Group #29596492 — subgroups

Subgroups — #29596492

Order 15616.

370 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 131 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	#21458922	C₂	1	1	yes	maximal, characteristic
2	#21447584	C₂	1	2	yes	maximal, complemented
3	#21445619	C₂	1	1	yes	maximal, characteristic
4	#21443907	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2113	C₂	1	1	yes	maximal, characteristic
6	#16840179	—	2	4	no	—
7	C₂×#14519717	—	2	2	no	—
8	C₂×#14519717	C₂²	1	2	yes	—
9	#16839746	C₄	1	1	yes	characteristic
10	#16839736	C₄	1	1	yes	characteristic
11	#16839717	C₂²	1	2	yes	—
12	#16839582	—	2	4	no	—
13	#16839533	C₄	1	2	yes	—
14	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×C₂²⋊C₁₆	—	2	2	no	—
18	C₂×#11758194	C₂×C₄	1	2	yes	—
19	#14519736	C₂×C₄	1	2	yes	—
20	#14519717	—	2	4	no	—
21	#14519717	—	2	2	no	—
22	#14519717	D₄	1	4	yes	—
23	C₂×#11758185	—	2	4	no	—
24	C₂×#11758185	—	2	2	no	—
25	#14519696	C₈	1	2	yes	—
26	#14519694	C₈	1	2	yes	—
27	#14519692	—	2	2	no	—
28	#14519687	—	2	2	no	—
29	C₈×Dic₆₁	—	2	2	no	—
30	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
31	#131×C₆₁	—	2	2	no	—
32	#131×C₆₁	—	2	2	no	—
33	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₆₁	Q₈	1	2	yes	—
35	C₂²×C₈×C₆₁	D₄	1	2	yes	—
36	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	M₃₂×C₆₁	—	2	2	no	—
38	C₂×C₁₆×C₆₁	—	2	1	no	—
39	C₂×C₁₆×C₆₁	—	2	1	no	—
40	C₂×C₁₆×C₆₁	—	2	2	no	—
41	#41×C₆₁	—	2	2	no	—
42	#41×C₆₁	—	2	2	no	—
43	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
44	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₆₁	M₁₆	1	2	yes	—
46	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₆₁	—	2	2	no	—
49	M₁₆×C₆₁	—	2	4	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₆₁	#41	1	2	yes	—
57	C₂²⋊Dic₆₁	—	2	2	no	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄×Dic₆₁	—	2	2	no	—
60	#11758194	—	2	2	no	—
61	#11758194	C₂×C₈	1	4	yes	—
62	C₂×#92418	—	2	2	no	—
63	C₁₆×C₆₁	—	4	1	no	—
64	C₁₆×C₆₁	—	4	1	no	—
65	#11758185	—	2	4	no	—
66	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
67	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
68	Q₈×C₆₁	—	2	2	no	—
69	D₄×C₆₁	—	2	4	no	—
70	D₄×C₆₁	—	2	2	no	—
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#131	1	2	yes	—
78	C₂×Dic₆₁	—	4	2	no	—
79	C₈×C₆₁	—	2	2	no	—
80	C₈×C₆₁	#120	1	1	yes	characteristic
81	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
82	C₈×C₆₁	#123	1	1	yes	characteristic
83	#92418	—	2	4	no	—
84	#12916	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	—	2	2	no	—
86	C₂²×C₆₁	—	4	1	no	—
87	C₂²×C₆₁	—	4	2	no	—
88	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₆₁	#435	1	2	yes	—
90	C₄×C₆₁	—	2	2	no	—
91	C₄×C₆₁	#385	1	1	yes	characteristic
92	C₄×C₆₁	#424	1	1	yes	characteristic
93	C₄×C₆₁	C₈×D₄	1	2	yes	—
