Group #36453509 — subgroups

Subgroups — #36453509

Order 22784.

538 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 171 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	#26028650	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151346	C₂	1	4	yes	maximal
4	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₄	1	4	yes	—
6	C₂×#15668012	C₂²	1	2	yes	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	#19151468	—	2	2	no	—
9	#19151468	—	2	1	no	—
10	#19151468	—	2	1	no	—
11	#19151353	—	2	2	no	—
12	#19151346	—	2	8	no	—
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×#97×C₈₉	—	2	1	no	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	—	2	1	no	—
17	C₂²×Dic₁₇₈	C₂×C₄	1	4	yes	—
18	C₂²×#10623494	—	2	2	no	—
19	#15668012	—	2	4	no	—
20	#15668012	—	2	4	no	—
21	#15668012	—	4	8	no	—
22	C₂×C₄⋊Dic₈₉	—	2	4	no	—
23	C₂×C₄⋊Dic₈₉	—	2	4	no	—
24	C₂×C₄⋊Dic₈₉	—	2	2	no	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₄⋊#10623494	—	2	4	no	—
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₈₉	D₄	1	2	yes	—
30	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	—	2	2	no	—
34	C₄⋊C₈×C₈₉	D₄	1	4	yes	—
35	#97×C₈₉	—	4	2	no	—
36	C₂×Dic₁₇₈	—	2	8	no	—
37	C₂×Dic₁₇₈	—	4	8	no	—
38	C₂²×Dic₈₉	—	2	4	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂×#10623494	—	4	4	no	—
41	C₄⋊Dic₈₉	—	4	8	no	—
42	C₄⋊Dic₈₉	—	4	8	no	—
43	C₄⋊Dic₈₉	—	4	4	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	—	2	4	no	—
52	C₂×C₈×C₈₉	—	4	2	no	—
53	C₂×C₈×C₈₉	—	4	2	no	—
54	C₄²×C₈₉	—	2	4	no	—
55	C₄²×C₈₉	—	2	2	no	—
56	C₄²×C₈₉	—	2	4	no	—
57	C₄²×C₈₉	SD₁₆	1	2	yes	—
58	C₄²×C₈₉	Q₁₆	1	2	yes	—
59	C₄²×C₈₉	C₂×D₄	1	2	yes	—
60	C₄²×C₈₉	#41	1	2	yes	—
61	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	#100	1	4	yes	—
68	C₂×C₄×C₈₉	#103	1	4	yes	—
69	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₈₉	#121	1	2	yes	—
71	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#137	1	2	yes	—
76	C₂×C₄×C₈₉	#137	1	4	yes	—
77	C₂×Dic₈₉	—	4	4	no	—
78	C₂×Dic₈₉	—	4	8	no	—
79	Dic₁₇₈	—	8	8	no	—
80	C₈×C₈₉	—	4	4	no	—
81	#10623494	—	8	2	no	—
82	C₂²×C₈₉	#390	1	1	yes	characteristic
83	C₂²×C₈₉	C₂×#100	1	2	yes	—
84	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
85	C₂²×C₈₉	#419	1	2	yes	—
86	C₂²×C₈₉	#419	1	1	yes	characteristic
87	C₄×C₈₉	—	2	4	no	—
88	C₄×C₈₉	—	2	4	no	—
89	C₄×C₈₉	—	2	2	no	—
90	C₄×C₈₉	#461	1	2	yes	—
91	C₄×C₈₉	#462	1	2	yes	—
92	C₄×C₈₉	#491	1	2	yes	—
93	C₄×C₈₉	#497	1	2	yes	—
94	Dic₈₉	—	8	4	no	—
95	#9702	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	#1548	1	2	yes	—
97	C₂×C₈₉	#1558	1	2	yes	—
98	C₂×C₈₉	#1960	1	1	yes	characteristic
99	C₂×C₈₉	#1965	1	1	yes	characteristic
100	C₂×C₈₉	#1981	1	1	yes	characteristic
