C₄⋊C₄

—

244

—

164

C₄⋊C₄

—

244

—

165

C₄²

C₂×C₄×D₆₁

yes

characteristic

166

C₄²

#11758222

yes

—

167

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

168

C₂³

#14519774

yes

characteristic

169

Q₈

—

488

—

170

Q₈

—

244

—

171

C₂×C₄

—

488

—

172

C₂×C₄

—

488

—

173

C₂×C₄

—

488

—

174

C₂×C₄

—

488

—

175

C₂×C₄

—

488

—

176

C₂×C₄

—

488

—

177

C₂×C₄

—

244

—

178

C₂×C₄

C₂×#11758198

yes

characteristic

179

C₂×C₄

C₄×D₂₄₄

yes

—

180

C₂×C₄

#14519751

yes

—

181

C₂×C₄

#14519765

yes

characteristic

182

C₂×C₄

C₂×#11758190

yes

—

183

C₂×C₄

C₂×#11758190

yes

—

184

C₂×C₄

C₂×Dic₂₄₄

yes

—

185

C₂×C₄

#14519799

yes

—

186

C₈

—

187

C₈

#14519702

yes

complemented

188

C₂²

#16839664

yes

characteristic

189

C₂²

#16840135

yes

—

190

C₂²

#16840135

yes

characteristic

191

C₂²

#16840142

yes

characteristic

192

C₂²

#16840146

yes

—

193

C₄

—

976

—

194

C₄

—

976

—

195

C₄

—

488

—

196

C₄

C₂×#14519702

yes

—

197

C₄

C₄×#11758190

yes

—

198

C₄

#16839991

yes

—

199

C₄

#16840113

yes

—

200

C₂

#21445406

yes

characteristic

201

C₂

#21445416

yes

—

202

C₂

#21445420

yes

characteristic

203

C₂

#21446621

yes

characteristic

204

C₂

#21446673

yes

—

	subgroup	G/H
center	C₂³	#14519774
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	#9819
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14519774

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₂×C₈²×C₆₁

C₂

socle, prime radical

C₂³×C₆₁

#127

cosocle

C₂×C₄²×C₆₁

C₂³

hypercenter

C₂×C₈²

D₆₁

hypocenter

C₆₁

#9819

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
8. #16840160	C₄	semidirect
84. C₈×C₆₁	#103	semidirect
120. C₆₁	#9819	semidirect
187. C₈	#14519702	semidirect

H ≅ G/N

type

8. #16840160

C₄

semidirect

84. C₈×C₆₁

#103

semidirect

120. C₆₁

#9819

semidirect

187. C₈

#14519702

semidirect

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

subgroup	index
C₂×#16840160	2
#21446780	2
#21446621	2
#21445976	2
C₂×C₈²×C₆₁	2
#9819	61

subgroup

index

C₂×C₈²×C₆₁

Subgroups — #29579747

Conjugacy classes of subgroups

Complements

Maximal subgroups