Group #36709852 — subgroups

Subgroups — #36709852

Order 22784.

826 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 219 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×#19152030	C₂	1	4	yes	maximal, complemented
2	#26030680	C₂	1	2	yes	maximal, complemented
3	#26022523	C₂	1	4	yes	maximal, complemented
4	#26022518	C₂	1	4	yes	maximal, complemented
5	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668034	—	2	2	no	—
7	C₂×#15668034	C₂²	1	4	yes	—
8	#19152148	—	2	2	no	—
9	#19152030	—	2	16	no	—
10	C₂×Dic₇₁₂	C₂²	1	4	yes	—
11	C₂×#15667957	C₂²	1	8	yes	—
12	C₂×D₁₄₂₄	C₂²	1	4	yes	—
13	#19151982	—	2	2	no	—
14	#19151982	C₂²	1	4	yes	—
15	#19151978	C₂²	1	2	yes	—
16	#19151975	C₂²	1	2	yes	—
17	#19151971	—	2	2	no	—
18	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#346×C₈₉	C₂²	1	4	yes	—
21	C₂×#13083574	—	2	4	no	—
22	#15668034	—	2	16	no	—
23	#15668034	—	4	8	no	—
24	C₂×Dic₃₅₆	—	2	2	no	—
25	C₂×Dic₃₅₆	C₂³	1	4	yes	—
26	C₂×#13083563	—	2	4	no	—
27	C₂×#13083563	—	2	4	no	—
28	C₂×D₇₁₂	—	2	2	no	—
29	C₂×D₇₁₂	C₂³	1	4	yes	—
30	#15668012	—	2	2	no	—
31	#15668009	—	2	2	no	—
32	#15668000	—	2	4	no	—
33	#15667986	—	4	4	no	—
34	C₄×D₃₅₆	—	4	4	no	—
35	Dic₇₁₂	—	2	8	no	—
36	#15667957	—	2	16	no	—
37	D₁₄₂₄	—	2	8	no	—
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₈₉	D₄	1	8	yes	complemented
42	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	C₄×C₈×C₈₉	C₂³	1	2	yes	—
45	#13083574	—	4	8	no	—
46	#13083574	—	4	8	no	—
47	C₂×Dic₁₇₈	—	2	4	no	—
48	C₂×D₃₅₆	—	2	4	no	—
49	C₂×D₃₅₆	—	4	4	no	—
50	C₂×C₂²⋊D₈₉	—	4	4	no	—
51	C₂×C₄×D₈₉	—	4	4	no	—
52	Dic₃₅₆	—	2	8	no	—
53	Dic₃₅₆	—	4	4	no	—
54	#13083563	—	4	8	no	—
55	#13083563	—	4	8	no	—
56	D₇₁₂	—	2	8	no	—
57	D₇₁₂	—	4	4	no	—
58	C₄⋊Dic₈₉	—	4	4	no	—
59	#13083551	—	4	8	no	—
60	C₂²×C₄×C₈₉	D₈	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	D₈	1	4	yes	—
63	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₄²×C₈₉	D₈	1	2	yes	—
68	C₄²×C₈₉	C₂×D₄	1	2	yes	—
69	C₁₆×C₈₉	C₂×D₄	1	8	yes	—
70	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₈₉	—	4	4	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	#127	1	2	yes	—
74	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
78	C₂²⋊D₈₉	—	8	8	no	—
79	C₂×Dic₈₉	—	4	4	no	—
80	D₃₅₆	—	4	4	no	—
81	D₃₅₆	—	4	4	no	—
82	D₃₅₆	—	8	8	no	—
83	C₄×D₈₉	—	8	8	no	—
84	Dic₁₇₈	—	4	4	no	—
85	Dic₁₇₈	—	4	4	no	—
86	C₈×C₈₉	#127	1	2	yes	—
87	C₈×C₈₉	C₂×D₈	1	4	yes	—
88	C₈×C₈₉	C₂²×D₄	1	2	yes	—
89	C₂²×C₈₉	—	2	2	no	—
90	C₂²×C₈₉	#493	1	2	yes	—
91	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
92	C₂×D₈₉	—	8	4	no	—
93	C₂×D₈₉	—	8	8	no	—
94	C₄×C₈₉	—	2	4	no	—
95	C₄×C₈₉	#493	1	2	yes	—
96	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
97	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
98	Dic₈₉	—	8	4	no	—
99	#33716	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	—	2	2	no	—
101	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
102	C₂×C₈₉	#3414	1	2	yes	—
103	D₈₉	—	16	4	no	—
104	C₂×#508	—	89	4	no	—
105	#3147	—	89	2	no	—
106	#2251	—	89	4	no	—
