Group #29580043 — subgroups

Subgroups — #29580043

Order 15616.

486 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₄×#14519769	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519688	C₂	1	1	yes	maximal, characteristic
3	#21443658	C₂	1	4	yes	maximal
4	C₆₁×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519769	C₄	1	1	yes	characteristic
6	C₂×#14519688	—	2	2	no	—
7	C₂×C₄×#92418	C₂²	1	2	yes	—
8	#16839660	C₄	1	2	yes	—
9	#16839646	—	2	2	no	—
10	#16839628	C₄	1	1	yes	characteristic
11	C₄×#11758196	—	2	2	no	—
12	C₄×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	—	2	2	no	—
17	#14519769	—	2	4	no	—
18	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×#11758196	—	2	2	no	—
21	C₂×#11758196	—	2	2	no	—
22	C₂×C₄×Dic₆₁	—	2	2	no	—
23	#14519688	—	2	8	no	—
24	C₄×#92418	—	2	2	no	—
25	C₄×#92418	Q₈	1	4	yes	—
26	#14519684	—	2	2	no	—
27	#14519684	—	2	2	no	—
28	#14519684	—	2	2	no	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	D₄	1	2	yes	—
33	C₂×C₄²×C₆₁	D₄	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	—	2	4	no	—
36	C₄⋊C₈×C₆₁	—	4	4	no	—
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	—	2	2	no	—
43	C₂²×Dic₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	—	4	4	no	—
46	C₄²×C₆₁	—	2	4	no	—
47	C₄²×C₆₁	—	2	2	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄²×C₆₁	—	2	2	no	—
51	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
52	C₄²×C₆₁	#41	1	2	yes	—
53	C₄⋊Dic₆₁	—	2	2	no	—
54	C₄⋊Dic₆₁	—	2	4	no	—
55	C₄⋊Dic₆₁	M₁₆	1	4	yes	—
56	#11758196	—	2	8	no	—
57	#11758196	—	4	4	no	—
58	C₄×Dic₆₁	—	4	2	no	—
59	C₄×Dic₆₁	—	4	2	no	—
60	C₂×#92418	—	2	8	no	—
61	C₂×#92418	—	2	4	no	—
62	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	#104	1	4	yes	—
70	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₆₁	#131	1	2	yes	—
76	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₆₁	#137	1	2	yes	—
78	C₂×Dic₆₁	—	2	4	no	—
79	C₂×Dic₆₁	—	4	2	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₂×Dic₆₁	—	4	2	no	—
82	C₂×Dic₆₁	—	4	2	no	—
83	C₂×Dic₆₁	—	4	2	no	—
84	C₈×C₆₁	—	8	2	no	—
85	#92418	—	2	8	no	—
86	#9876	—	61	1	no	maximal, Sylow
87	C₂²×C₆₁	#391	1	1	yes	characteristic
88	C₂²×C₆₁	#407	1	1	yes	characteristic
89	C₂²×C₆₁	#408	1	1	yes	characteristic
90	C₂²×C₆₁	#417	1	2	yes	—
91	C₂²×C₆₁	C₂×#104	1	2	yes	—
92	C₄×C₆₁	—	2	2	no	—
93	C₄×C₆₁	—	2	4	no	—
94	C₄×C₆₁	—	4	2	no	—
95	C₄×C₆₁	#446	1	4	yes	—
96	C₄×C₆₁	#490	1	1	yes	characteristic
97	C₄×C₆₁	#491	1	1	yes	characteristic
98	C₄×C₆₁	#501	1	2	yes	—
99	Dic₆₁	—	4	4	no	—
100	Dic₆₁	—	8	2	no	—
101	C₄×#125	—	61	1	no	—
102	#1842	D₆₁	1	1	yes	characteristic
103	C₄×#97	—	61	1	no	—
104	#1279	—	61	4	no	—
105	C₂×C₆₁	#1483	1	2	yes	—
106	C₂×C₆₁	#1489	1	2	yes	—
