Group #36707805 — subgroups

Subgroups — #36707805

Order 22784.

646 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#26029119	C₂	1	2	yes	maximal
3	#26022122	C₂	1	4	yes	maximal, complemented
4	#26022110	C₂	1	4	yes	maximal, complemented
5	#26021971	C₂	1	2	yes	maximal
6	#26021848	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
9	#19152075	—	2	2	no	—
10	C₂×Dic₇₁₂	C₄	1	8	yes	complemented
11	C₂×Dic₇₁₂	C₂²	1	2	yes	—
12	C₂×Dic₇₁₂	C₂²	1	2	yes	—
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	#19151704	—	2	4	no	—
15	#19151692	C₂²	1	8	yes	—
16	#19151606	—	2	2	no	—
17	#19151553	C₂²	1	1	yes	characteristic
18	#19151401	C₂²	1	8	yes	—
19	#19151373	C₂²	1	2	yes	—
20	C₁₆×Dic₈₉	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	#427×C₈₉	C₂²	1	2	yes	—
23	#367×C₈₉	C₂²	1	2	yes	—
24	#366×C₈₉	C₂²	1	2	yes	—
25	C₂²×Dic₁₇₈	—	2	2	no	—
26	C₂×Dic₃₅₆	—	2	2	no	—
27	C₂×Dic₃₅₆	—	2	4	no	—
28	C₂×Dic₃₅₆	C₂×C₄	1	8	yes	—
29	C₂×Dic₃₅₆	C₂³	1	4	yes	—
30	C₂×#13083558	D₄	1	1	yes	characteristic
31	C₄×Dic₁₇₈	—	4	4	no	—
32	#15667977	—	2	8	no	—
33	#15667971	D₄	1	1	yes	characteristic
34	#15667968	—	4	4	no	—
35	Dic₇₁₂	C₂×C₄	1	8	yes	—
36	Dic₇₁₂	C₂×C₄	1	8	yes	—
37	#15667952	—	2	2	no	—
38	C₂×#13083549	C₂³	1	2	yes	—
39	#15667935	—	2	8	no	—
40	#15667922	D₄	1	2	yes	—
41	C₈×Dic₈₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	#117×C₈₉	—	2	2	no	—
44	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
45	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
47	#107×C₈₉	C₂³	1	4	yes	—
48	#106×C₈₉	—	2	2	no	—
49	C₂×Dic₁₇₈	—	2	8	no	—
50	C₂×Dic₁₇₈	—	2	4	no	—
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×Dic₁₇₈	—	4	4	no	—
53	C₂²×Dic₈₉	—	4	2	no	—
54	Dic₃₅₆	—	2	8	no	—
55	Dic₃₅₆	—	2	4	no	—
56	Dic₃₅₆	C₂²×C₄	1	8	yes	—
57	#13083558	—	2	2	no	—
58	C₂×#10623494	C₂×D₄	1	2	yes	—
59	C₄⋊Dic₈₉	D₈	1	2	yes	—
60	#13083553	—	4	8	no	—
61	C₂²⋊Dic₈₉	D₈	1	2	yes	—
62	C₄×Dic₈₉	—	4	8	no	—
63	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
64	#13083549	—	2	2	no	—
65	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	#41	1	2	yes	—
69	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₈₉	—	2	4	no	—
71	C₂²⋊C₄×C₈₉	—	4	2	no	—
72	C₄²×C₈₉	—	4	2	no	—
73	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
74	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
75	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	—	4	4	no	—
77	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#135	1	2	yes	—
80	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₈₉	—	4	4	no	—
82	C₂×Dic₈₉	—	4	8	no	—
83	C₂×Dic₈₉	C₂×D₈	1	4	yes	—
84	Dic₁₇₈	—	2	8	no	—
85	Dic₁₇₈	—	4	4	no	—
86	Dic₁₇₈	—	4	8	no	—
87	C₈×C₈₉	C₄×D₄	1	2	yes	—
88	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
90	#10623494	—	2	2	no	—
91	C₂²×C₈₉	—	2	2	no	—
92	C₂²×C₈₉	C₄×D₈	1	2	yes	—
93	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₈₉	—	8	4	no	—
95	C₄×C₈₉	C₄×D₈	1	2	yes	—
96	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
97	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
98	Dic₈₉	—	2	4	no	—
99	Dic₈₉	—	4	8	no	—
100	#33458	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	—	2	2	no	—
