Group #26020681 — subgroups

Subgroups — #26020681

Order 11392.

432 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂³×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667933	C₂	1	2	yes	maximal, complemented
3	#19151458	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151360	C₂	1	2	yes	maximal, complemented
5	C₂×#121×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	—	2	4	no	—
7	C₂²×#10623494	—	2	2	no	—
8	C₂²×#10623494	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
11	#15667933	—	2	4	no	—
12	#15667919	—	2	1	no	—
13	C₂²×D₄×C₈₉	—	2	1	no	—
14	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
16	#121×C₈₉	—	2	2	no	—
17	#121×C₈₉	C₄	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
20	C₂²×Dic₈₉	—	2	2	no	—
21	C₂×#10623494	—	2	8	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₂×#10623494	—	2	2	no	—
24	C₂×#10623494	D₄	1	4	yes	complemented
25	C₄⋊Dic₈₉	—	2	2	no	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	C₂²⋊Dic₈₉	—	4	2	no	—
28	C₂⁴×C₈₉	—	2	1	no	—
29	C₂⁴×C₈₉	D₄	1	1	yes	characteristic
30	C₂×D₄×C₈₉	—	2	4	no	—
31	C₂×D₄×C₈₉	—	2	2	no	—
32	C₂×D₄×C₈₉	—	2	1	no	—
33	C₂×D₄×C₈₉	—	2	2	no	—
34	C₂×D₄×C₈₉	—	4	2	no	—
35	C₂×D₄×C₈₉	—	4	2	no	—
36	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₈₉	—	2	2	no	—
38	C₂²×C₄×C₈₉	—	2	2	no	—
39	C₂²×C₄×C₈₉	—	2	1	no	—
40	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	D₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
46	C₂²⋊C₄×C₈₉	—	2	4	no	—
47	C₂³×C₈₉	—	2	2	no	—
48	C₂³×C₈₉	—	2	2	no	—
49	C₂³×C₈₉	—	2	2	no	—
50	C₂³×C₈₉	—	4	2	no	—
51	C₂³×C₈₉	—	4	1	no	—
52	C₂³×C₈₉	—	4	1	no	—
53	C₂³×C₈₉	—	4	2	no	—
54	C₂³×C₈₉	C₂²⋊C₄	1	2	yes	—
55	C₂³×C₈₉	D₈	1	2	yes	—
56	C₂³×C₈₉	SD₁₆	1	2	yes	—
57	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
58	D₄×C₈₉	—	2	4	no	—
59	D₄×C₈₉	—	4	8	no	—
60	D₄×C₈₉	—	4	1	no	—
61	D₄×C₈₉	—	4	1	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	4	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	—	4	1	no	—
68	C₂×C₄×C₈₉	—	4	1	no	—
69	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#41	1	2	yes	—
74	C₂×Dic₈₉	—	4	2	no	—
75	C₂×Dic₈₉	—	4	2	no	—
76	C₂×Dic₈₉	—	4	2	no	—
77	#10623494	—	2	4	no	—
78	C₂²×C₈₉	—	2	2	no	—
79	C₂²×C₈₉	—	2	2	no	—
80	C₂²×C₈₉	—	2	4	no	—
81	C₂²×C₈₉	—	2	4	no	—
82	C₂²×C₈₉	—	4	2	no	—
83	C₂²×C₈₉	—	4	2	no	—
84	C₂²×C₈₉	—	4	2	no	—
85	C₂²×C₈₉	—	8	2	no	—
86	C₂²×C₈₉	—	8	2	no	—
87	C₂²×C₈₉	#102	1	4	yes	complemented
88	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₈₉	#121	1	2	yes	—
90	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
91	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
92	C₂²×C₈₉	#135	1	2	yes	—
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	—	4	2	no	—
95	C₄×C₈₉	C₄×D₄	1	2	yes	—
96	C₄×C₈₉	#120	1	1	yes	characteristic
97	C₄×C₈₉	#123	1	1	yes	characteristic
98	Dic₈₉	—	8	2	no	—
99	C₂×C₈₉	—	2	4	no	—
100	C₂×C₈₉	—	8	2	no	—
101	C₂×C₈₉	#386	1	1	yes	characteristic
102	C₂×C₈₉	C₂×#102	1	1	yes	characteristic
103	C₂×C₈₉	#416	1	1	yes	characteristic