107	#2250	—	89	4	no	—
108	C₂×#346	D₈₉	1	1	yes	characteristic, complemented
109	C₈₉	#33716	1	1	yes	characteristic, complemented, Sylow
110	C₂×#135	—	89	4	no	—
111	C₂×#135	—	178	2	no	—
112	#532	—	178	2	no	—
113	#508	—	178	16	no	—
114	C₂×Q₃₂	—	89	4	no	—
115	C₂×SD₃₂	—	89	8	no	—
116	C₂×D₁₆	—	89	4	no	—
117	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
118	#495	—	89	4	no	—
119	#495	—	178	2	no	—
120	#494	—	89	2	no	—
121	#493	—	89	2	no	—
122	#492	—	178	2	no	—
123	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
124	#346	C₂×D₈₉	1	4	yes	—
125	C₂×#41	—	178	4	no	—
126	#135	—	356	8	no	—
127	#135	—	178	16	no	—
128	C₂×Q₁₆	—	89	4	no	—
129	C₂×Q₁₆	—	178	2	no	—
130	C₂×SD₁₆	—	178	4	no	—
131	C₂×SD₁₆	—	178	4	no	—
132	C₂×D₈	—	89	4	no	—
133	C₂×D₈	—	178	2	no	—
134	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	178	2	no	—
137	#127	—	178	2	no	—
138	#124	—	178	4	no	—
139	#121	—	356	4	no	—
140	C₄×D₄	—	356	4	no	—
141	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
142	Q₃₂	—	178	8	no	—
143	SD₃₂	—	178	16	no	—
144	D₁₆	—	178	8	no	—
145	C₂×C₁₆	D₃₅₆	1	8	yes	complemented
146	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
147	C₄×C₈	D₃₅₆	1	2	yes	—
148	C₄×C₈	C₂²×D₈₉	1	2	yes	—
149	#41	—	356	8	no	—
150	#41	—	356	8	no	—
151	C₂×Q₈	—	178	4	no	—
152	C₂×D₄	—	356	4	no	—
153	C₂×D₄	—	356	4	no	—
154	C₂×D₄	—	178	4	no	—
155	C₂²×C₄	—	356	4	no	—
156	C₂²×C₄	D₇₁₂	1	2	yes	—
157	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
158	Q₁₆	—	356	4	no	—
159	Q₁₆	—	178	8	no	—
160	SD₁₆	—	356	8	no	—
161	SD₁₆	—	356	8	no	—
162	D₈	—	356	4	no	—
163	D₈	—	178	8	no	—
164	C₂×C₈	D₇₁₂	1	4	yes	—
165	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
166	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
167	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
168	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
169	C₄⋊C₄	—	356	4	no	—
170	C₂²⋊C₄	—	356	8	no	—
171	C₄²	D₇₁₂	1	2	yes	—
172	C₄²	C₂×D₃₅₆	1	2	yes	—
173	C₁₆	C₂×D₃₅₆	1	8	yes	—
174	C₂³	—	356	4	no	—
175	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
176	Q₈	—	356	4	no	—
177	Q₈	—	356	4	no	—
178	D₄	—	356	4	no	—
179	D₄	—	356	4	no	—
180	D₄	—	712	8	no	—
181	D₄	—	712	8	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	356	4	no	—
184	C₂×C₄	—	712	8	no	—
185	C₂×C₄	#15668009	1	2	yes	—
186	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
187	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
188	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
190	C₈	#15668009	1	2	yes	—
191	C₈	C₂×D₇₁₂	1	4	yes	—
192	C₈	C₂²×D₃₅₆	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	712	4	no	—
195	C₂²	—	712	8	no	—
196	C₂²	#19151975	1	2	yes	—
197	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	712	4	no	—
200	C₄	#19151975	1	2	yes	—
201	C₄	C₂×#15668009	1	1	yes	characteristic
202	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	1424	4	no	—
205	C₂	C₂×#19151975	1	1	yes	characteristic
206	C₂	#26033388	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152030

