Group #36472591 — subgroups

Subgroups — #36472591

Order 22784.

450 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 147 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	#26028965	C₂	1	1	yes	maximal, characteristic
2	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151381	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15668018	C₂²	1	2	yes	—
6	#19152148	C₄	1	1	yes	characteristic
7	C₂×#15667952	C₂²	1	2	yes	—
8	#19151677	—	2	2	no	—
9	C₈×D₃₅₆	—	2	2	no	—
10	#19151622	C₄	1	1	yes	characteristic
11	#19151564	C₄	1	2	yes	—
12	C₄⋊#13083549	—	2	2	no	—
13	#19151381	—	2	8	no	—
14	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₈₉	—	2	2	no	—
17	C₂×#13083574	C₂×C₄	1	2	yes	—
18	#15668018	—	2	4	no	—
19	#15668018	—	2	4	no	—
20	C₂×#13083558	C₂×C₄	1	2	yes	—
21	C₂×#13083557	—	2	2	no	—
22	C₂×C₈×D₈₉	—	2	2	no	—
23	#15668012	C₈	1	1	yes	characteristic
24	#15668009	C₈	1	1	yes	characteristic
25	#15668000	C₈	1	2	yes	—
26	#15667986	—	2	2	no	—
27	C₄×D₃₅₆	—	2	2	no	—
28	#15667952	—	2	4	no	—
29	C₂×#13083549	—	2	2	no	—
30	C₄⋊#10623494	—	2	2	no	—
31	#15667924	—	2	4	no	—
32	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
38	C₄×C₈×C₈₉	D₄	1	2	yes	—
39	C₄×C₈×C₈₉	D₄	1	2	yes	—
40	#13083574	—	2	4	no	—
41	#13083574	—	2	4	no	—
42	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
43	C₂×D₃₅₆	—	2	2	no	—
44	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
45	C₂×C₂²⋊D₈₉	—	2	2	no	—
46	C₂×C₄×D₈₉	—	2	2	no	—
47	#13083558	—	2	4	no	—
48	#13083557	—	4	4	no	—
49	C₂×#10623494	—	2	2	no	—
50	C₈×D₈₉	—	4	2	no	—
51	C₈×D₈₉	—	4	2	no	—
52	C₄⋊Dic₈₉	—	2	2	no	—
53	#13083551	—	2	4	no	—
54	#13083549	—	4	2	no	—
55	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	#41	1	2	yes	—
63	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
66	C₁₆×C₈₉	—	2	4	no	—
67	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×D₈₉	—	2	2	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#131	1	2	yes	—
76	C₂²⋊D₈₉	—	4	4	no	—
77	C₂×Dic₈₉	—	2	2	no	—
78	D₃₅₆	—	2	4	no	—
79	D₃₅₆	—	4	4	no	—
80	C₄×D₈₉	—	4	2	no	—
81	C₄×D₈₉	—	4	2	no	—
82	Dic₁₇₈	—	2	4	no	—
83	C₈×C₈₉	—	2	2	no	—
84	C₈×C₈₉	#121	1	2	yes	—
85	C₈×C₈₉	#124	1	1	yes	characteristic
86	C₈×C₈₉	#127	1	1	yes	characteristic
87	#10623494	—	4	2	no	—
88	C₂²×C₈₉	—	2	2	no	—
89	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
90	C₂²×C₈₉	#436	1	2	yes	—
91	C₂×D₈₉	—	4	4	no	—
92	C₂×D₈₉	—	4	2	no	—
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	#350	1	4	yes	complemented
95	C₄×C₈₉	#390	1	1	yes	characteristic
96	C₄×C₈₉	#408	1	1	yes	characteristic
97	C₄×C₈₉	C₈×D₄	1	2	yes	—
98	Dic₈₉	—	4	2	no	—
99	#13191	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	—	2	2	no	—
101	C₂×C₈₉	#1954	1	1	yes	characteristic
102	C₂×C₈₉	C₂×#350	1	2	yes	—
103	D₈₉	—	8	2	no	—
104	#2931	—	89	1	no	—
105	#2150	—	89	1	no	—
106	C₂×#350	—	89	4	no	—
107	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
108	C₈₉	#13191	1	1	yes	characteristic, complemented, Sylow
109	C₂×#131	—	89	2	no	—
