Group #36473009 — subgroups

Subgroups — #36473009

Order 22784.

366 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 147 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×#19152000	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic
3	#26018343	C₂	1	2	yes	maximal
4	#26018343	C₂	1	2	yes	maximal
5	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668004	—	2	1	no	—
7	C₂²×#13083549	C₂²	1	2	yes	—
8	#19152000	—	2	2	no	—
9	#19152000	C₄	1	4	yes	complemented
10	C₂×#15667946	C₂²	1	1	yes	characteristic
11	C₂×#15667946	C₂²	1	1	yes	characteristic
12	C₂×#15667944	—	2	1	no	—
13	C₄⋊#13083549	—	2	2	no	—
14	C₂×#107×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
17	#340×C₈₉	—	2	1	no	—
18	#15668004	—	2	4	no	—
19	#15668004	—	4	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	1	no	—
21	C₂×C₄⋊Dic₈₉	—	2	1	no	—
22	C₂×#13083553	—	4	1	no	—
23	C₂×#13083553	—	4	1	no	—
24	C₂×#13083549	—	2	2	no	—
25	C₂×#13083549	—	2	4	no	—
26	#15667946	—	2	1	no	—
27	#15667946	—	2	1	no	—
28	#15667946	C₂×C₄	1	2	yes	—
29	#15667946	C₂×C₄	1	2	yes	—
30	#15667944	—	2	4	no	—
31	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄²×C₈₉	—	4	1	no	—
35	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
36	#107×C₈₉	—	2	2	no	—
37	#107×C₈₉	Q₈	1	4	yes	—
38	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₈₉	D₄	1	2	yes	—
40	#95×C₈₉	—	4	1	no	—
41	C₂²×Dic₈₉	—	4	1	no	—
42	C₂²×Dic₈₉	—	4	1	no	—
43	C₄⋊Dic₈₉	—	2	2	no	—
44	C₄⋊Dic₈₉	—	2	2	no	—
45	C₄⋊Dic₈₉	—	4	1	no	—
46	C₄⋊Dic₈₉	—	4	1	no	—
47	#13083553	—	4	4	no	—
48	#13083553	—	4	4	no	—
49	#13083549	—	4	2	no	—
50	C₂²×C₄×C₈₉	—	4	2	no	—
51	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₈₉	#41	1	2	yes	—
60	C₄⋊C₄×C₈₉	—	2	4	no	—
61	C₄⋊C₄×C₈₉	—	4	2	no	—
62	C₄²×C₈₉	—	8	2	no	—
63	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₈₉	SD₁₆	1	2	yes	—
65	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	4	4	no	—
70	C₂×C₄×C₈₉	—	8	2	no	—
71	C₂×C₄×C₈₉	—	8	2	no	—
72	C₂×C₄×C₈₉	#102	1	2	yes	—
73	C₂×C₄×C₈₉	#102	1	2	yes	—
74	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#137	1	2	yes	—
78	C₂×Dic₈₉	—	4	2	no	—
79	C₂×Dic₈₉	—	4	2	no	—
80	C₂×Dic₈₉	—	8	2	no	—
81	C₂×Dic₈₉	—	8	1	no	—
82	C₂×Dic₈₉	—	8	1	no	—
83	C₈×C₈₉	—	2	2	no	—
84	C₈×C₈₉	C₄×Q₈	1	2	yes	—
85	C₈×C₈₉	#124	1	1	yes	characteristic
86	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
87	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
88	C₂²×C₈₉	#474	1	2	yes	—
89	C₂²×C₈₉	#508	1	2	yes	—
90	C₂²×C₈₉	#509	1	1	yes	characteristic
91	C₂²×C₈₉	#510	1	1	yes	characteristic
92	C₄×C₈₉	—	2	2	no	—
93	C₄×C₈₉	—	2	2	no	—
94	C₄×C₈₉	—	8	4	no	—
95	C₄×C₈₉	#391	1	1	yes	characteristic
96	C₄×C₈₉	#419	1	1	yes	characteristic
97	C₄×C₈₉	#475	1	2	yes	—
98	Dic₈₉	—	8	2	no	—
99	Dic₈₉	—	8	2	no	—
100	#13277	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	#1988	1	1	yes	characteristic
102	C₂×C₈₉	#2138	1	1	yes	characteristic
103	C₂×C₈₉	#2139	1	1	yes	characteristic
104	C₂×C₈₉	#2228	1	2	yes	—
105	C₂×C₈₉	#2229	1	2	yes	—
106	C₂×#499	—	89	1	no	—
