Group #36470787 — subgroups

Subgroups — #36470787

Order 22784.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	#26022007	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018334	C₂	1	1	yes	maximal, characteristic
4	#26018333	C₂	1	2	yes	maximal
5	C₈₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
12	C₄²×Dic₈₉	C₄	1	1	yes	characteristic
13	#19151416	—	2	4	no	—
14	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
17	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₈₉	—	2	2	no	—
19	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
21	#15667952	—	2	2	no	—
22	#15667952	—	2	2	no	—
23	C₂×#13083549	—	2	2	no	—
24	C₄×#10623494	—	2	2	no	—
25	C₈×Dic₈₉	—	2	2	no	—
26	C₈×Dic₈₉	—	2	2	no	—
27	C₈×Dic₈₉	C₈	1	2	yes	—
28	C₈×Dic₈₉	C₈	1	2	yes	—
29	C₈×Dic₈₉	D₄	1	2	yes	—
30	C₈×Dic₈₉	Q₈	1	2	yes	—
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₈₉	—	2	2	no	—
35	M₃₂×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₄×C₈×C₈₉	—	2	2	no	—
38	C₂²×Dic₈₉	—	2	1	no	—
39	C₂²×Dic₈₉	M₁₆	1	2	yes	—
40	C₂×#10623494	—	2	1	no	—
41	C₂×#10623494	—	2	1	no	—
42	C₂×#10623494	—	2	1	no	—
43	C₂×#10623494	—	2	1	no	—
44	C₂×#10623494	C₄⋊C₄	1	2	yes	—
45	C₂×#10623494	C₂×C₈	1	2	yes	—
46	C₄×Dic₈₉	—	2	8	no	—
47	C₄×Dic₈₉	—	2	1	no	—
48	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
49	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
50	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
51	C₄×Dic₈₉	M₁₆	1	2	yes	—
52	#13083549	—	4	2	no	—
53	C₂²×C₄×C₈₉	—	2	1	no	—
54	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	—	2	1	no	—
56	C₂×C₈×C₈₉	—	2	1	no	—
57	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₉	#41	1	2	yes	—
61	C₂×C₈×C₈₉	#41	1	2	yes	—
62	C₄²×C₈₉	—	2	1	no	—
63	C₄²×C₈₉	M₁₆	1	2	yes	—
64	C₁₆×C₈₉	—	4	2	no	—
65	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₉	—	2	4	no	—
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
70	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₈₉	—	2	4	no	—
75	C₂×Dic₈₉	—	2	2	no	—
76	C₂×Dic₈₉	—	2	4	no	—
77	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₈₉	C₄⋊C₈	1	2	yes	—
79	C₈×C₈₉	—	2	2	no	—
80	C₈×C₈₉	#124	1	2	yes	—
81	C₈×C₈₉	#125	1	2	yes	—
82	#10623494	—	2	2	no	—
83	#10623494	—	2	2	no	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₈₉	#432	1	1	yes	characteristic
87	C₂²×C₈₉	#432	1	1	yes	characteristic
88	C₄×C₈₉	—	2	4	no	—
89	C₄×C₈₉	#382	1	1	yes	characteristic
90	C₄×C₈₉	#423	1	1	yes	characteristic
91	C₄×C₈₉	#431	1	1	yes	characteristic
92	C₄×C₈₉	#431	1	1	yes	characteristic
93	Dic₈₉	—	2	4	no	—
94	Dic₈₉	—	2	2	no	—
95	Dic₈₉	#364	1	4	yes	—
96	#12883	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	—	2	1	no	—
98	C₂×C₈₉	—	2	1	no	—
99	C₂×C₈₉	#1831	1	1	yes	characteristic
100	C₂×C₈₉	C₂×#364	1	2	yes	—
101	#2150	—	89	2	no	—