110	#532	—	89	1	no	—
111	C₂×M₃₂	—	89	2	no	—
112	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
113	#436	—	178	2	no	—
114	C₈×D₄	—	178	2	no	—
115	#423	—	89	1	no	—
116	#408	—	89	2	no	—
117	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
118	C₄⋊C₁₆	—	178	2	no	—
119	#350	—	178	8	no	—
120	C₄×C₁₆	—	2	2	no	—
121	C₂×#41	—	89	2	no	—
122	#131	—	178	4	no	—
123	#131	—	178	4	no	—
124	C₂×M₁₆	—	89	2	no	—
125	C₂×M₁₆	—	178	2	no	—
126	C₂²×C₈	—	178	2	no	—
127	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
129	C₄⋊Q₈	—	89	1	no	—
130	#127	—	89	1	no	—
131	#124	—	89	2	no	—
132	#121	—	178	2	no	—
133	C₄×D₄	—	178	2	no	—
134	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
135	M₃₂	—	178	4	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	178	2	no	—
139	C₂×C₁₆	D₃₅₆	1	4	yes	complemented
140	C₄⋊C₈	—	178	2	no	—
141	C₂²⋊C₈	—	178	4	no	—
142	C₄×C₈	D₃₅₆	1	2	yes	—
143	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
144	#41	—	178	4	no	—
145	#41	—	178	4	no	—
146	C₂×Q₈	—	89	2	no	—
147	C₂×D₄	—	89	2	no	—
148	C₂×D₄	—	178	2	no	—
149	C₂×D₄	—	178	2	no	—
150	C₂²×C₄	—	178	2	no	—
151	C₂²×C₄	#13083557	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
153	M₁₆	—	356	4	no	—
154	M₁₆	—	178	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	356	2	no	—
157	C₂×C₈	—	356	2	no	—
158	C₂×C₈	—	178	2	no	—
159	C₂×C₈	#13083551	1	2	yes	—
160	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
161	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
162	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
163	C₂×C₈	#13083574	1	2	yes	—
164	C₄⋊C₄	—	178	2	no	—
165	C₂²⋊C₄	—	178	4	no	—
166	C₄²	#13083551	1	2	yes	—
167	C₄²	C₈×D₈₉	1	1	yes	characteristic
168	C₄²	C₈×D₈₉	1	1	yes	characteristic
169	C₁₆	—	2	4	no	—
170	C₁₆	—	356	2	no	—
171	C₂³	—	178	2	no	—
172	C₂³	C₂×#13083557	1	1	yes	characteristic
173	Q₈	—	178	4	no	—
174	D₄	—	356	4	no	—
175	D₄	—	356	4	no	—
176	D₄	—	178	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	356	2	no	—
179	C₂×C₄	—	356	2	no	—
180	C₂×C₄	—	178	2	no	—
181	C₂×C₄	#15667924	1	2	yes	—
182	C₂×C₄	#15667924	1	2	yes	—
183	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
184	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
185	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
186	C₂×C₄	#15668018	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	356	2	no	—
189	C₈	#15667986	1	2	yes	—
190	C₈	#15668000	1	1	yes	characteristic
191	C₈	#15668009	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	356	2	no	—
194	C₂²	—	356	4	no	—
195	C₂²	C₂×#15667924	1	1	yes	characteristic
196	C₂²	#19151677	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	356	2	no	—
199	C₄	#19151381	1	4	yes	complemented
200	C₄	#19151500	1	1	yes	characteristic
201	C₄	#19151564	1	1	yes	characteristic
202	C₄	C₈×D₃₅₆	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	712	2	no	—
205	C₂	#26021046	1	1	yes	characteristic
206	C₂	C₂×#19151381	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26028965

