Group #29598535 — subgroups

Subgroups — #29598535

Order 15616.

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

normal subgroups only

This group has 386 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	C₂×C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
3	#21443896	C₂	1	4	yes	maximal
4	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₂₂	C₄	1	1	yes	characteristic
6	C₂²×#11758185	C₂²	1	2	yes	—
7	C₈×Dic₁₂₂	—	2	2	no	—
8	C₈×Dic₁₂₂	—	2	2	no	—
9	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
10	C₂×#14519706	C₄	1	2	yes	—
11	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
12	#16839538	—	2	2	no	—
13	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×C₄⋊C₁₆	—	2	2	no	—
16	C₂²×Dic₁₂₂	C₈	1	1	yes	characteristic
17	C₂²×#92418	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₂₂	—	2	2	no	—
19	C₄×Dic₁₂₂	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
21	C₂×#11758196	C₈	1	2	yes	—
22	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
23	C₂×#11758185	—	2	2	no	—
24	C₂×#11758185	—	2	2	no	—
25	C₂×#11758185	Q₈	1	4	yes	complemented
26	C₄⋊#92418	—	2	2	no	—
27	#14519706	—	2	4	no	—
28	C₈×Dic₆₁	—	2	2	no	—
29	C₈×Dic₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₁	—	2	2	no	—
34	C₂×C₁₆×C₆₁	—	2	4	no	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
41	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
42	C₂×Dic₁₂₂	—	2	2	no	—
43	C₂×Dic₁₂₂	—	2	4	no	—
44	C₂×C₈×C₆₁	—	2	2	no	—
45	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₆₁	#41	1	2	yes	—
50	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₆₁	M₁₆	1	2	yes	—
52	C₄⋊Dic₆₁	—	2	2	no	—
53	#11758196	—	2	4	no	—
54	C₄×Dic₆₁	—	2	2	no	—
55	C₄×Dic₆₁	—	2	2	no	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	—	2	4	no	—
58	C₁₆×C₆₁	—	4	2	no	—
59	#11758185	—	2	4	no	—
60	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#131	1	2	yes	—
69	C₂×Dic₆₁	—	2	2	no	—
70	C₂×Dic₆₁	—	2	2	no	—
71	C₂×Dic₆₁	—	2	2	no	—
72	C₂×Dic₆₁	—	2	4	no	—
73	C₂×Dic₆₁	M₃₂	1	4	yes	—
74	Dic₁₂₂	—	2	8	no	—
75	C₈×C₆₁	—	2	2	no	—
76	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₆₁	#124	1	1	yes	characteristic
78	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
79	#92418	—	4	2	no	—
80	#13264	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₂²×C₆₁	C₈×Q₈	1	2	yes	—
83	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
84	C₂²×C₆₁	#504	1	2	yes	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	2	2	no	—
87	C₄×C₆₁	#391	1	1	yes	characteristic
88	C₄×C₆₁	#408	1	1	yes	characteristic
89	C₄×C₆₁	#446	1	2	yes	—
90	Dic₆₁	—	2	4	no	—
91	Dic₆₁	—	4	2	no	—
92	C₂×C₈×Q₈	—	61	1	no	—
93	C₂×C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
94	C₂×#346	—	61	1	no	—
95	#1374	—	61	4	no	—
96	C₂×C₆₁	#1984	1	1	yes	characteristic
97	C₂×C₆₁	#2112	1	1	yes	characteristic