107	C₂×C₄⋊C₁₆	—	89	1	no	—
108	C₄×#107	D₈₉	1	1	yes	characteristic
109	#1380	—	89	2	no	—
110	#1380	—	89	2	no	—
111	C₈₉	#13277	1	1	yes	characteristic, complemented, Sylow
112	C₂×#125	—	178	1	no	—
113	C₂²×C₁₆	—	89	2	no	—
114	#499	—	89	4	no	—
115	#499	—	178	2	no	—
116	C₂×#107	—	89	1	no	—
117	C₂×#107	—	89	1	no	—
118	C₂×#107	C₂×D₈₉	1	2	yes	—
119	C₂×#106	—	178	1	no	—
120	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
121	C₄×C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₁₆	—	178	2	no	—
123	#340	—	2	1	no	—
124	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
126	#125	—	356	2	no	—
127	#125	—	178	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	356	1	no	—
130	C₂×C₄⋊C₄	—	356	1	no	—
131	C₂×C₄⋊C₄	—	178	1	no	—
132	C₂×C₄⋊C₄	—	178	1	no	—
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
135	C₂×C₁₆	—	178	2	no	—
136	C₂×C₁₆	—	178	4	no	—
137	#107	—	2	2	no	—
138	#107	—	89	2	no	—
139	#107	—	89	2	no	—
140	#107	—	178	1	no	—
141	#107	—	178	1	no	—
142	#107	Dic₁₇₈	1	4	yes	—
143	#106	—	178	4	no	—
144	C₄×C₈	C₄×D₈₉	1	2	yes	—
145	C₄×C₈	D₃₅₆	1	2	yes	—
146	#95	—	4	1	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	356	1	no	—
149	C₂²×C₄	—	356	1	no	—
150	C₂²×C₄	#13083559	1	1	yes	characteristic
151	C₂²×C₄	#13083559	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	#13083551	1	2	yes	—
155	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
156	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
157	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
158	C₂×C₈	#13083574	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	356	1	no	—
162	C₄⋊C₄	—	356	1	no	—
163	C₄⋊C₄	—	356	4	no	—
164	C₄⋊C₄	—	356	4	no	—
165	C₄⋊C₄	—	178	2	no	—
166	C₄⋊C₄	—	178	2	no	—
167	C₄²	—	8	2	no	—
168	C₄²	#13083551	1	2	yes	—
169	C₄²	#13083561	1	2	yes	—
170	C₁₆	—	356	2	no	—
171	C₂³	C₂×#13083559	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	356	2	no	—
179	C₂×C₄	—	356	2	no	—
180	C₂×C₄	—	712	2	no	—
181	C₂×C₄	—	712	1	no	—
182	C₂×C₄	—	712	1	no	—
183	C₂×C₄	#15667932	1	2	yes	—
184	C₂×C₄	#15667932	1	2	yes	—
185	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
186	C₂×C₄	#15667991	1	2	yes	—
187	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
188	C₂×C₄	#15668042	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	C₄×Dic₁₇₈	1	2	yes	—
191	C₈	#15668000	1	1	yes	characteristic
192	C₈	#15668012	1	1	yes	characteristic
193	C₂²	C₂×#15667932	1	1	yes	characteristic
194	C₂²	#19151878	1	2	yes	—
195	C₂²	#19152028	1	2	yes	—
196	C₂²	#19152034	1	1	yes	characteristic
197	C₂²	#19152039	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	8	4	no	—
201	C₄	—	712	2	no	—
202	C₄	—	712	2	no	—
203	C₄	#19151504	1	1	yes	characteristic
204	C₄	#19151606	1	1	yes	characteristic
205	C₄	#19151883	1	2	yes	—
206	C₂	#26021204	1	1	yes	characteristic
207	C₂	#26021957	1	1	yes	characteristic
208	C₂	#26021962	1	1	yes	characteristic
209	C₂	#26022409	1	2	yes	—
210	C₂	#26022416	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152000