104	C₂×C₈₉	#465	1	2	yes	—
105	C₂×C₈₉	#466	1	2	yes	—
106	#1893	—	89	1	no	maximal, Sylow
107	C₈₉	#1893	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₈	—	89	1	no	—
109	C₂×#121	D₈₉	1	1	yes	characteristic
110	C₂×#102	—	89	2	no	—
111	#380	—	89	1	no	—
112	#340	—	89	2	no	—
113	C₂²×D₄	—	2	1	no	—
114	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
115	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
116	C₂²×C₈	—	89	2	no	—
117	C₂²×C₈	—	178	4	no	—
118	C₂²×C₈	—	178	2	no	—
119	#121	—	2	2	no	—
120	#121	Dic₈₉	1	4	yes	complemented
121	C₂×C₄⋊C₄	—	89	2	no	—
122	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
123	C₂×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₂²⋊C₄	—	178	1	no	—
125	#102	—	178	4	no	—
126	#95	—	178	1	no	—
127	C₂⁴	—	2	1	no	—
128	C₂⁴	C₂²⋊D₈₉	1	1	yes	characteristic
129	C₂×D₄	—	2	4	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	2	1	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	—	4	2	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	178	2	no	—
140	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
141	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
143	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
144	C₂×C₈	—	89	4	no	—
145	C₂×C₈	—	178	8	no	—
146	C₂×C₈	—	178	2	no	—
147	C₂×C₈	—	178	2	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	178	2	no	—
150	C₄⋊C₄	—	178	2	no	—
151	C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₂²⋊C₄	—	356	2	no	—
154	C₂³	—	2	2	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	4	2	no	—
158	C₂³	—	4	1	no	—
159	C₂³	—	4	1	no	—
160	C₂³	—	4	2	no	—
161	C₂³	C₂²⋊Dic₈₉	1	2	yes	—
162	C₂³	#13083559	1	2	yes	—
163	C₂³	#13083562	1	2	yes	—
164	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
165	D₄	—	2	4	no	—
166	D₄	—	4	8	no	—
167	D₄	—	4	1	no	—
168	D₄	—	4	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	356	2	no	—
177	C₂×C₄	—	356	2	no	—
178	C₂×C₄	—	356	2	no	—
179	C₂×C₄	C₂²⋊Dic₈₉	1	2	yes	—
180	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
181	C₂×C₄	D₄×D₈₉	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
183	C₂×C₄	#13083573	1	2	yes	—
184	C₈	—	178	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	4	no	—
187	C₂²	—	2	4	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	8	2	no	—
193	C₂²	—	8	2	no	—
194	C₂²	#15667933	1	4	yes	complemented
195	C₂²	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
196	C₂²	#15667984	1	2	yes	—
197	C₂²	C₂×#13083559	1	1	yes	characteristic
198	C₂²	C₂×#13083562	1	1	yes	characteristic
199	C₂²	#15668032	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	712	2	no	—
203	C₄	D₄×Dic₈₉	1	2	yes	—
204	C₄	#15667980	1	1	yes	characteristic
205	C₄	#15667993	1	1	yes	characteristic
206	C₂	—	2	4	no	—
207	C₂	—	8	2	no	—
208	C₂	#19151478	1	1	yes	characteristic
209	C₂	C₂×#15667933	1	1	yes	characteristic
210	C₂	#19151594	1	1	yes	characteristic
211	C₂	#19151825	1	2	yes	—
212	C₂	#19151832	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#10623494