102	C₄²×C₈	—	89	1	no	—
103	#1376	—	89	2	no	—
104	#1376	—	89	1	no	—
105	#1376	D₈₉	1	1	yes	characteristic
106	C₈₉	#12883	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₃₂	—	89	2	no	—
108	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
109	C₂×C₄×C₈	—	89	1	no	—
110	C₂×C₄×C₈	—	89	1	no	—
111	C₂×C₄×C₈	—	89	1	no	—
112	C₂×C₄×C₈	—	89	1	no	—
113	C₂×C₄×C₈	—	89	1	no	—
114	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	C₄³	—	89	1	no	—
116	C₄⋊C₁₆	—	178	4	no	—
117	C₂²×C₈	—	89	1	no	—
118	C₂²×C₈	—	89	1	no	—
119	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
121	C₂×C₄²	—	89	1	no	—
122	C₂×C₄²	—	89	1	no	—
123	C₂×C₄²	—	178	2	no	—
124	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	178	2	no	—
128	M₃₂	—	178	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	178	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	89	2	no	—
133	C₄×C₈	—	89	2	no	—
134	C₄×C₈	—	89	2	no	—
135	C₄×C₈	—	89	2	no	—
136	C₄×C₈	—	178	2	no	—
137	C₄×C₈	—	178	2	no	—
138	C₄×C₈	—	178	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	89	2	no	—
141	C₂²×C₄	—	178	1	no	—
142	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	89	2	no	—
146	C₂×C₈	—	89	2	no	—
147	C₂×C₈	—	178	1	no	—
148	C₂×C₈	—	178	1	no	—
149	C₂×C₈	—	178	1	no	—
150	C₂×C₈	—	178	1	no	—
151	C₂×C₈	C₈×D₈₉	1	2	yes	—
152	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
153	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
154	C₂×C₈	#13083573	1	2	yes	—
155	C₂×C₈	#13083574	1	2	yes	—
156	C₄²	—	2	1	no	—
157	C₄²	—	89	1	no	—
158	C₄²	—	89	1	no	—
159	C₄²	—	89	2	no	—
160	C₄²	—	89	2	no	—
161	C₄²	—	178	8	no	—
162	C₄²	—	178	1	no	—
163	C₄²	#13083557	1	2	yes	—
164	C₁₆	—	4	2	no	—
165	C₁₆	—	356	2	no	—
166	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	89	2	no	—
170	C₂×C₄	—	89	2	no	—
171	C₂×C₄	—	178	4	no	—
172	C₂×C₄	—	178	4	no	—
173	C₂×C₄	—	178	2	no	—
174	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
175	C₂×C₄	#15667968	1	1	yes	characteristic
176	C₂×C₄	#15667969	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
178	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
179	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	178	2	no	—
182	C₈	—	178	2	no	—
183	C₈	#15667998	1	2	yes	—
184	C₈	#15668002	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₄⋊C₈×D₈₉	1	1	yes	characteristic
187	C₂²	#19151652	1	1	yes	characteristic
188	C₂²	#19151653	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	89	4	no	—
191	C₄	—	178	4	no	—
192	C₄	—	178	2	no	—
193	C₄	#19151462	1	1	yes	characteristic
194	C₄	#19151620	1	1	yes	characteristic
195	C₄	#19151648	1	1	yes	characteristic
196	C₄	#19151649	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#26020367	1	1	yes	characteristic
200	C₂	#364×D₈₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022007