C₂

yes

maximal, characteristic

C₂×C₄⋊#13083549

C₂

yes

maximal, characteristic

#26018343

C₂

yes

maximal

#26018343

C₂

yes

maximal

C₄×#107×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668004

—

C₂²×#13083549

C₂²

yes

—

#19152000

—

#19152000

C₄

yes

complemented

C₂×#15667946

C₂²

yes

characteristic

C₂×#15667946

C₂²

yes

characteristic

C₂×#15667944

—

C₄⋊#13083549

—

C₂×#107×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₈₉

—

#340×C₈₉

—

#15668004

—

#15668004

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

—

—

C₂×C₄

yes

—

#15667946

C₂×C₄

yes

—

#15667944

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

#107×C₈₉

—

#107×C₈₉

Q₈

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#95×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄×Q₈

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#508

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

100

#13277

—

maximal, Sylow

101

C₂×C₈₉

#1988

yes

characteristic

102

C₂×C₈₉

#2138

yes

characteristic

103

C₂×C₈₉

#2139

yes

characteristic

104

C₂×C₈₉

#2228

yes

—

105

C₂×C₈₉

#2229

yes

—

106

C₂×#499

—

107

C₂×C₄⋊C₁₆

—

108

C₄×#107

D₈₉

yes

characteristic

109

#1380

—

110

#1380

—

111

C₈₉

#13277

yes

characteristic, complemented, Sylow

112

C₂×#125

—

178

—

113

C₂²×C₁₆

—

114

#499

—

115

#499

—

178

—

116

C₂×#107

—

117

C₂×#107

—

118

C₂×#107

C₂×D₈₉

yes

—

119

C₂×#106

—

178

—

120

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

121

C₄×C₄⋊C₄

—

122

C₄⋊C₁₆

—

178

—

123

#340

—

124

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

125

C₂²×C₈

C₂²×D₈₉

yes

characteristic

126

#125

—

356

—

127

#125

—

178

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

356

—

130

C₂×C₄⋊C₄

—

356

—

131

C₂×C₄⋊C₄

—

178

—

132

C₂×C₄⋊C₄

—

178

—

133

C₂×C₄²

—

134

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

135

C₂×C₁₆

—

178

—

136

C₂×C₁₆

—

178

—

137

#107

—

138

#107

—

139

#107

—

140

#107

—

178

—

141

#107

—

178

—

142

#107

Dic₁₇₈

yes

—

143

#106

—

178

—

144

C₄×C₈

C₄×D₈₉

yes

—

145

C₄×C₈

D₃₅₆

yes

—

146

#95

—

147

C₂²×C₄

—

148

C₂²×C₄

—

356

—

149

C₂²×C₄

—

356

—

150

C₂²×C₄

#13083559

yes

characteristic

151

C₂²×C₄

#13083559

yes

characteristic

152

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

153

C₂×C₈

—

154

C₂×C₈

#13083551

yes

—

155

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

156

C₂×C₈

C₂×D₃₅₆

yes

characteristic

157

C₂×C₈

C₂×Dic₁₇₈

yes

—

158

C₂×C₈

#13083574

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

356

—

162

C₄⋊C₄

—

356

—

163

C₄⋊C₄

—

356

—

164

C₄⋊C₄

—

356

—

165

C₄⋊C₄

—

178

—

166

C₄⋊C₄

—

178

—

167

C₄²

—

168

C₄²

#13083551

yes

—

169

C₄²

#13083561

yes

—

170

C₁₆

—

356

—

171

C₂³

C₂×#13083559

yes

characteristic

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

356

—

179

C₂×C₄

—

356

—

180

C₂×C₄

—

712

—

181

C₂×C₄

—

712

—

182

C₂×C₄

—

712

—

183

C₂×C₄

#15667932

yes

—

184

C₂×C₄

#15667932

yes

—

185

C₂×C₄

C₂×#13083551

yes

characteristic

186

C₂×C₄

#15667991

yes

—

187

C₂×C₄

C₂×#13083561

yes

characteristic

188

C₂×C₄

#15668042

yes

—

189

C₈

—

190

C₈

C₄×Dic₁₇₈

yes

—

191

C₈

#15668000

yes

characteristic

192

C₈

#15668012

yes

characteristic

193

C₂²

C₂×#15667932

yes

characteristic

194

C₂²

#19151878

yes

—

195

C₂²

#19152028

yes

—

196

C₂²

#19152034

yes

characteristic

197

C₂²

#19152039

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

712

—

202

C₄

—

712

—

203

C₄

#19151504

yes

characteristic

204

C₄

#19151606

yes

characteristic

205

C₄

#19151883

yes

—

206

C₂

#26021204

yes

characteristic

207

C₂

#26021957

yes

characteristic

208

C₂

#26021962

yes

characteristic

209

C₂

#26022409

yes

—

210

C₂

#26022416

yes

—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#13277
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

subgroup

G/H

center

C₂³

C₂×#13083559

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₄×#107×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

N	H ≅ G/N	type
9. #19152000	C₄	semidirect
111. C₈₉	#13277	semidirect

H ≅ G/N

type

9. #19152000

C₄

semidirect

111. C₈₉

#13277

semidirect

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

subgroup	index
C₂×#19152000	2
C₂×C₄⋊#13083549	2
#26018343	2
C₄×#107×C₈₉	2
#13277	89

subgroup

index

C₄×#107×C₈₉

Subgroups — #36473009

Conjugacy classes of subgroups

Complements

Maximal subgroups