C₂

yes

maximal, characteristic, complemented

C₂×#15667933

C₂

yes

maximal, complemented

#19151458

C₂

yes

maximal, characteristic, complemented

#19151360

C₂

yes

maximal, complemented

C₂×#121×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

—

C₂²×#10623494

—

C₂²×#10623494

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

—

#15667933

—

#15667919

—

C₂²×D₄×C₈₉

—

C₂²×D₄×C₈₉

C₂²

yes

characteristic

C₂³×C₄×C₈₉

C₂²

yes

characteristic

#121×C₈₉

—

#121×C₈₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₉

—

C₂²×Dic₈₉

—

—

—

—

D₄

yes

complemented

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

—

C₂⁴×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₄

yes

—

C₂³×C₈₉

D₈

yes

—

C₂³×C₈₉

SD₁₆

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#102

yes

complemented

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#121

yes

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂²×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

#135

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

—

100

C₂×C₈₉

—

101

C₂×C₈₉

#386

yes

characteristic

102

C₂×C₈₉

C₂×#102

yes

characteristic

103

C₂×C₈₉

#416

yes

characteristic

104

C₂×C₈₉

#465

yes

—

105

C₂×C₈₉

#466

yes

—

106

#1893

—

maximal, Sylow

107

C₈₉

#1893

yes

characteristic, complemented, Sylow

108

C₂³×C₈

—

109

C₂×#121

D₈₉

yes

characteristic

110

C₂×#102

—

111

#380

—

112

#340

—

113

C₂²×D₄

—

114

C₂²×D₄

C₂×D₈₉

yes

characteristic

115

C₂³×C₄

C₂×D₈₉

yes

characteristic

116

C₂²×C₈

—

117

C₂²×C₈

—

178

—

118

C₂²×C₈

—

178

—

119

#121

—

120

#121

Dic₈₉

yes

complemented

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

123

C₂×C₂²⋊C₄

—

124

C₂×C₂²⋊C₄

—

178

—

125

#102

—

178

—

126

#95

—

178

—

127

C₂⁴

—

128

C₂⁴

C₂²⋊D₈₉

yes

characteristic

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

C₂×Dic₈₉

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

178

—

140

C₂²×C₄

C₂×Dic₈₉

yes

—

141

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

142

C₂²×C₄

C₂²⋊D₈₉

yes

—

143

C₂²×C₄

C₂²×D₈₉

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

178

—

146

C₂×C₈

—

178

—

147

C₂×C₈

—

178

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

178

—

150

C₄⋊C₄

—

178

—

151

C₄⋊C₄

C₂×Dic₈₉

yes

—

152

C₂²⋊C₄

—

153

C₂²⋊C₄

—

356

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

—

161

C₂³

C₂²⋊Dic₈₉

yes

—

162

C₂³

#13083559

yes

—

163

C₂³

#13083562

yes

—

164

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

165

D₄

—

166

D₄

—

167

D₄

—

168

D₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

356

—

177

C₂×C₄

—

356

—

178

C₂×C₄

—

356

—

179

C₂×C₄

C₂²⋊Dic₈₉

yes

—

180

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

181

C₂×C₄

D₄×D₈₉

yes

—

182

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

183

C₂×C₄

#13083573

yes

—

184

C₈

—

178

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

#15667933

yes

complemented

195

C₂²

C₂×C₂²⋊Dic₈₉

yes

characteristic

196

C₂²

#15667984

yes

—

197

C₂²

C₂×#13083559

yes

characteristic

198

C₂²

C₂×#13083562

yes

characteristic

199

C₂²

#15668032

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

712

—

203

C₄

D₄×Dic₈₉

yes

—

204

C₄

#15667980

yes

characteristic

205

C₄

#15667993

yes

characteristic

206

C₂

—

207

C₂

—

208

C₂

#19151478

yes

characteristic

209

C₂

C₂×#15667933

yes

characteristic

210

C₂

#19151594

yes

characteristic

211

C₂

#19151825

yes

—

212

C₂

#19151832

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting, prime radical	C₂×#121×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×#121	D₈₉
hypocenter	C₈₉	#1893
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting, prime radical

C₂×#121×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

N	H ≅ G/N	type
1. C₂³×#10623494	C₂	semidirect
2. C₂×#15667933	C₂	semidirect
3. #19151458	C₂	semidirect
4. #19151360	C₂	semidirect
17. #121×C₈₉	C₄	semidirect
24. C₂×#10623494	D₄	semidirect
87. C₂²×C₈₉	#102	semidirect
107. C₈₉	#1893	semidirect
120. #121	Dic₈₉	semidirect
194. C₂²	#15667933	semidirect

H ≅ G/N

type

1. C₂³×#10623494

C₂

semidirect

2. C₂×#15667933

C₂

semidirect

3. #19151458

C₂

semidirect

4. #19151360

C₂

semidirect

17. #121×C₈₉

C₄

semidirect

24. C₂×#10623494

D₄

semidirect

87. C₂²×C₈₉

#102

semidirect

107. C₈₉

#1893

semidirect

120. #121

Dic₈₉

semidirect

194. C₂²

#15667933

semidirect

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

subgroup	index
C₂³×#10623494	2
C₂×#15667933	2
#19151458	2
#19151360	2
C₂×#121×C₈₉	2
#1893	89

subgroup

index

C₂×#121×C₈₉

Subgroups — #26020681

Conjugacy classes of subgroups

Complements

Maximal subgroups