Group #29835514 — subgroups

Subgroups — #29835514

Order 15616.

482 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 187 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	#21456331	C₂	1	2	yes	maximal
2	#21448104	C₂	1	4	yes	maximal, complemented
3	#21448103	C₂	1	4	yes	maximal, complemented
4	#21447603	C₂	1	1	yes	maximal, characteristic
5	#21447601	C₂	1	2	yes	maximal
6	C₂×#16839537	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#2158	C₂	1	1	yes	maximal, characteristic
8	#16840315	—	2	2	no	—
9	C₂²×#11758185	C₂²	1	1	yes	characteristic
10	#16840166	C₂²	1	4	yes	—
11	#16840163	C₂²	1	4	yes	—
12	#16840162	—	2	2	no	—
13	#16840159	—	2	2	no	—
14	#16839800	C₂²	1	2	yes	—
15	#16839798	—	2	2	no	—
16	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
17	#16839590	C₂²	1	2	yes	—
18	#16839589	C₂²	1	4	yes	—
19	#16839588	C₂²	1	2	yes	—
20	#16839587	C₂²	1	4	yes	—
21	#16839537	C₂²	1	2	yes	—
22	#16839537	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#427	C₂²	1	2	yes	—
25	C₆₁×#367	C₂²	1	2	yes	—
26	C₆₁×#366	C₂²	1	2	yes	—
27	C₂²×#92418	D₄	1	1	yes	characteristic
28	#14519776	—	2	4	no	—
29	#14519768	—	2	4	no	—
30	#14519754	—	4	4	no	—
31	C₄×Dic₁₂₂	—	4	4	no	—
32	#14519736	—	2	2	no	—
33	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
34	C₂×#11758185	Q₈	1	4	yes	complemented
35	C₂×#11758185	C₂³	1	2	yes	—
36	#14519711	C₂³	1	4	yes	—
37	#14519710	—	2	4	no	—
38	#14519709	—	2	2	no	—
39	#14519708	—	2	4	no	—
40	C₈×Dic₆₁	D₄	1	2	yes	—
41	C₈×Dic₆₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	#117×C₆₁	—	2	2	no	—
44	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
45	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
47	#107×C₆₁	C₂³	1	4	yes	—
48	#106×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₆₁	D₈	1	2	yes	—
51	C₂×Dic₁₂₂	—	4	4	no	—
52	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₆₁	—	2	4	no	—
57	C₂²⋊C₄×C₆₁	—	4	2	no	—
58	C₄²×C₆₁	—	4	2	no	—
59	C₂²⋊Dic₆₁	—	4	2	no	—
60	C₄⋊Dic₆₁	—	2	4	no	—
61	C₄⋊Dic₆₁	—	4	4	no	—
62	#11758196	—	4	4	no	—
63	#11758196	—	4	4	no	—
64	C₄×Dic₆₁	—	4	2	no	—
65	C₄×Dic₆₁	D₈	1	2	yes	—
66	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
67	C₂×#92418	Q₁₆	1	4	yes	—
68	C₂×#92418	C₂×D₄	1	2	yes	—
69	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
70	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
71	#11758185	C₂×Q₈	1	4	yes	—
72	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	—	4	4	no	—
74	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₆₁	—	4	2	no	—
79	C₂×Dic₆₁	—	4	4	no	—
80	C₂×Dic₆₁	C₂×D₈	1	4	yes	—
81	Dic₁₂₂	—	8	8	no	—
82	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
83	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
85	#92418	C₂×Q₁₆	1	4	yes	—
86	#33728	—	61	1	no	maximal, Sylow
87	C₂²×C₆₁	—	2	2	no	—
88	C₂²×C₆₁	#500	1	2	yes	—
89	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
90	C₄×C₆₁	—	8	4	no	—
91	C₄×C₆₁	#500	1	2	yes	—
92	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
94	Dic₆₁	—	2	4	no	—
95	Dic₆₁	—	8	4	no	—