C₂

yes

maximal, complemented

#26030680

C₂

yes

maximal, complemented

#26022523

C₂

yes

maximal, complemented

#26022518

C₂

yes

maximal, complemented

C₂×#346×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668034

—

C₂×#15668034

C₂²

yes

—

#19152148

—

#19152030

—

C₂×Dic₇₁₂

C₂²

yes

—

C₂×#15667957

C₂²

yes

—

C₂×D₁₄₂₄

C₂²

yes

—

#19151982

—

#19151982

C₂²

yes

—

#19151978

C₂²

yes

—

#19151975

C₂²

yes

—

#19151971

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₂²

yes

—

C₂×#13083574

—

#15668034

—

#15668034

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

C₂³

yes

—

C₂×#13083563

—

C₂×#13083563

—

C₂×D₇₁₂

—

C₂×D₇₁₂

C₂³

yes

—

—

—

—

—

C₄×D₃₅₆

—

Dic₇₁₂

—

#15667957

—

D₁₄₂₄

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

complemented

C₂×C₁₆×C₈₉

C₂³

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083563

—

#13083563

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄²×C₈₉

D₈

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₁₆×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

#127

yes

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

C₂²×D₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

#493

yes

—

C₂²×C₈₉

C₂²×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#493

yes

—

C₄×C₈₉

C₂×#127

yes

characteristic

C₄×C₈₉

C₂²×D₈

yes

characteristic

Dic₈₉

—

#33716

—

maximal, Sylow

100

C₂×C₈₉

—

101

C₂×C₈₉

C₂×#493

yes

characteristic

102

C₂×C₈₉

#3414

yes

—

103

D₈₉

—

104

C₂×#508

—

105

#3147

—

106

#2251

—

107

#2250

—

108

C₂×#346

D₈₉

yes

characteristic, complemented

109

C₈₉

#33716

yes

characteristic, complemented, Sylow

110

C₂×#135

—

111

C₂×#135

—

178

—

112

#532

—

178

—

113

#508

—

178

—

114

C₂×Q₃₂

—

115

C₂×SD₃₂

—

116

C₂×D₁₆

—

117

C₂²×C₁₆

C₂×D₈₉

yes

—

118

#495

—

119

#495

—

178

—

120

#494

—

121

#493

—

122

#492

—

178

—

123

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

124

#346

C₂×D₈₉

yes

—

125

C₂×#41

—

178

—

126

#135

—

356

—

127

#135

—

178

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

178

—

130

C₂×SD₁₆

—

178

—

131

C₂×SD₁₆

—

178

—

132

C₂×D₈

—

133

C₂×D₈

—

178

—

134

C₂²×C₈

D₃₅₆

yes

characteristic

135

C₂²×C₈

C₂²×D₈₉

yes

characteristic

136

C₄⋊Q₈

—

178

—

137

#127

—

178

—

138

#124

—

178

—

139

#121

—

356

—

140

C₄×D₄

—

356

—

141

C₂×C₄²

D₃₅₆

yes

characteristic

142

Q₃₂

—

178

—

143

SD₃₂

—

178

—

144

D₁₆

—

178

—

145

C₂×C₁₆

D₃₅₆

yes

complemented

146

C₂×C₁₆

C₂²×D₈₉

yes

—

147

C₄×C₈

D₃₅₆

yes

—

