Group #36445122 — subgroups

Subgroups — #36445122

Order 22784.

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

normal subgroups only

This group has 412 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	#26018634	C₂	1	2	yes	maximal
2	#26018544	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151319	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018090	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2922	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₂²	1	2	yes	—
8	Q₈⋊#10623494	C₄	1	4	yes	—
9	D₄⋊#10623494	C₄	1	4	yes	—
10	#19151320	—	2	2	no	—
11	#19151319	—	2	2	no	—
12	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
13	#433×C₈₉	—	2	2	no	—
14	#431×C₈₉	—	2	2	no	—
15	#424×C₈₉	C₄	1	2	yes	—
16	#408×C₈₉	C₄	1	2	yes	—
17	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×#10623494	—	2	2	no	—
20	C₄⋊#10623494	C₂×C₄	1	4	yes	—
21	C₂²⋊#10623494	—	2	2	no	—
22	C₄×#10623494	—	2	2	no	—
23	C₄×#10623494	—	2	2	no	—
24	C₄×#10623494	C₂×C₄	1	4	yes	—
25	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₄×D₄×C₈₉	—	2	2	no	—
30	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
31	#117×C₈₉	—	2	1	no	—
32	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	—	2	1	no	—
34	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₈₉	—	2	4	no	—
36	C₂²⋊C₈×C₈₉	—	2	4	no	—
37	#97×C₈₉	—	2	1	no	—
38	C₄×C₈×C₈₉	—	2	1	no	—
39	C₄×C₈×C₈₉	D₄	1	2	yes	—
40	C₄×C₈×C₈₉	D₄	1	2	yes	—
41	C₂×#10623494	—	2	4	no	—
42	C₂×#10623494	—	2	4	no	—
43	C₂×#10623494	—	4	4	no	—
44	#41×C₈₉	—	2	4	no	—
45	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
46	C₂×D₄×C₈₉	—	2	1	no	—
47	C₂×D₄×C₈₉	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	—	2	1	no	—
49	C₂²×C₄×C₈₉	—	2	2	no	—
50	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	#41	1	2	yes	—
52	M₁₆×C₈₉	—	2	4	no	—
53	C₂×C₈×C₈₉	—	2	4	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
56	C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
58	C₂²⋊C₄×C₈₉	—	2	1	no	—
59	C₂²⋊C₄×C₈₉	—	2	2	no	—
60	C₄²×C₈₉	—	2	1	no	—
61	C₄²×C₈₉	—	2	2	no	—
62	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
63	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₈₉	—	2	1	no	—
66	C₂³×C₈₉	#117	1	1	yes	characteristic
67	Q₈×C₈₉	—	2	2	no	—
68	D₄×C₈₉	—	2	2	no	—
69	D₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	2	4	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	—	4	4	no	—
75	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₈₉	#131	1	4	yes	—
80	C₈×C₈₉	—	2	4	no	—
81	C₈×C₈₉	—	4	2	no	—
82	#10623494	—	4	4	no	—
83	#10623494	—	4	4	no	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	—	4	1	no	—
86	C₂²×C₈₉	—	4	2	no	—
87	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₈₉	#433	1	2	yes	—
89	C₄×C₈₉	—	2	2	no	—
90	C₄×C₈₉	—	4	1	no	—
91	C₄×C₈₉	—	4	2	no	—
92	C₄×C₈₉	#405	1	2	yes	—
93	C₄×C₈₉	#408	1	2	yes	—
94	C₄×C₈₉	#444	1	4	yes	—
95	#8188	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	—	2	2	no	—
97	C₂×C₈₉	—	8	1	no	—
98	C₂×C₈₉	#1750	1	1	yes	characteristic
99	C₂×C₈₉	#1765	1	2	yes	—
100	#2922	D₈₉	1	1	yes	characteristic
101	#1476	—	89	2	no	—
102	#1452	—	89	1	no	—
103	C₂×#322	—	89	1	no	—
104	#1278	—	89	2	no	—
105	C₈₉	#8188	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
107	#433	—	2	2	no	—
108	#431	—	2	2	no	—
109	#431	—	89	2	no	—
110	#424	Dic₈₉	1	2	yes	—
111	#408	Dic₈₉	1	2	yes	—
112	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
113	C₂×C₄×C₈	—	89	2	no	—
114	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	Q₈⋊C₈	—	89	4	no	—
116	D₄⋊C₈	—	89	4	no	—
117	#322	—	178	2	no	—
118	#322	—	178	2	no	—
119	C₂×#41	C₂×Dic₈₉	1	1	yes	characteristic
120	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
121	C₂²×C₈	—	178	2	no	—
122	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
123	C₄×Q₈	C₄×D₈₉	1	2	yes	—
124	C₄×D₄	—	2	2	no	—
125	C₄×D₄	C₄×D₈₉	1	2	yes	—
126	#117	—	2	1	no	—
127	#117	C₂×Dic₈₉	1	1	yes	characteristic
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
130	C₄⋊C₈	—	2	4	no	—
131	C₄⋊C₈	—	89	4	no	—
132	C₂²⋊C₈	—	2	4	no	—
133	C₂²⋊C₈	—	178	2	no	—
134	#97	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	89	4	no	—
137	C₄×C₈	—	178	2	no	—
138	C₄×C₈	—	178	2	no	—
139	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
140	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
141	#41	—	2	4	no	—
142	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	C₄×Dic₈₉	1	1	yes	characteristic
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
148	C₂²×C₄	#13083574	1	2	yes	—
149	M₁₆	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	356	4	no	—
153	C₂×C₈	—	178	4	no	—
154	C₂×C₈	—	178	4	no	—
155	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
158	C₂²⋊C₄	—	2	1	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	1	no	—
162	C₄²	C₂×C₄×D₈₉	1	2	yes	—
163	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
164	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₂³	—	2	1	no	—
166	C₂³	#15667971	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
176	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
177	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
178	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
179	C₂×C₄	#15668019	1	4	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	4	2	no	—
182	C₈	—	356	4	no	—
183	C₈	—	356	4	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	1	no	—
186	C₂²	—	4	2	no	—
187	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
188	C₂²	#19151659	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	4	1	no	—
191	C₄	—	4	2	no	—
192	C₄	#19151554	1	2	yes	—
193	C₄	#19151565	1	2	yes	—
194	C₄	#19151718	1	4	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	8	1	no	—
197	C₂	#26020018	1	1	yes	characteristic
198	C₂	#26020094	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26018634