96	#3160	—	61	2	no	—
97	#2255	—	61	4	no	—
98	#2255	—	61	4	no	—
99	#2158	—	61	1	no	—
100	#2158	—	61	2	no	—
101	#2158	D₆₁	1	1	yes	characteristic
102	C₂×#346	—	61	1	no	—
103	C₂×C₆₁	—	2	2	no	—
104	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
105	C₂×C₆₁	#3414	1	2	yes	—
106	#533	—	122	2	no	—
107	C₂²×C₁₆	—	61	1	no	—
108	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
109	#500	—	61	4	no	—
110	#499	—	61	4	no	—
111	#499	—	122	2	no	—
112	#498	—	122	2	no	—
113	#427	—	61	2	no	—
114	#427	—	122	2	no	—
115	#427	C₂×D₆₁	1	2	yes	—
116	C₂×C₄×C₈	—	61	1	no	—
117	#367	—	61	2	no	—
118	#367	—	61	4	no	—
119	#367	C₂×D₆₁	1	2	yes	—
120	#366	—	61	2	no	—
121	#366	—	61	4	no	—
122	#366	C₂×D₆₁	1	2	yes	—
123	#346	—	61	2	no	—
124	#346	—	61	2	no	—
125	C₆₁	#33728	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	61	1	no	—
127	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
128	C₄⋊Q₈	—	122	4	no	—
129	#125	—	122	4	no	—
130	C₂²⋊Q₈	—	244	4	no	—
131	C₄×Q₈	—	244	4	no	—
132	#117	—	2	2	no	—
133	#117	—	122	2	no	—
134	C₂×C₄²	—	61	1	no	—
135	C₂×C₁₆	—	61	4	no	—
136	C₂×C₁₆	—	61	2	no	—
137	C₂×C₁₆	Dic₁₂₂	1	4	yes	complemented
138	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
139	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
140	#107	—	61	4	no	—
141	#107	—	122	4	no	—
142	#107	C₂²×D₆₁	1	4	yes	—
143	#106	—	2	2	no	—
144	#106	—	122	4	no	—
145	#106	—	122	2	no	—
146	C₄×C₈	—	61	2	no	—
147	C₄×C₈	—	61	2	no	—
148	C₂×Q₈	—	244	4	no	—
149	C₂²×C₄	—	61	2	no	—
150	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
151	C₂×C₈	—	61	2	no	—
152	C₂×C₈	—	61	4	no	—
153	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
154	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₆₁	1	2	yes	—
156	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	122	4	no	—
159	C₄⋊C₄	—	244	4	no	—
160	C₄⋊C₄	—	244	4	no	—
161	C₄⋊C₄	—	244	4	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	244	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	61	2	no	—
166	C₄²	—	61	2	no	—
167	C₄²	—	244	2	no	—
168	C₁₆	—	61	4	no	—
169	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
170	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
171	C₂³	D₈×D₆₁	1	1	yes	characteristic
172	Q₈	—	488	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	61	4	no	—
175	C₂×C₄	—	244	2	no	—
176	C₂×C₄	—	244	4	no	—
177	C₂×C₄	#14519776	1	2	yes	—
178	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
180	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
181	C₈	—	61	4	no	—
182	C₈	#14519776	1	2	yes	—
183	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
184	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	#16840166	1	2	yes	—
187	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	122	4	no	—
190	C₄	—	488	4	no	—
191	C₄	#16840166	1	2	yes	—
192	C₄	C₂×#14519776	1	1	yes	characteristic
193	C₄	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₂×#16840166	1	1	yes	characteristic
196	C₂	#21458952	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21456331