94	Dic₆₁	—	8	2	no	—
95	#3407	—	61	1	no	—
96	#2153	—	61	2	no	—
97	#2113	D₆₁	1	1	yes	characteristic
98	#1757	—	61	1	no	—
99	#1379	—	61	2	no	—
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	—	2	1	no	—
102	C₂×C₆₁	—	4	2	no	—
103	C₂×C₆₁	#1850	1	1	yes	characteristic
104	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
105	#504	—	122	4	no	—
106	C₂×M₃₂	—	61	2	no	—
107	C₂×M₃₂	—	122	2	no	—
108	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
109	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
110	#413	—	61	1	no	—
111	#410	—	61	1	no	—
112	#405	—	61	2	no	—
113	#364	—	122	4	no	—
114	C₂²⋊C₁₆	—	2	2	no	—
115	#344	—	61	2	no	—
116	C₆₁	#12916	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	C₄×D₆₁	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	—	61	2	no	—
121	C₂×M₁₆	Dic₁₂₂	1	2	yes	—
122	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
124	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
125	#117	—	61	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	61	4	no	—
128	M₃₂	—	122	4	no	—
129	M₃₂	—	122	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	122	4	no	—
134	C₂×C₁₆	—	122	2	no	—
135	#101	—	61	2	no	—
136	#100	—	61	2	no	—
137	#99	—	122	2	no	—
138	#97	—	122	2	no	—
139	C₄×C₈	—	122	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₈×D₆₁	1	1	yes	characteristic
143	C₂×D₄	C₈×D₆₁	1	1	yes	characteristic
144	C₂×D₄	#11758189	1	2	yes	—
145	C₂²×C₄	#11758196	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	61	4	no	—
150	M₁₆	—	122	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	122	2	no	—
154	C₂×C₈	#11758196	1	2	yes	—
155	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
156	C₂×C₈	D₄×D₆₁	1	2	yes	—
157	C₂×C₈	#11758224	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
159	C₄⋊C₄	—	122	2	no	—
160	C₂²⋊C₄	—	122	2	no	—
161	C₄²	—	122	2	no	—
162	C₁₆	—	4	1	no	—
163	C₁₆	—	4	1	no	—
164	C₁₆	—	122	4	no	—
165	C₂³	#14519706	1	2	yes	—
166	C₂³	C₂×#11758189	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	244	2	no	—
173	C₂×C₄	#14519706	1	2	yes	—
174	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
175	C₂×C₄	#14519737	1	2	yes	—
176	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
177	C₂×C₄	#14519785	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	122	4	no	—
180	C₈	#14519745	1	1	yes	characteristic
181	C₈	#14519754	1	2	yes	—
182	C₈	#14519758	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×#14519706	1	1	yes	characteristic
187	C₂²	#16839830	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	488	2	no	—
190	C₄	#16839634	1	1	yes	characteristic
191	C₄	#16839790	1	1	yes	characteristic
192	C₄	#16839825	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#21445996	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21458922