102	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
103	C₂×C₈₉	#3417	1	2	yes	—
104	C₂²×Q₃₂	—	89	1	no	—
105	#2950	—	89	2	no	—
106	#2176	—	89	4	no	—
107	C₄×Q₃₂	—	89	4	no	—
108	#2158	D₈₉	1	1	yes	characteristic
109	#2141	—	89	2	no	—
110	#2108	—	89	1	no	—
111	C₈₉	#33458	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	—	89	2	no	—
113	#519	—	178	2	no	—
114	C₂×Q₃₂	—	89	8	no	—
115	C₂×Q₃₂	—	89	2	no	—
116	C₂×Q₃₂	—	89	2	no	—
117	C₂×M₃₂	—	89	1	no	—
118	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
119	#441	—	178	4	no	—
120	C₄×Q₁₆	—	89	8	no	—
121	#427	C₂×D₈₉	1	2	yes	—
122	#419	—	178	2	no	—
123	#405	—	89	1	no	—
124	#367	C₂×D₈₉	1	2	yes	—
125	#366	C₂×D₈₉	1	2	yes	—
126	#358	—	89	8	no	—
127	#346	—	89	2	no	—
128	C₄×C₁₆	—	89	2	no	—
129	C₂²×Q₈	—	178	2	no	—
130	C₂×Q₁₆	—	89	8	no	—
131	C₂×Q₁₆	—	89	4	no	—
132	C₂×Q₁₆	—	178	2	no	—
133	C₂×Q₁₆	—	178	4	no	—
134	C₂×M₁₆	—	89	1	no	—
135	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
136	C₄×Q₈	—	356	4	no	—
137	C₄×Q₈	—	178	8	no	—
138	#117	—	2	2	no	—
139	#117	—	89	1	no	—
140	#117	—	356	4	no	—
141	Q₃₂	—	89	8	no	—
142	Q₃₂	—	89	8	no	—
143	M₃₂	—	178	2	no	—
144	C₂×C₁₆	—	89	2	no	—
145	C₂×C₁₆	C₄×D₈₉	1	4	yes	—
146	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
148	#107	C₂²×D₈₉	1	4	yes	—
149	#106	—	2	2	no	—
150	#103	—	178	8	no	—
151	#97	—	89	2	no	—
152	C₄×C₈	—	89	2	no	—
153	C₂×Q₈	—	356	2	no	—
154	C₂×Q₈	—	356	4	no	—
155	C₂×Q₈	—	178	8	no	—
156	C₂×Q₈	—	178	4	no	—
157	C₂²×C₄	—	356	2	no	—
158	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
159	Q₁₆	—	89	8	no	—
160	Q₁₆	—	178	8	no	—
161	Q₁₆	—	178	4	no	—
162	M₁₆	—	178	2	no	—
163	C₂×C₈	—	89	2	no	—
164	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
165	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
166	C₂×C₈	#13083575	1	2	yes	—
167	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	89	2	no	—
170	C₄⋊C₄	—	356	8	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	89	2	no	—
173	C₄²	—	4	2	no	—
174	C₄²	—	89	2	no	—
175	C₄²	—	356	8	no	—
176	C₁₆	—	178	2	no	—
177	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
178	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
179	C₂³	D₈×D₈₉	1	1	yes	characteristic
180	Q₈	—	356	8	no	—
181	Q₈	—	356	4	no	—
182	Q₈	—	178	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	89	4	no	—
185	C₂×C₄	—	356	8	no	—
186	C₂×C₄	—	356	4	no	—
187	C₂×C₄	#15667974	1	2	yes	—
188	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
189	C₂×C₄	#15668035	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
191	C₈	—	178	2	no	—
192	C₈	#15667974	1	2	yes	—
193	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
194	C₈	C₂×#13083575	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	#19151680	1	2	yes	—
197	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
198	C₄	—	8	4	no	—
199	C₄	—	356	8	no	—
200	C₄	—	178	4	no	—
201	C₄	#19151680	1	2	yes	—
202	C₄	C₂×#15667974	1	1	yes	characteristic
203	C₄	C₂×#15668035	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×#19151680	1	1	yes	characteristic
206	C₂	#26033402	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₁₂