107	C₂×C₆₁	#1985	1	1	yes	characteristic
108	C₂×C₆₁	#1992	1	1	yes	characteristic
109	C₂×C₆₁	#1994	1	1	yes	characteristic
110	C₂×#125	—	61	1	no	—
111	C₂×C₄⋊C₈	—	2	2	no	—
112	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
113	C₂×#97	—	122	2	no	—
114	C₂×C₄×C₈	—	61	2	no	—
115	#389	—	61	2	no	—
116	#387	—	122	2	no	—
117	#382	—	61	1	no	—
118	C₄×C₄⋊C₄	—	61	2	no	—
119	C₄×C₄⋊C₄	—	122	2	no	—
120	C₄³	C₂×D₆₁	1	1	yes	characteristic
121	C₆₁	#9876	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	122	2	no	—
124	#125	—	122	4	no	—
125	C₂×C₄⋊C₄	—	61	2	no	—
126	C₂×C₄⋊C₄	—	61	2	no	—
127	C₂×C₄⋊C₄	—	122	2	no	—
128	C₂×C₄⋊C₄	—	122	2	no	—
129	C₂×C₄²	—	122	2	no	—
130	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
132	C₂×C₄²	D₂₄₄	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
134	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	4	4	no	—
137	#97	—	122	8	no	—
138	C₄×C₈	—	61	4	no	—
139	C₄×C₈	—	122	2	no	—
140	#95	—	122	2	no	—
141	#95	—	122	2	no	—
142	#95	—	122	2	no	—
143	C₂²×C₄	—	122	2	no	—
144	C₂²×C₄	—	122	2	no	—
145	C₂²×C₄	#11758198	1	2	yes	—
146	C₂²×C₄	#11758198	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	122	8	no	—
153	C₂×C₈	—	122	4	no	—
154	C₄⋊C₄	—	61	4	no	—
155	C₄⋊C₄	—	122	8	no	—
156	C₄⋊C₄	—	122	2	no	—
157	C₄⋊C₄	—	122	4	no	—
158	C₄⋊C₄	—	244	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	244	2	no	—
165	C₄²	—	244	2	no	—
166	C₄²	#11758224	1	2	yes	—
167	C₄²	Q₈×D₆₁	1	2	yes	—
168	C₂³	C₂×#11758198	1	1	yes	characteristic
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	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	122	4	no	—
176	C₂×C₄	—	244	2	no	—
177	C₂×C₄	—	244	2	no	—
178	C₂×C₄	—	244	2	no	—
179	C₂×C₄	—	244	2	no	—
180	C₂×C₄	—	244	2	no	—
181	C₂×C₄	#14519703	1	4	yes	—
182	C₂×C₄	#14519742	1	2	yes	—
183	C₂×C₄	#14519757	1	2	yes	—
184	C₂×C₄	#14519764	1	1	yes	characteristic
185	C₂×C₄	#14519775	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
187	C₂×C₄	#14519783	1	2	yes	—
188	C₂×C₄	#14519803	1	2	yes	—
189	C₂×C₄	#14519808	1	2	yes	—
190	C₈	—	8	2	no	—
191	C₈	—	122	8	no	—
192	C₂²	#16839667	1	1	yes	characteristic
193	C₂²	#16839725	1	1	yes	characteristic
194	C₂²	#16839728	1	1	yes	characteristic
195	C₂²	#16839761	1	2	yes	—
196	C₂²	C₂×#14519703	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	488	2	no	—
201	C₄	—	244	4	no	—
202	C₄	#16839891	1	4	yes	—
203	C₄	#16840126	1	1	yes	characteristic
204	C₄	#16840130	1	1	yes	characteristic
205	C₄	#16840168	1	2	yes	—
206	C₂	#21444230	1	2	yes	—
207	C₂	#21444256	1	2	yes	—
208	C₂	#21446758	1	1	yes	characteristic
209	C₂	#21446787	1	1	yes	characteristic
210	C₂	#21446795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519769