C₂

yes

maximal, characteristic

#21447584

C₂

yes

maximal, complemented

#21445619

C₂

yes

maximal, characteristic

#21443907

C₂

yes

maximal, complemented

C₆₁×#2113

C₂

yes

maximal, characteristic

#16840179

—

C₂×#14519717

—

C₂×#14519717

C₂²

yes

—

#16839746

C₄

yes

characteristic

#16839736

C₄

yes

characteristic

#16839717

C₂²

yes

—

#16839582

—

#16839533

C₄

yes

—

C₂×#131×C₆₁

C₂²

yes

characteristic

C₂×M₃₂×C₆₁

C₂²

yes

characteristic

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×C₂²⋊C₁₆

—

C₂×#11758194

C₂×C₄

yes

—

#14519736

C₂×C₄

yes

—

#14519717

—

#14519717

—

#14519717

D₄

yes

—

C₂×#11758185

—

C₂×#11758185

—

#14519696

C₈

yes

—

#14519694

C₈

yes

—

#14519692

—

#14519687

—

C₈×Dic₆₁

—

C₂×#41×C₆₁

C₂×C₄

yes

characteristic

#131×C₆₁

—

#131×C₆₁

—

C₂×M₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₁

Q₈

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₈

yes

characteristic

C₂×D₄×C₆₁

C₂×C₈

yes

characteristic

C₂×D₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₁

#41

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

#11758194

—

#11758194

C₂×C₈

yes

—

C₂×#92418

—

C₁₆×C₆₁

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₄⋊C₈

yes

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#120

yes

characteristic

C₈×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₆₁

#435

yes

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

C₈×D₄

yes

—

Dic₆₁

—

#3407

—

#2153

—

#2113

D₆₁

yes

characteristic

#1757

—

#1379

—

100

C₂×C₆₁

—

101

C₂×C₆₁

—

102

C₂×C₆₁

—

103

C₂×C₆₁

#1850

yes

characteristic

104

C₂×#131

C₂×D₆₁

yes

characteristic

105

#504

—

122

—

106

C₂×M₃₂

—

107

C₂×M₃₂

—

122

—

108

C₂×M₃₂

C₂×D₆₁

yes

characteristic

109

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

110

#413

—

111

#410

—

112

#405

—

113

#364

—

122

—

114

C₂²⋊C₁₆

—

115

#344

—

116

C₆₁

#12916

yes

characteristic, complemented, Sylow

117

C₂×#41

C₄×D₆₁

yes

characteristic

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

Dic₁₂₂

yes

—

122

C₂×M₁₆

C₄×D₆₁

yes

characteristic

123

C₂²×C₈

C₂²⋊D₆₁

yes

—

124

C₂²×C₈

C₂²×D₆₁

yes

characteristic

125

#117

—

126

M₃₂

—

127

M₃₂

—

128

M₃₂

—

122

—

129

M₃₂

—

122

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

122

—

134

C₂×C₁₆

—

122

—

135

#101

—

136

#100

—

137

#99

—

122

—

138

#97

—

122

—

139

C₄×C₈

—

122

—

140

#41

—

141

#41

—

142

C₂×Q₈

C₈×D₆₁

yes

characteristic

143

C₂×D₄

C₈×D₆₁

yes

characteristic

144

C₂×D₄

#11758189

yes

—

145

C₂²×C₄

#11758196

yes

—

146

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

122

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

122

—

154

C₂×C₈

#11758196

yes

—

155

C₂×C₈

C₂×Dic₁₂₂

yes

characteristic

156

C₂×C₈

D₄×D₆₁

yes

—

157

C₂×C₈

#11758224

yes

—

158

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

159

C₄⋊C₄

—

122

—

160

C₂²⋊C₄

—

122

—

161

C₄²

—

122

—

162

C₁₆

—

163

C₁₆

—

164

C₁₆

—

122

—

165

C₂³

#14519706

yes

—

166

C₂³

C₂×#11758189

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

244

—

173

C₂×C₄

#14519706

yes

—

174

C₂×C₄

C₂×#11758196

yes

characteristic

175

C₂×C₄

#14519737

yes

—

176

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

177

C₂×C₄

#14519785

yes

—

178

C₈

—

179

C₈

—

122

—

180

C₈

#14519745

yes

characteristic

181

C₈

#14519754

yes

—

182

C₈

#14519758

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×#14519706

yes

characteristic

187

C₂²

#16839830

yes

—

188

C₄

—

189

C₄

—

488

—

190

C₄

#16839634

yes

characteristic

191

C₄

#16839790

yes

characteristic

192

C₄

#16839825

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#21445996

yes

characteristic

	subgroup	G/H
center	C₈	#14519745
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#2113	C₂
socle	C₂×C₆₁	#1850
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#2113	D₆₁
hypocenter	C₆₁	#12916
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2113

subgroup

G/H

center

C₈

#14519745

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₆₁

Fitting

C₆₁×#2113

C₂

socle

C₂×C₆₁

#1850

prime radical

C₂×#41×C₆₁

C₂×C₄

cosocle

C₂²×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2113

N	H ≅ G/N	type
2. #21447584	C₂	semidirect
4. #21443907	C₂	semidirect
116. C₆₁	#12916	semidirect

H ≅ G/N

type

2. #21447584

C₂

semidirect

4. #21443907

C₂

semidirect

116. C₆₁

#12916

semidirect

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

subgroup	index
#21458922	2
#21447584	2
#21445619	2
#21443907	2
C₆₁×#2113	2
#12916	61

subgroup

index

C₆₁×#2113

Subgroups — #29596492

Conjugacy classes of subgroups

Complements

Maximal subgroups