C₂

yes

maximal, characteristic

#26029119

C₂

yes

maximal

#26022122

C₂

yes

maximal, complemented

#26022110

C₂

yes

maximal, complemented

#26021971

C₂

yes

maximal

#26021848

C₂

yes

maximal, characteristic

C₈₉×#2158

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₆

C₂²

yes

—

#19152075

—

C₂×Dic₇₁₂

C₄

yes

complemented

C₂×Dic₇₁₂

C₂²

yes

—

C₂×Dic₇₁₂

C₂²

yes

—

C₂×#15667952

C₂²

yes

characteristic

#19151704

—

#19151692

C₂²

yes

—

#19151606

—

#19151553

C₂²

yes

characteristic

#19151401

C₂²

yes

—

#19151373

C₂²

yes

—

C₁₆×Dic₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

#427×C₈₉

C₂²

yes

—

#367×C₈₉

C₂²

yes

—

#366×C₈₉

C₂²

yes

—

C₂²×Dic₁₇₈

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

C₂×C₄

yes

—

C₂×Dic₃₅₆

C₂³

yes

—

C₂×#13083558

D₄

yes

characteristic

C₄×Dic₁₇₈

—

#15667977

—

#15667971

D₄

yes

characteristic

#15667968

—

Dic₇₁₂

C₂×C₄

yes

—

Dic₇₁₂

C₂×C₄

yes

—

#15667952

—

C₂×#13083549

C₂³

yes

—

#15667935

—

#15667922

D₄

yes

—

C₈×Dic₈₉

C₂³

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#117×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