98	C₂×C₆₁	#2113	1	1	yes	characteristic
99	C₂×C₆₁	#2183	1	4	yes	—
100	C₂×C₄×Q₈	—	61	1	no	—
101	C₂²×C₁₆	—	61	2	no	—
102	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
103	C₈×Q₈	—	122	2	no	—
104	C₈×Q₈	—	122	2	no	—
105	C₂×C₄⋊C₈	—	61	2	no	—
106	C₂×C₄⋊C₈	—	61	1	no	—
107	C₂×C₄×C₈	—	61	2	no	—
108	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
109	C₄⋊C₁₆	—	2	2	no	—
110	#346	—	122	2	no	—
111	C₆₁	#13264	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	61	1	no	—
113	C₂²×C₈	—	61	2	no	—
114	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
116	C₄×Q₈	—	122	2	no	—
117	C₄×Q₈	—	122	2	no	—
118	C₂×C₄⋊C₄	—	61	2	no	—
119	C₂×C₄⋊C₄	—	61	1	no	—
120	C₂×C₄²	—	61	2	no	—
121	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
122	C₂×C₁₆	—	2	4	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	61	4	no	—
125	C₂×C₁₆	—	122	2	no	—
126	C₂×C₁₆	—	122	2	no	—
127	C₄⋊C₈	—	122	4	no	—
128	C₄⋊C₈	—	122	2	no	—
129	C₄×C₈	—	122	2	no	—
130	C₄×C₈	—	122	2	no	—
131	C₄×C₈	D₂₄₄	1	2	yes	—
132	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
133	C₂×Q₈	—	122	4	no	—
134	C₂×Q₈	—	122	2	no	—
135	C₂²×C₄	—	61	2	no	—
136	C₂²×C₄	—	61	2	no	—
137	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
138	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	122	2	no	—
142	C₂×C₈	—	122	4	no	—
143	C₂×C₈	#11758198	1	2	yes	—
144	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
145	C₂×C₈	#11758224	1	2	yes	—
146	C₂×C₈	Q₈×D₆₁	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
148	C₄⋊C₄	—	122	4	no	—
149	C₄⋊C₄	—	122	2	no	—
150	C₄²	—	122	2	no	—
151	C₄²	—	122	2	no	—
152	C₄²	#11758189	1	2	yes	—
153	C₄²	#11758198	1	2	yes	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	122	4	no	—
156	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
157	Q₈	—	122	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	61	4	no	—
161	C₂×C₄	—	122	2	no	—
162	C₂×C₄	—	122	2	no	—
163	C₂×C₄	—	122	2	no	—
164	C₂×C₄	—	122	4	no	—
165	C₂×C₄	#14519689	1	2	yes	—
166	C₂×C₄	#14519689	1	2	yes	—
167	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
168	C₂×C₄	#14519742	1	2	yes	—
169	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
170	C₂×C₄	#14519785	1	2	yes	—
171	C₈	—	2	2	no	—
172	C₈	—	244	2	no	—
173	C₈	#14519757	1	2	yes	—
174	C₈	#14519764	1	1	yes	characteristic
175	C₈	#14519775	1	1	yes	characteristic
176	C₂²	C₂×#14519689	1	1	yes	characteristic
177	C₂²	#16839886	1	2	yes	—
178	C₂²	M₃₂×D₆₁	1	2	yes	—
179	C₂²	#16840178	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	122	4	no	—
183	C₄	—	244	2	no	—
184	C₄	#16839667	1	1	yes	characteristic
185	C₄	#16839730	1	1	yes	characteristic
186	C₄	#16839889	1	2	yes	—
187	C₂	#21446754	1	1	yes	characteristic
188	C₂	#21447429	1	1	yes	characteristic
189	C₂	#21447432	1	1	yes	characteristic
190	C₂	#21447726	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₂₂

