Group #37573434 — subgroups

Subgroups — #37573434

Order 24448.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15690277	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259257	C₂	1	1	yes	maximal, characteristic
3	#27132671	C₂	1	1	yes	maximal, characteristic
4	C₂×#20259241	C₂	1	1	yes	maximal, characteristic
5	C₂×#20259240	C₂	1	1	yes	maximal, characteristic
6	#27132619	C₂	1	1	yes	maximal, characteristic
7	#419×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
9	C₂×#15690277	—	2	1	no	—
10	C₂×#15690277	—	2	1	no	—
11	C₂×#15690277	—	2	2	no	—
12	C₂×#15690277	C₄	1	4	yes	complemented
13	C₂×#15690271	C₂²	1	1	yes	characteristic
14	C₂×#15690271	C₂²	1	1	yes	characteristic
15	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
16	#20259257	—	2	2	no	—
17	#20259245	—	2	2	no	—
18	#20259241	—	2	2	no	—
19	#20259240	—	2	2	no	—
20	#20259233	—	2	1	no	—
21	C₄×C₁₉₁⋊C₈	—	2	1	no	—
22	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
24	#117×C₁₉₁	C₂²	1	1	yes	characteristic
25	#103×C₁₉₁	—	2	2	no	—
26	C₂×Dic₃₈₂	—	2	1	no	—
27	C₂×Dic₃₈₂	—	2	1	no	—
28	C₂×Dic₃₈₂	—	2	1	no	—
29	C₂×Dic₃₈₂	—	2	1	no	—
30	C₂×Dic₃₈₂	—	2	1	no	—
31	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
32	C₂²×Dic₁₉₁	—	2	1	no	—
33	#15690277	—	2	4	no	—
34	#15690277	—	2	4	no	—
35	#15690271	—	2	1	no	—
36	#15690271	—	2	1	no	—
37	#15690271	—	2	1	no	—
38	#15690271	D₄	1	2	yes	—
39	C₂×C₁₉₁⋊C₈	—	2	1	no	—
40	C₂×C₁₉₁⋊C₈	—	2	1	no	—
41	C₂×C₁₉₁⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₁₉₁⋊C₈	D₄	1	2	yes	—
45	C₂×Q₈×C₁₉₁	—	2	1	no	—
46	C₂×Q₈×C₁₉₁	—	2	1	no	—
47	C₂×Q₈×C₁₉₁	—	2	1	no	—
48	C₂×Q₈×C₁₉₁	—	2	1	no	—
49	C₂×Q₈×C₁₉₁	—	2	1	no	—
50	C₂×Q₈×C₁₉₁	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₁₉₁	—	2	1	no	—
52	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
53	M₁₆×C₁₉₁	—	2	1	no	—
54	M₁₆×C₁₉₁	—	2	1	no	—
55	C₂×C₈×C₁₉₁	—	2	1	no	—
56	C₄⋊C₄×C₁₉₁	D₄	1	2	yes	—
57	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
58	C₄²×C₁₉₁	—	2	1	no	—
59	Dic₃₈₂	—	2	2	no	—
60	Dic₃₈₂	—	2	2	no	—
61	Dic₃₈₂	—	4	1	no	—
62	Dic₃₈₂	—	4	1	no	—
63	C₂×Dic₁₉₁	—	2	2	no	—
64	C₂×Dic₁₉₁	—	4	1	no	—
65	C₂×Dic₁₉₁	—	4	1	no	—
66	C₁₉₁⋊C₈	—	2	2	no	—
67	C₁₉₁⋊C₈	—	4	1	no	—
68	C₁₉₁⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₁₉₁⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₂³×C₁₉₁	#41	1	1	yes	characteristic
71	Q₈×C₁₉₁	—	2	2	no	—
72	Q₈×C₁₉₁	—	2	2	no	—
73	Q₈×C₁₉₁	—	4	1	no	—
74	Q₈×C₁₉₁	—	4	1	no	—
75	C₂×C₄×C₁₉₁	—	2	2	no	—
76	C₂×C₄×C₁₉₁	—	2	2	no	—
77	C₂×C₄×C₁₉₁	—	4	1	no	—
78	C₂×C₄×C₁₉₁	—	4	1	no	—
79	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
85	C₈×C₁₉₁	—	4	1	no	—
86	C₂²×C₁₉₁	—	2	2	no	—