C₂³

yes

—

#106×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

C₂²×C₄

yes

—

#13083558

—

C₂×#10623494

C₂×D₄

yes

—

C₄⋊Dic₈₉

D₈

yes

—

#13083553

—

C₂²⋊Dic₈₉

D₈

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×D₄

yes

—

#13083549

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₁₆×C₈₉

C₂²×C₄

yes

—

C₁₆×C₈₉

C₂²×C₄

yes

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₈

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₄×D₄

yes

—

C₈×C₈₉

C₂³×C₄

yes

characteristic

C₈×C₈₉

C₂×#41

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×D₈

yes

—

C₂²×C₈₉

C₂²×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₄×D₈

yes

—

C₄×C₈₉

C₂×C₄×D₄

yes

characteristic

C₄×C₈₉

C₂×#135

yes

characteristic

Dic₈₉

—

Dic₈₉

—

100

#33458

—

maximal, Sylow

101

C₂×C₈₉

—

102

C₂×C₈₉

C₂×C₄×D₈

yes

characteristic

103

C₂×C₈₉

#3417

yes

—

104

C₂²×Q₃₂

—

105

#2950

—

106

#2176

—

107

C₄×Q₃₂

—

108

#2158

D₈₉

yes

characteristic

109

#2141

—

110

#2108

—

111

C₈₉

#33458

yes

characteristic, complemented, Sylow

112

C₂²×Q₁₆

—

113

#519

—

178

—

114

C₂×Q₃₂

—

115

C₂×Q₃₂

—

116

C₂×Q₃₂

—

117

C₂×M₃₂

—

118

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

119

#441

—

178

—

120

C₄×Q₁₆

—

121

#427

C₂×D₈₉

yes

—

122

#419

—

178

—

123

#405

—

124

#367

C₂×D₈₉

yes

—

125

#366

C₂×D₈₉

yes

—

126

#358

—

127

#346

—

128

C₄×C₁₆

—

129

C₂²×Q₈

—

178

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

178

—

133

C₂×Q₁₆

—

178

—

134

C₂×M₁₆

—

135

C₂²×C₈

C₂²×D₈₉

yes

characteristic

136

C₄×Q₈

—

356

—

137

C₄×Q₈

—

178

—

138

#117

—

139

#117

—

140

#117

—

356

—

141

Q₃₂

—

142

Q₃₂

—

143

M₃₂

—

178

—

144

C₂×C₁₆

—

145

C₂×C₁₆

C₄×D₈₉

yes

—

146

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

147

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

148

#107

C₂²×D₈₉

yes

—

149

#106

—

150

#103

—

178

—

151

#97

—

152

C₄×C₈

—

153

C₂×Q₈

—

356

—

154

C₂×Q₈

—

356

—

155

C₂×Q₈

—

178

—

156

C₂×Q₈

—

178

—

157

C₂²×C₄

—

356

—

158

C₂²×C₄

D₄×D₈₉

yes

characteristic

159

Q₁₆

—

160

Q₁₆

—

178

—

161

Q₁₆

—

178

—

162

M₁₆

—

178

—

163

C₂×C₈

—

164

C₂×C₈

C₂×C₄×D₈₉

yes

—

165

C₂×C₈

D₄×D₈₉

yes

characteristic

166

C₂×C₈

#13083575

yes

—

167

C₂×C₈

C₂³×D₈₉

yes

characteristic

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

356

—

171

C₂²⋊C₄

—

172

C₂²⋊C₄

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

356

—

176

C₁₆

—

178

—

177

C₁₆

C₂×C₄×D₈₉

yes

—

178

C₁₆

C₂×C₄×D₈₉

yes

—

179

C₂³

D₈×D₈₉

yes

characteristic

180

Q₈

—

356

—

181

Q₈

—

356

—

182

Q₈

—

178

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

356

—

186

C₂×C₄

—

356

—

187

C₂×C₄

#15667974

yes

—

188

C₂×C₄

D₈×D₈₉

yes

characteristic

189

C₂×C₄

#15668035

yes

—

190

C₂×C₄

C₂×D₄×D₈₉

yes

characteristic

191

C₈

—

178

—

192

C₈

#15667974

yes

—

193

C₈

C₂²×C₄×D₈₉

yes

characteristic

194

C₈

C₂×#13083575

yes

characteristic

195

C₂²

—

196

C₂²

#19151680

yes

—

197

C₂²

C₂×D₈×D₈₉

yes

characteristic

198

C₄

—

199

C₄

—

356

—

200

C₄

—

178

—

201

C₄

#19151680

yes

—

202

C₄

C₂×#15667974

yes

characteristic

203

C₄

C₂×#15668035

yes

characteristic

204

C₂

—

205

C₂

C₂×#19151680

yes

characteristic

206

C₂

#26033402

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2158	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2158	D₈₉
hypocenter	C₈₉	#33458
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂²

C₂×D₈×D₈₉

commutator

C₈×C₈₉

C₂³×C₄

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

C₈₉×#2158

C₂

socle

C₂²×C₈₉

C₂²×D₈

prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂×C₈×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
3. #26022122	C₂	semidirect
4. #26022110	C₂	semidirect
10. C₂×Dic₇₁₂	C₄	semidirect
111. C₈₉	#33458	semidirect

H ≅ G/N

type

3. #26022122

C₂

semidirect

4. #26022110

C₂

semidirect

10. C₂×Dic₇₁₂

C₄

semidirect

111. C₈₉

#33458

semidirect

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

subgroup	index
C₂²×Dic₇₁₂	2
#26029119	2
#26022122	2
#26022110	2
#26021971	2
#26021848	2
C₈₉×#2158	2
#33458	89

subgroup

index

C₂²×Dic₇₁₂

C₈₉×#2158

Subgroups — #36707805

Conjugacy classes of subgroups

Complements

Maximal subgroups