C₂

yes

maximal

C₄×C₈×Dic₈₉

C₂

yes

maximal, characteristic

#26018334

C₂

yes

maximal, characteristic

#26018333

C₂

yes

maximal

C₈₉×#1376

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₂×C₄×#10623494

C₄

yes

characteristic

C₂×C₈×Dic₈₉

C₄

yes

characteristic

C₂×C₈×Dic₈₉

C₄

yes

characteristic

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

C₄²×Dic₈₉

C₄

yes

characteristic

#19151416

—

C₂×M₃₂×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

characteristic

C₂×C₄×Dic₈₉

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

—

—

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

C₈×Dic₈₉

C₈

yes

—

C₈×Dic₈₉

C₈

yes

—

C₈×Dic₈₉

D₄

yes

—

C₈×Dic₈₉

Q₈

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

M₃₂×C₈₉

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

M₁₆

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10623494

C₂×C₈

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₄⋊C₄

yes

—

C₄×Dic₈₉

C₂×C₈

yes

characteristic

C₄×Dic₈₉

C₂×C₈

yes

characteristic

C₄×Dic₈₉

M₁₆

yes

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

M₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄⋊C₈

yes

—

C₂×Dic₈₉

C₄⋊C₈

yes

—

C₈×C₈₉

—

C₈×C₈₉

#124

yes

—

C₈×C₈₉

#125

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#364

yes

—

#12883

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1831

yes

characteristic

100

C₂×C₈₉

C₂×#364

yes

—

101

#2150

—

102

C₄²×C₈

—

103

#1376

—

104

#1376

—

105

#1376

D₈₉

yes

characteristic

106

C₈₉

#12883

yes

characteristic, complemented, Sylow

107

C₂×M₃₂

—

108

C₂×M₃₂

C₂×D₈₉

yes

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

115

C₄³

—

116

C₄⋊C₁₆

—

178

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₈₉

yes

characteristic

120

C₂²×C₈

C₂²×D₈₉

yes

characteristic

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

178

—

124

C₂×C₄²

C₄×D₈₉

yes

characteristic

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

178

—

128

M₃₂

—

178

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

178

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

178

—

137

C₄×C₈

—

178

—

138

C₄×C₈

—

178

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

178

—

142

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

178

—

148

C₂×C₈

—

178

—

149

C₂×C₈

—

178

—

150

C₂×C₈

—

178

—

151

C₂×C₈

C₈×D₈₉

yes

—

152

C₂×C₈

D₄×D₈₉

yes

characteristic

153

C₂×C₈

Q₈×D₈₉

yes

characteristic

154

C₂×C₈

#13083573

yes

—

155

C₂×C₈

#13083574

yes

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

178

—

162

C₄²

—

178

—

163

C₄²

#13083557

yes

—

164

C₁₆

—

165

C₁₆

—

356

—

166

C₂³

M₁₆×D₈₉

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

178

—

172

C₂×C₄

—

178

—

173

C₂×C₄

—

178

—

174

C₂×C₄

C₄⋊C₄×D₈₉

yes

characteristic

175

C₂×C₄

#15667968

yes

characteristic

176

C₂×C₄

#15667969

yes

characteristic

177

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

178

C₂×C₄

M₁₆×D₈₉

yes

characteristic

179

C₂×C₄

C₂×#13083557

yes

characteristic

180

C₈

—

181

C₈

—

178

—

182

C₈

—

178

—

183

C₈

#15667998

yes

—

184

C₈

#15668002

yes

—

185

C₂²

—

186

C₂²

C₄⋊C₈×D₈₉

yes

characteristic

187

C₂²

#19151652

yes

characteristic

188

C₂²

#19151653

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

178

—

192

C₄

—

178

—

193

C₄

#19151462

yes

characteristic

194

C₄

#19151620

yes

characteristic

195

C₄

#19151648

yes

characteristic

196

C₄

#19151649

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#26020367

yes

characteristic

200

C₂

#364×D₈₉

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1376	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1376	D₈₉
hypocenter	C₈₉	#12883
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1376

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₈

prime radical

C₂³×C₈₉

C₂×M₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
106. C₈₉	#12883	semidirect

H ≅ G/N

type

106. C₈₉

#12883

semidirect

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

subgroup	index
#26022007	2
C₄×C₈×Dic₈₉	2
#26018334	2
#26018333	2
C₈₉×#1376	2
#12883	89

subgroup

index

C₄×C₈×Dic₈₉

C₈₉×#1376

Subgroups — #36470787

Conjugacy classes of subgroups

Complements

Maximal subgroups