C₂

yes

maximal, characteristic

#26022008

C₂

yes

maximal, characteristic, complemented

C₂×#19151381

C₂

yes

maximal, complemented

C₂×C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668018

C₂²

yes

—

#19152148

C₄

yes

characteristic

C₂×#15667952

C₂²

yes

—

#19151677

—

C₈×D₃₅₆

—

#19151622

C₄

yes

characteristic

#19151564

C₄

yes

—

C₄⋊#13083549

—

#19151381

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

—

C₂×#13083574

C₂×C₄

yes

—

#15668018

—

#15668018

—

C₂×#13083558

C₂×C₄

yes

—

C₂×#13083557

—

C₂×C₈×D₈₉

—

#15668012

C₈

yes

characteristic

#15668009

C₈

yes

characteristic

#15668000

C₈

yes

—

#15667986

—

C₄×D₃₅₆

—

—

—

—

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

complemented

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

C₂×C₈

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂×C₈

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

#13083558

—

#13083557

—

C₂×#10623494

—

C₈×D₈₉

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083549

—

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

#131

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

#121

yes

—

C₈×C₈₉

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₈₉

#436

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₈×D₄

yes

—

Dic₈₉

—

#13191

—

maximal, Sylow

100

C₂×C₈₉

—

101

C₂×C₈₉

#1954

yes

characteristic

102

C₂×C₈₉

C₂×#350

yes

—

103

D₈₉

—

104

#2931

—

105

#2150

—

106

C₂×#350

—

107

C₂×C₄×C₁₆

D₈₉

yes

characteristic, complemented

108

C₈₉

#13191

yes

characteristic, complemented, Sylow

109

C₂×#131

—

110

#532

—

111

C₂×M₃₂

—

112

C₂²×C₁₆

C₂×D₈₉

yes

—

113

#436

—

178

—

114

C₈×D₄

—

178

—

115

#423

—

116

#408

—

117

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

118

C₄⋊C₁₆

—

178

—

119

#350

—

178

—

120

C₄×C₁₆

—

121

C₂×#41

—

122

#131

—

178

—

123

#131

—

178

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

178

—

126

C₂²×C₈

—

178

—

127

C₂²×C₈

C₄×D₈₉

yes

characteristic

128

C₂²×C₈

C₂²×D₈₉

yes

characteristic

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

178

—

133

C₄×D₄

—

178

—

134

C₂×C₄²

C₄×D₈₉

yes

characteristic

135

M₃₂

—

178

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

178

—

139

C₂×C₁₆

D₃₅₆

yes

complemented

140

C₄⋊C₈

—

178

—

141

C₂²⋊C₈

—

178

—

142

C₄×C₈

D₃₅₆

yes

—

143

C₄×C₈

C₂²⋊D₈₉

yes

—

144

#41

—

178

—

145

#41

—

178

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

178

—

149

C₂×D₄

—

178

—

150

C₂²×C₄

—

178

—

151

C₂²×C₄

#13083557

yes

—

152

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

153

M₁₆

—

356

—

154

M₁₆

—

178

—

155

C₂×C₈

—

156

C₂×C₈

—

356

—

157

C₂×C₈

—

356

—

158

C₂×C₈

—

178

—

159

C₂×C₈

#13083551

yes

—

160

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

161

C₂×C₈

C₂×D₃₅₆

yes

—

162

C₂×C₈

C₂×D₃₅₆

yes

characteristic

163

C₂×C₈

#13083574

yes

—

164

C₄⋊C₄

—

178

—

165

C₂²⋊C₄

—

178

—

166

C₄²

#13083551

yes

—

167

C₄²

C₈×D₈₉

yes

characteristic

168

C₄²

C₈×D₈₉

yes

characteristic

169

C₁₆

—

170

C₁₆

—

356

—

171

C₂³

—

178

—

172

C₂³

C₂×#13083557

yes

characteristic

173

Q₈

—

178

—

174

D₄

—

356

—

175

D₄

—

356

—

176

D₄

—

178

—

177

C₂×C₄

—

178

C₂×C₄

—

356

—

179

C₂×C₄

—

356

—

180

C₂×C₄

—

178

—

181

C₂×C₄

#15667924

yes

—

182

C₂×C₄

#15667924

yes

—

183

C₂×C₄

C₂×#13083551

yes

characteristic

184

C₂×C₄

C₄×D₃₅₆

yes

—

185

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

186

C₂×C₄

#15668018

yes

—

187

C₈

—

188

C₈

—

356

—

189

C₈

#15667986

yes

—

190

C₈

#15668000

yes

characteristic

191

C₈

#15668009

yes

characteristic

192

C₂²

—

193

C₂²

—

356

—

194

C₂²

—

356

—

195

C₂²

C₂×#15667924

yes

characteristic

196

C₂²

#19151677

yes

—

197

C₄

—

198

C₄

—

356

—

199

C₄

#19151381

yes

complemented

200

C₄

#19151500

yes

characteristic

201

C₄

#19151564

yes

characteristic

202

C₄

C₈×D₃₅₆

yes

—

203

C₂

—

204

C₂

—

712

—

205

C₂

#26021046

yes

characteristic

206

C₂

C₂×#19151381

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	#19152148	C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#13191
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₈

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×C₄×C₁₆×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₈

prime radical

#19152148

C₄

cosocle

C₂²×C₈×C₈₉

C₂³

hypercenter

C₂×C₄×C₁₆

D₈₉

hypocenter

C₈₉

#13191

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
2. #26022008	C₂	semidirect
3. C₂×#19151381	C₂	semidirect
4. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
37. C₂×C₁₆×C₈₉	D₄	semidirect
94. C₄×C₈₉	#350	semidirect
107. C₂×C₄×C₁₆	D₈₉	semidirect
108. C₈₉	#13191	semidirect
139. C₂×C₁₆	D₃₅₆	semidirect
199. C₄	#19151381	semidirect

H ≅ G/N

type

2. #26022008

C₂

semidirect

3. C₂×#19151381

C₂

semidirect

4. C₂×C₄×C₁₆×C₈₉

C₂

semidirect

37. C₂×C₁₆×C₈₉

D₄

semidirect

94. C₄×C₈₉

#350

semidirect

107. C₂×C₄×C₁₆

D₈₉

semidirect

108. C₈₉

#13191

semidirect

139. C₂×C₁₆

D₃₅₆

semidirect

199. C₄

#19151381

semidirect

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

subgroup	index
#26028965	2
#26022008	2
C₂×#19151381	2
C₂×C₄×C₁₆×C₈₉	2
#13191	89

subgroup

index

C₂×C₄×C₁₆×C₈₉

Subgroups — #36472591

Conjugacy classes of subgroups

Complements

Maximal subgroups