101	#2868	—	89	1	no	—
102	#1831	—	89	1	no	—
103	#1767	D₈₉	1	1	yes	characteristic
104	C₂×#333	—	89	4	no	—
105	C₈₉	#9702	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄⋊Q₈	—	89	4	no	—
107	C₂×C₄⋊Q₈	—	89	2	no	—
108	C₂×C₄⋊C₈	—	89	2	no	—
109	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
110	C₂×#97	—	2	1	no	—
111	#384	—	178	2	no	—
112	#384	—	178	1	no	—
113	#384	—	178	1	no	—
114	C₄³	C₂×D₈₉	1	1	yes	characteristic
115	#336	—	2	1	no	—
116	#336	—	178	2	no	—
117	#333	—	178	8	no	—
118	C₂²×Q₈	—	89	4	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	178	2	no	—
121	C₄⋊Q₈	—	356	8	no	—
122	C₄⋊Q₈	—	178	4	no	—
123	C₄⋊Q₈	—	178	4	no	—
124	C₂×C₄⋊C₄	—	178	4	no	—
125	C₂×C₄⋊C₄	—	178	4	no	—
126	C₂×C₄⋊C₄	—	178	2	no	—
127	C₂×C₄⋊C₄	—	178	2	no	—
128	C₂×C₄²	C₄×D₈₉	1	2	yes	—
129	C₂×C₄²	D₃₅₆	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
131	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	178	4	no	—
135	C₄⋊C₈	D₃₅₆	1	4	yes	—
136	#97	—	4	2	no	—
137	C₂×Q₈	—	356	8	no	—
138	C₂×Q₈	—	178	8	no	—
139	C₂²×C₄	—	178	4	no	—
140	C₂²×C₄	#13083551	1	2	yes	—
141	C₂²×C₄	#13083551	1	2	yes	—
142	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	356	2	no	—
149	C₂×C₈	—	356	4	no	—
150	C₄⋊C₄	—	356	8	no	—
151	C₄⋊C₄	—	356	4	no	—
152	C₄⋊C₄	—	356	2	no	—
153	C₄⋊C₄	—	356	2	no	—
154	C₄⋊C₄	—	356	8	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	#13083562	1	2	yes	—
159	C₄²	#13083564	1	2	yes	—
160	C₄²	C₂×D₃₅₆	1	2	yes	—
161	C₄²	#13083574	1	2	yes	—
162	C₂³	C₂×#13083551	1	1	yes	characteristic
163	Q₈	—	712	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	356	4	no	—
170	C₂×C₄	—	356	8	no	—
171	C₂×C₄	#15667928	1	4	yes	—
172	C₂×C₄	#15667935	1	4	yes	—
173	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
174	C₂×C₄	#15667986	1	2	yes	—
175	C₂×C₄	#15668000	1	1	yes	characteristic
176	C₂×C₄	#15668009	1	1	yes	characteristic
177	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
178	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
179	C₂×C₄	#15668042	1	2	yes	—
180	C₂×C₄	#15668045	1	4	yes	—
181	C₈	—	4	4	no	—
182	C₈	—	712	2	no	—
183	C₂²	#19151500	1	1	yes	characteristic
184	C₂²	C₂×#15667928	1	2	yes	—
185	C₂²	C₂×#15667935	1	1	yes	characteristic
186	C₂²	#19151606	1	1	yes	characteristic
187	C₂²	#19151608	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	712	4	no	—
192	C₄	#19151803	1	2	yes	—
193	C₄	#19151808	1	2	yes	—
194	C₄	#19151967	1	2	yes	—
195	C₄	#19151992	1	2	yes	—
196	C₂	#26018934	1	2	yes	—
197	C₂	#26018977	1	2	yes	—
198	C₂	#26021072	1	1	yes	characteristic
199	C₂	#26021092	1	1	yes	characteristic
200	C₂	#26021175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26028650