C₂

yes

maximal

#21448104

C₂

yes

maximal, complemented

#21448103

C₂

yes

maximal, complemented

#21447603

C₂

yes

maximal, characteristic

#21447601

C₂

yes

maximal

C₂×#16839537

C₂

yes

maximal, characteristic

C₆₁×#2158

C₂

yes

maximal, characteristic

#16840315

—

C₂²×#11758185

C₂²

yes

characteristic

#16840166

C₂²

yes

—

#16840163

C₂²

yes

—

#16840162

—

#16840159

—

#16839800

C₂²

yes

—

#16839798

—

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839590

C₂²

yes

—

#16839589

C₂²

yes

—

#16839588

C₂²

yes

—

#16839587

C₂²

yes

—

#16839537

C₂²

yes

—

#16839537

C₂²

yes

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₆₁×#427

C₂²

yes

—

C₆₁×#367

C₂²

yes

—

C₆₁×#366

C₂²

yes

—

C₂²×#92418

D₄

yes

—

—

—

C₄×Dic₁₂₂

—

#14519736

—

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×#11758185

Q₈

yes

complemented

C₂×#11758185

C₂³

yes

—

#14519711

C₂³

yes

—

#14519710

—

#14519709

—

#14519708

—

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

C₂³

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

—

C₂×C₁₆×C₆₁

Q₈

yes

complemented

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

#107×C₆₁

C₂³

yes

—

#106×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

D₈

yes

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₈

yes

—

C₄×Dic₆₁

C₂×D₄

yes

—

C₂×#92418

Q₁₆

yes

—

C₂×#92418

C₂×D₄

yes

—

C₁₆×C₆₁

C₂×Q₈

yes

—

C₁₆×C₆₁

C₂×Q₈

yes

—

#11758185

C₂×Q₈

yes

—

C₂³×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₁₆

yes

—

C₂×C₄×C₆₁

C₂²×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₈

yes

—

Dic₁₂₂

—

C₈×C₆₁

C₄⋊Q₈

yes

—

C₈×C₆₁

C₂²×Q₈

yes

characteristic

C₈×C₆₁

C₂²×Q₈

yes

characteristic

#92418

C₂×Q₁₆

yes

—

#33728

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

#500

yes

—

C₂²×C₆₁

C₂²×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#500

yes

—

C₄×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

—

—

—

100

#2158

—

101

#2158

D₆₁

yes

characteristic

102

C₂×#346

—

103

C₂×C₆₁

—

104

C₂×C₆₁

C₂×#500

yes

characteristic

105

C₂×C₆₁

#3414

yes

—

106

#533

—

122

—

107

C₂²×C₁₆

—

108

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

109

#500

—

110

#499

—

111

#499

—

122

—

112

#498

—

122

—

113

#427

—

114

#427

—

122

—

115

#427

C₂×D₆₁

yes

—

116

C₂×C₄×C₈

—

117

#367

—

118

#367

—

119

#367

C₂×D₆₁

yes

—

120

#366

—

121

#366

—

122

#366

C₂×D₆₁

yes

—

123

#346

—

124

#346

—

125

C₆₁

#33728

yes

characteristic, complemented, Sylow

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₆₁

yes

characteristic

128

C₄⋊Q₈

—

122

—

129

#125

—

122

—

130

C₂²⋊Q₈

—

244

—

131

C₄×Q₈

—

244

—

132

#117

—

133

#117

—

122

—

134

C₂×C₄²

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

Dic₁₂₂

yes

complemented

138

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

139

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

140

#107

—

141

#107

—

122

—

142

#107

C₂²×D₆₁

yes

—

143

#106

—

144

#106

—

122

—

145

#106

—

122

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₂×Q₈

—

244

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₆₁

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂×Dic₁₂₂

yes

—

154

C₂×C₈

D₄×D₆₁

yes

characteristic

155

C₂×C₈

Q₈×D₆₁

yes

—

156

C₂×C₈

C₂³×D₆₁

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

122

—

159

C₄⋊C₄

—

244

—

160

C₄⋊C₄

—

244

—

161

C₄⋊C₄

—

244

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

244

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

244

—

168

C₁₆

—

169

C₁₆

C₂×Dic₁₂₂

yes

—

170

C₁₆

C₂×Dic₁₂₂

yes

—

171

C₂³

D₈×D₆₁

yes

characteristic

172

Q₈

—

488

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

244

—

176

C₂×C₄

—

244

—

177

C₂×C₄

#14519776

yes

—

178

C₂×C₄

D₈×D₆₁

yes

characteristic

179

C₂×C₄

Q₁₆×D₆₁

yes

—

180

C₂×C₄

C₂×D₄×D₆₁

yes

characteristic

181

C₈

—

182

C₈

#14519776

yes

—

183

C₈

C₂×Q₈×D₆₁

yes

characteristic

184

C₈

C₂²×Dic₁₂₂

yes

characteristic

185

C₂²

—

186

C₂²

#16840166

yes

—

187

C₂²

C₂×D₈×D₆₁

yes

characteristic

188

C₄

—

189

C₄

—

122

—

190

C₄

—

488

—

191

C₄

#16840166

yes

—

192

C₄

C₂×#14519776

yes

characteristic

193

C₄

C₂×Q₁₆×D₆₁

yes

characteristic

194

C₂

—

195

C₂

C₂×#16840166

yes

characteristic

196

C₂

#21458952

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2158	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#2158	D₆₁
hypocenter	C₆₁	#33728
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₆₁

commutator, cosocle

C₂×C₈×C₆₁

C₂⁴

Frattini

C₂×C₈

C₂³×D₆₁

Fitting

C₆₁×#2158

C₂

socle

C₂²×C₆₁

C₂²×D₈

prime radical

C₂³×C₆₁

C₂×D₈

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
2. #21448104	C₂	semidirect
3. #21448103	C₂	semidirect
34. C₂×#11758185	Q₈	semidirect
44. C₂×C₁₆×C₆₁	Q₈	semidirect
125. C₆₁	#33728	semidirect
137. C₂×C₁₆	Dic₁₂₂	semidirect

H ≅ G/N

type

2. #21448104

C₂

semidirect

3. #21448103

C₂

semidirect

34. C₂×#11758185

Q₈

semidirect

44. C₂×C₁₆×C₆₁

Q₈

semidirect

125. C₆₁

#33728

semidirect

137. C₂×C₁₆

Dic₁₂₂

semidirect

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

subgroup	index
#21456331	2
#21448104	2
#21448103	2
#21447603	2
#21447601	2
C₂×#16839537	2
C₆₁×#2158	2
#33728	61

subgroup

index

C₆₁×#2158

Subgroups — #29835514

Conjugacy classes of subgroups

Complements

Maximal subgroups