C₂

yes

maximal, characteristic

C₂×#16839538

C₂

yes

maximal, characteristic

#21443896

C₂

yes

maximal

C₂×C₆₁×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₂₂

C₄

yes

characteristic

C₂²×#11758185

C₂²

yes

—

C₈×Dic₁₂₂

—

C₈×Dic₁₂₂

—

C₂×C₄⋊#92418

C₄

yes

characteristic

C₂×#14519706

C₄

yes

—

C₂×C₈×Dic₆₁

C₂²

yes

—

#16839538

—

C₂²×C₁₆×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₂²×Dic₁₂₂

C₈

yes

characteristic

C₂²×#92418

C₂×C₄

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

—

C₂×C₄⋊Dic₆₁

C₈

yes

characteristic

C₂×#11758196

C₈

yes

—

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

C₂×#11758185

—

C₂×#11758185

—

C₂×#11758185

Q₈

yes

complemented

C₄⋊#92418

—

#14519706

—

C₈×Dic₆₁

—

C₈×Dic₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

C₂×C₈

yes

—

C₂²×Dic₆₁

C₂×C₈

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×Q₈

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

#131

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

M₃₂

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂²⋊Q₈

yes

—

C₈×C₆₁

#124

yes

characteristic

C₈×C₆₁

C₄⋊Q₈

yes

—

—

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₁

C₈×Q₈

yes

—

C₂²×C₆₁

C₂×M₃₂

yes

—

C₂²×C₆₁

#504

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

Dic₆₁

—

C₂×C₈×Q₈

—

C₂×C₄⋊C₁₆

D₆₁

yes

characteristic

C₂×#346

—

#1374

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

100

C₂×C₄×Q₈

—

101

C₂²×C₁₆

—

102

C₂²×C₁₆

C₂×D₆₁

yes

—

103

C₈×Q₈

—

122

—

104

C₈×Q₈

—

122

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

109

C₄⋊C₁₆

—

110

#346

—

122

—

111

C₆₁

#13264

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₆₁

yes

characteristic

115

C₂²×C₈

C₂²×D₆₁

yes

characteristic

116

C₄×Q₈

—

122

—

117

C₄×Q₈

—

122

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₄×D₆₁

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

122

—

126

C₂×C₁₆

—

122

—

127

C₄⋊C₈

—

122

—

128

C₄⋊C₈

—

122

—

129

C₄×C₈

—

122

—

130

C₄×C₈

—

122

—

131

C₄×C₈

D₂₄₄

yes

—

132

C₄×C₈

C₂²⋊D₆₁

yes

—

133

C₂×Q₈

—

122

—

134

C₂×Q₈

—

122

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₈×D₆₁

yes

characteristic

138

C₂²×C₄

C₈×D₆₁

yes

characteristic

139

C₂²×C₄

C₂×C₄×D₆₁

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

122

—

142

C₂×C₈

—

122

—

143

C₂×C₈

#11758198

yes

—

144

C₂×C₈

C₂×D₂₄₄

yes

characteristic

145

C₂×C₈

#11758224

yes

—

146

C₂×C₈

Q₈×D₆₁

yes

—

147

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

148

C₄⋊C₄

—

122

—

149

C₄⋊C₄

—

122

—

150

C₄²

—

122

—

151

C₄²

—

122

—

152

C₄²

#11758189

yes

—

153

C₄²

#11758198

yes

—

154

C₁₆

—

155

C₁₆

—

122

—

156

C₂³

C₂×C₈×D₆₁

yes

characteristic

157

Q₈

—

122

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

122

—

162

C₂×C₄

—

122

—

163

C₂×C₄

—

122

—

164

C₂×C₄

—

122

—

165

C₂×C₄

#14519689

yes

—

166

C₂×C₄

#14519689

yes

—

167

C₂×C₄

C₂×#11758198

yes

characteristic

168

C₂×C₄

#14519742

yes

—

169

C₂×C₄

C₂×#11758189

yes

characteristic

170

C₂×C₄

#14519785

yes

—

171

C₈

—

172

C₈

—

244

—

173

C₈

#14519757

yes

—

174

C₈

#14519764

yes

characteristic

175

C₈

#14519775

yes

characteristic

176

C₂²

C₂×#14519689

yes

characteristic

177

C₂²

#16839886

yes

—

178

C₂²

M₃₂×D₆₁

yes

—

179

C₂²

#16840178

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

122

—

183

C₄

—

244

—

184

C₄

#16839667

yes

characteristic

185

C₄

#16839730

yes

characteristic

186

C₄

#16839889

yes

—

187

C₂

#21446754

yes

characteristic

188

C₂

#21447429

yes

characteristic

189

C₂

#21447432

yes

characteristic

190

C₂

#21447726

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂²×C₈
Fitting	C₂×C₆₁×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	#13264
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₆₁

commutator, socle, prime radical

C₂³×C₆₁

C₂²×C₈

Fitting

C₂×C₆₁×C₄⋊C₁₆

C₂

cosocle

C₂²×C₈×C₆₁

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
25. C₂×#11758185	Q₈	semidirect
111. C₆₁	#13264	semidirect

H ≅ G/N

type

25. C₂×#11758185

Q₈

semidirect

111. C₆₁

#13264

semidirect

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

subgroup	index
C₂×C₈×Dic₁₂₂	2
C₂×#16839538	2
#21443896	2
C₂×C₆₁×C₄⋊C₁₆	2
#13264	61

subgroup

index

C₂×C₈×Dic₁₂₂

C₂×#16839538

#21443896

C₂×C₆₁×C₄⋊C₁₆

#13264

Subgroups — #29598535

Conjugacy classes of subgroups

Complements

Maximal subgroups