C₂

yes

maximal, characteristic

#26020369

C₂

yes

maximal, characteristic

C₂×#19151346

C₂

yes

maximal

C₈₉×#1767

C₂

yes

maximal, characteristic

C₂×#15668012

C₄

yes

—

C₂×#15668012

C₂²

yes

—

C₂×C₄⋊#10623494

C₂²

yes

—

—

—

—

—

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×#97×C₈₉

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×Dic₁₇₈

C₂×C₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₄⋊#10623494

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

D₄

yes

—

#97×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#100

yes

—

C₂×C₄×C₈₉

#103

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

#137

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

C₂×#100

yes

—

C₂²×C₈₉

C₂×#103

yes

characteristic

C₂²×C₈₉

#419

yes

—

C₂²×C₈₉

#419

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#461

yes

—

C₄×C₈₉

#462

yes

—

C₄×C₈₉

#491

yes

—

C₄×C₈₉

#497

yes

—

Dic₈₉

—

#9702

—

maximal, Sylow

C₂×C₈₉

#1548

yes

—

C₂×C₈₉

#1558

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

100

C₂×C₈₉

#1981

yes

characteristic

101

#2868

—

102

#1831

—

103

#1767

D₈₉

yes

characteristic

104

C₂×#333

—

105

C₈₉

#9702

yes

characteristic, complemented, Sylow

106

C₂×C₄⋊Q₈

—

107

C₂×C₄⋊Q₈

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

110

C₂×#97

—

111

#384

—

178

—

112

#384

—

178

—

113

#384

—

178

—

114

C₄³

C₂×D₈₉

yes

characteristic

115

#336

—

116

#336

—

178

—

117

#333

—

178

—

118

C₂²×Q₈

—

119

C₂²×C₈

—

120

C₂²×C₈

—

178

—

121

C₄⋊Q₈

—

356

—

122

C₄⋊Q₈

—

178

—

123

C₄⋊Q₈

—

178

—

124

C₂×C₄⋊C₄

—

178

—

125

C₂×C₄⋊C₄

—

178

—

126

C₂×C₄⋊C₄

—

178

—

127

C₂×C₄⋊C₄

—

178

—

128

C₂×C₄²

C₄×D₈₉

yes

—

129

C₂×C₄²

D₃₅₆

yes

—

130

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

131

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

132

C₂×C₄²

C₂²×D₈₉

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

178

—

135

C₄⋊C₈

D₃₅₆

yes

—

136

#97

—

137

C₂×Q₈

—

356

—

138

C₂×Q₈

—

178

—

139

C₂²×C₄

—

178

—

140

C₂²×C₄

#13083551

yes

—

141

C₂²×C₄

#13083551

yes

—

142

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

143

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

144

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

356

—

149

C₂×C₈

—

356

—

150

C₄⋊C₄

—

356

—

151

C₄⋊C₄

—

356

—

152

C₄⋊C₄

—

356

—

153

C₄⋊C₄

—

356

—

154

C₄⋊C₄

—

356

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#13083562

yes

—

159

C₄²

#13083564

yes

—

160

C₄²

C₂×D₃₅₆

yes

—

161

C₄²

#13083574

yes

—

162

C₂³

C₂×#13083551

yes

characteristic

163

Q₈

—

712

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

356

—

170

C₂×C₄

—

356

—

171

C₂×C₄

#15667928

yes

—

172

C₂×C₄

#15667935

yes

—

173

C₂×C₄

C₄×D₃₅₆

yes

—

174

C₂×C₄

#15667986

yes

—

175

C₂×C₄

#15668000

yes

characteristic

176

C₂×C₄

#15668009

yes

characteristic

177

C₂×C₄

C₂×#13083562

yes

characteristic

178

C₂×C₄

C₂×#13083564

yes

characteristic

179

C₂×C₄

#15668042

yes

—

180

C₂×C₄

#15668045

yes

—

181

C₈

—

182

C₈

—

712

—

183

C₂²

#19151500

yes

characteristic

184

C₂²

C₂×#15667928

yes

—

185

C₂²

C₂×#15667935

yes

characteristic

186

C₂²

#19151606

yes

characteristic

187

C₂²

#19151608

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

712

—

192

C₄

#19151803

yes

—

193

C₄

#19151808

yes

—

194

C₄

#19151967

yes

—

195

C₄

#19151992

yes

—

196

C₂

#26018934

yes

—

197

C₂

#26018977

yes

—

198

C₂

#26021072

yes

characteristic

199

C₂

#26021092

yes

characteristic

200

C₂

#26021175

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1767	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1767	D₈₉
hypocenter	C₈₉	#9702
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×#13083551

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1767

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
105. C₈₉	#9702	semidirect

H ≅ G/N

type

105. C₈₉

#9702

semidirect

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

subgroup	index
#26028650	2
#26020369	2
C₂×#19151346	2
C₈₉×#1767	2
#9702	89

subgroup

index

C₈₉×#1767

Subgroups — #36453509

Conjugacy classes of subgroups

Complements

Maximal subgroups