148

C₄×C₈

C₂²×D₈₉

yes

—

149

#41

—

356

—

150

#41

—

356

—

151

C₂×Q₈

—

178

—

152

C₂×D₄

—

356

—

153

C₂×D₄

—

356

—

154

C₂×D₄

—

178

—

155

C₂²×C₄

—

356

—

156

C₂²×C₄

D₇₁₂

yes

—

157

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

158

Q₁₆

—

356

—

159

Q₁₆

—

178

—

160

SD₁₆

—

356

—

161

SD₁₆

—

356

—

162

D₈

—

356

—

163

D₈

—

178

—

164

C₂×C₈

D₇₁₂

yes

—

165

C₂×C₈

C₂×D₃₅₆

yes

characteristic

166

C₂×C₈

C₂×D₃₅₆

yes

—

167

C₂×C₈

C₂×D₃₅₆

yes

—

168

C₂×C₈

C₂³×D₈₉

yes

characteristic

169

C₄⋊C₄

—

356

—

170

C₂²⋊C₄

—

356

—

171

C₄²

D₇₁₂

yes

—

172

C₄²

C₂×D₃₅₆

yes

—

173

C₁₆

C₂×D₃₅₆

yes

—

174

C₂³

—

356

—

175

C₂³

C₂×D₇₁₂

yes

characteristic

176

Q₈

—

356

—

177

Q₈

—

356

—

178

D₄

—

356

—

179

D₄

—

356

—

180

D₄

—

712

—

181

D₄

—

712

—

182

C₂×C₄

—

183

C₂×C₄

—

356

—

184

C₂×C₄

—

712

—

185

C₂×C₄

#15668009

yes

—

186

C₂×C₄

C₂×D₇₁₂

yes

—

187

C₂×C₄

C₂×D₇₁₂

yes

—

188

C₂×C₄

C₂×D₇₁₂

yes

characteristic

189

C₂×C₄

C₂²×D₃₅₆

yes

characteristic

190

C₈

#15668009

yes

—

191

C₈

C₂×D₇₁₂

yes

—

192

C₈

C₂²×D₃₅₆

yes

—

193

C₂²

—

194

C₂²

—

712

—

195

C₂²

—

712

—

196

C₂²

#19151975

yes

—

197

C₂²

C₂²×D₇₁₂

yes

characteristic

198

C₄

—

199

C₄

—

712

—

200

C₄

#19151975

yes

—

201

C₄

C₂×#15668009

yes

characteristic

202

C₄

C₂²×D₇₁₂

yes

characteristic

203

C₂

—

204

C₂

—

1424

—

205

C₂

C₂×#19151975

yes

characteristic

206

C₂

#26033388

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#33716
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₇₁₂

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

C₂×#346×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₈

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
1. C₂×#19152030	C₂	semidirect
2. #26030680	C₂	semidirect
3. #26022523	C₂	semidirect
4. #26022518	C₂	semidirect
5. C₂×#346×C₈₉	C₂	semidirect
41. C₂×C₁₆×C₈₉	D₄	semidirect
108. C₂×#346	D₈₉	semidirect
109. C₈₉	#33716	semidirect
145. C₂×C₁₆	D₃₅₆	semidirect

H ≅ G/N

type

1. C₂×#19152030

C₂

semidirect

2. #26030680

C₂

semidirect

3. #26022523

C₂

semidirect

4. #26022518

C₂

semidirect

5. C₂×#346×C₈₉

C₂

semidirect

41. C₂×C₁₆×C₈₉

D₄

semidirect

108. C₂×#346

D₈₉

semidirect

109. C₈₉

#33716

semidirect

145. C₂×C₁₆

D₃₅₆

semidirect

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

subgroup	index
C₂×#19152030	2
#26030680	2
#26022523	2
#26022518	2
C₂×#346×C₈₉	2
#33716	89

subgroup

index

C₂×#346×C₈₉

Subgroups — #36709852

Conjugacy classes of subgroups

Complements

Maximal subgroups