C₂

yes

maximal

#26018544

C₂

yes

maximal, characteristic, complemented

C₂×#19151319

C₂

yes

maximal, characteristic, complemented

#26018090

C₂

yes

maximal, complemented

C₈₉×#2922

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

—

Q₈⋊#10623494

C₄

yes

—

D₄⋊#10623494

C₄

yes

—

#19151320

—

#19151319

—

C₄×#41×C₈₉

C₂²

yes

characteristic

#433×C₈₉

—

#431×C₈₉

—

#424×C₈₉

C₄

yes

—

#408×C₈₉

C₄

yes

—

C₄×M₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

—

C₄⋊#10623494

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#41×C₈₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₄×Q₈×C₈₉

C₂×C₄

yes

—

C₄×D₄×C₈₉

—

C₄×D₄×C₈₉

C₂×C₄

yes

—

#117×C₈₉

—

#117×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₄²

yes

characteristic

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₄²

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₄²

yes

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

#117

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#433

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#405

yes

—

C₄×C₈₉

#408

yes

—

C₄×C₈₉

#444

yes

—

#8188

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1750

yes

characteristic

C₂×C₈₉

#1765

yes

—

100

#2922

D₈₉

yes

characteristic

101

#1476

—

102

#1452

—

103

C₂×#322

—

104

#1278

—

105

C₈₉

#8188

yes

characteristic, complemented, Sylow

106

C₄×#41

C₂×D₈₉

yes

characteristic

107

#433

—

108

#431

—

109

#431

—

110

#424

Dic₈₉

yes

—

111

#408

Dic₈₉

yes

—

112

C₄×M₁₆

C₂×D₈₉

yes

characteristic

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

115

Q₈⋊C₈

—

116

D₄⋊C₈

—

117

#322

—

178

—

118

#322

—

178

—

119

C₂×#41

C₂×Dic₈₉

yes

characteristic

120

C₂×M₁₆

C₂×Dic₈₉

yes

—

121

C₂²×C₈

—

178

—

122

C₂²×C₈

C₂×Dic₈₉

yes

—

123

C₄×Q₈

C₄×D₈₉

yes

—

124

C₄×D₄

—

125

C₄×D₄

C₄×D₈₉

yes

—

126

#117

—

127

#117

C₂×Dic₈₉

yes

characteristic

128

C₂×C₄²

—

129

C₂×C₄²

C₂²×D₈₉

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₂²⋊C₈

—

133

C₂²⋊C₈

—

178

—

134

#97

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

178

—

138

C₄×C₈

—

178

—

139

C₄×C₈

C₂²⋊D₈₉

yes

—

140

C₄×C₈

C₂²⋊D₈₉

yes

—

141

#41

—

142

C₂×Q₈

C₄×Dic₈₉

yes

characteristic

143

C₂×D₄

—

144

C₂×D₄

C₄×Dic₈₉

yes

characteristic

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

148

C₂²×C₄

#13083574

yes

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

356

—

153

C₂×C₈

—

178

—

154

C₂×C₈

—

178

—

155

C₂×C₈

C₂²⋊Dic₈₉

yes

—

156

C₄⋊C₄

—

157

C₄⋊C₄

C₄×Dic₈₉

yes

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₄²

—

162

C₄²

C₂×C₄×D₈₉

yes

—

163

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

164

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

165

C₂³

—

166

C₂³

#15667971

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₂×C₄×Dic₈₉

yes

characteristic

176

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

177

C₂×C₄

C₄×C₂²⋊D₈₉

yes

—

178

C₂×C₄

C₄×C₂²⋊D₈₉

yes

—

179

C₂×C₄

#15668019

yes

—

180

C₈

—

181

C₈

—

182

C₈

—

356

—

183

C₈

—

356

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₄×C₂²⋊Dic₈₉

yes

characteristic

188

C₂²

#19151659

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#19151554

yes

—

193

C₄

#19151565

yes

—

194

C₄

#19151718

yes

—

195

C₂

—

196

C₂

—

197

C₂

#26020018

yes

characteristic

198

C₂

#26020094

yes

—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2922	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2922	D₈₉
hypocenter	C₈₉	#8188
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #2922

subgroup

G/H

center

C₄²

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#2922

C₂

socle

C₂²×C₈₉

C₄×C₂²⋊C₄

prime radical

C₂×#41×C₈₉

C₂×C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #2922

N	H ≅ G/N	type
2. #26018544	C₂	semidirect
3. C₂×#19151319	C₂	semidirect
4. #26018090	C₂	semidirect
105. C₈₉	#8188	semidirect

H ≅ G/N

type

2. #26018544

C₂

semidirect

3. C₂×#19151319

C₂

semidirect

4. #26018090

C₂

semidirect

105. C₈₉

#8188

semidirect

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

subgroup	index
#26018634	2
#26018544	2
C₂×#19151319	2
#26018090	2
C₈₉×#2922	2
#8188	89

subgroup

index

C₈₉×#2922

Subgroups — #36445122

Conjugacy classes of subgroups

Complements

Maximal subgroups