C₂

yes

maximal, characteristic

C₄×#14519688

C₂

yes

maximal, characteristic

#21443658

C₂

yes

maximal

C₆₁×#1842

C₂

yes

maximal, characteristic

C₂×#14519769

C₄

yes

characteristic

C₂×#14519688

—

C₂×C₄×#92418

C₂²

yes

—

#16839660

C₄

yes

—

#16839646

—

#16839628

C₄

yes

characteristic

C₄×#11758196

—

C₄×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊C₈×C₆₁

—

C₂×C₄⋊C₈×C₆₁

C₂²

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

#14519769

—

C₂×C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×#11758196

—

C₂×#11758196

—

C₂×C₄×Dic₆₁

—

#14519688

—

C₄×#92418

—

C₄×#92418

Q₈

yes

—

#14519684

—

#14519684

—

#14519684

—

C₂²×C₈×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

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×Q₈

yes

—

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

M₁₆

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

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

#104

yes

—

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

#131

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₆₁

#137

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#104

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#446

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

100

Dic₆₁

—

101

C₄×#125

—

102

#1842

D₆₁

yes

characteristic

103

C₄×#97

—

104

#1279

—

105

C₂×C₆₁

#1483

yes

—

106

C₂×C₆₁

#1489

yes

—

107

C₂×C₆₁

#1985

yes

characteristic

108

C₂×C₆₁

#1992

yes

characteristic

109

C₂×C₆₁

#1994

yes

characteristic

110

C₂×#125

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄⋊C₈

C₂×D₆₁

yes

—

113

C₂×#97

—

122

—

114

C₂×C₄×C₈

—

115

#389

—

116

#387

—

122

—

117

#382

—

118

C₄×C₄⋊C₄

—

119

C₄×C₄⋊C₄

—

122

—

120

C₄³

C₂×D₆₁

yes

characteristic

121

C₆₁

#9876

yes

characteristic, complemented, Sylow

122

C₂²×C₈

—

123

C₂²×C₈

—

122

—

124

#125

—

122

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

122

—

128

C₂×C₄⋊C₄

—

122

—

129

C₂×C₄²

—

122

—

130

C₂×C₄²

C₄×D₆₁

yes

characteristic

131

C₂×C₄²

C₄×D₆₁

yes

characteristic

132

C₂×C₄²

D₂₄₄

yes

—

133

C₂×C₄²

C₂²⋊D₆₁

yes

—

134

C₂×C₄²

C₂²×D₆₁

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#97

—

122

—

138

C₄×C₈

—

139

C₄×C₈

—

122

—

140

#95

—

122

—

141

#95

—

122

—

142

#95

—

122

—

143

C₂²×C₄

—

122

—

144

C₂²×C₄

—

122

—

145

C₂²×C₄

#11758198

yes

—

146

C₂²×C₄

#11758198

yes

—

147

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

148

C₂²×C₄

C₂×D₂₄₄

yes

characteristic

149

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

122

—

153

C₂×C₈

—

122

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

122

—

156

C₄⋊C₄

—

122

—

157

C₄⋊C₄

—

122

—

158

C₄⋊C₄

—

244

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

244

—

165

C₄²

—

244

—

166

C₄²

#11758224

yes

—

167

C₄²

Q₈×D₆₁

yes

—

168

C₂³

C₂×#11758198

yes

characteristic

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

122

—

176

C₂×C₄

—

244

—

177

C₂×C₄

—

244

—

178

C₂×C₄

—

244

—

179

C₂×C₄

—

244

—

180

C₂×C₄

—

244

—

181

C₂×C₄

#14519703

yes

—

182

C₂×C₄

#14519742

yes

—

183

C₂×C₄

#14519757

yes

—

184

C₂×C₄

#14519764

yes

characteristic

185

C₂×C₄

#14519775

yes

characteristic

186

C₂×C₄

M₁₆×D₆₁

yes

—

187

C₂×C₄

#14519783

yes

—

188

C₂×C₄

#14519803

yes

—

189

C₂×C₄

#14519808

yes

—

190

C₈

—

191

C₈

—

122

—

192

C₂²

#16839667

yes

characteristic

193

C₂²

#16839725

yes

characteristic

194

C₂²

#16839728

yes

characteristic

195

C₂²

#16839761

yes

—

196

C₂²

C₂×#14519703

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

488

—

201

C₄

—

244

—

202

C₄

#16839891

yes

—

203

C₄

#16840126

yes

characteristic

204

C₄

#16840130

yes

characteristic

205

C₄

#16840168

yes

—

206

C₂

#21444230

yes

—

207

C₂

#21444256

yes

—

208

C₂

#21446758

yes

characteristic

209

C₂

#21446787

yes

characteristic

210

C₂

#21446795

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1842	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1842	D₆₁
hypocenter	C₆₁	#9876
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₆₁

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1842

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
121. C₆₁	#9876	semidirect

H ≅ G/N

type

121. C₆₁

#9876

semidirect

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

subgroup	index
C₄×#14519769	2
C₄×#14519688	2
#21443658	2
C₆₁×#1842	2
#9876	61

subgroup

index

C₆₁×#1842

Subgroups — #29580043

Conjugacy classes of subgroups

Complements

Maximal subgroups