87	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₉₁	#121	1	1	yes	characteristic
89	C₂²×C₁₉₁	#124	1	1	yes	characteristic
90	C₄×C₁₉₁	—	4	2	no	—
91	C₄×C₁₉₁	—	4	2	no	—
92	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₁₉₁	C₄×D₄	1	1	yes	characteristic
94	C₄×C₁₉₁	#121	1	1	yes	characteristic
95	C₄×C₁₉₁	#127	1	1	yes	characteristic
96	Dic₁₉₁	—	4	2	no	—
97	C₂×C₁₉₁	—	2	1	no	—
98	C₂×C₁₉₁	—	2	1	no	—
99	C₂×C₁₉₁	#390	1	1	yes	characteristic
100	C₂×C₁₉₁	#473	1	2	yes	—
101	C₁₉₁	#1979	1	1	yes	characteristic, complemented, Sylow
102	#1979	—	191	1	no	maximal, Sylow
103	C₂²×Q₁₆	—	191	1	no	—
104	C₂×#108	—	191	1	no	—
105	#419	—	191	1	no	—
106	#419	D₁₉₁	1	1	yes	characteristic
107	C₂×#101	—	191	1	no	—
108	C₂×#101	—	191	1	no	—
109	#405	—	191	1	no	—
110	C₂²×Q₈	—	191	1	no	—
111	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
112	C₂×Q₁₆	—	382	1	no	—
113	C₂×Q₁₆	—	382	1	no	—
114	C₂×Q₁₆	—	382	2	no	—
115	C₂×Q₁₆	—	191	4	no	—
116	C₂×M₁₆	—	191	1	no	—
117	C₂×M₁₆	—	191	1	no	—
118	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
119	C₂²×C₈	—	191	1	no	—
120	#117	C₂×D₁₉₁	1	1	yes	characteristic
121	#108	—	382	2	no	—
122	#103	—	2	2	no	—
123	#103	—	382	2	no	—
124	#101	—	382	2	no	—
125	#101	—	382	2	no	—
126	#97	—	382	1	no	—
127	C₄×C₈	—	382	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	382	1	no	—
134	C₂×Q₈	—	382	1	no	—
135	C₂×Q₈	—	382	1	no	—
136	C₂×Q₈	—	382	1	no	—
137	C₂×Q₈	—	382	1	no	—
138	C₂×Q₈	—	191	2	no	—
139	C₂×Q₈	C₄×D₁₉₁	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	382	1	no	—
142	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
143	Q₁₆	—	382	4	no	—
144	Q₁₆	—	382	4	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	382	1	no	—
148	M₁₆	—	382	1	no	—
149	M₁₆	—	382	1	no	—
150	M₁₆	—	191	2	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	382	1	no	—
153	C₂×C₈	—	382	1	no	—
154	C₂×C₈	—	191	2	no	—
155	C₂×C₈	—	191	1	no	—
156	C₂×C₈	—	191	1	no	—
157	C₂×C₈	—	191	2	no	—
158	C₄⋊C₄	C₂²⋊D₁₉₁	1	2	yes	—
159	C₂²⋊C₄	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₂³	#15690286	1	1	yes	characteristic
162	Q₈	—	2	2	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	4	1	no	—
166	Q₈	—	382	2	no	—
167	Q₈	—	382	2	no	—
168	Q₈	—	764	1	no	—
169	Q₈	—	764	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	382	2	no	—
175	C₂×C₄	—	764	1	no	—
176	C₂×C₄	—	764	1	no	—
177	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
178	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
179	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
180	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
182	C₂×C₄	#15690287	1	1	yes	characteristic
183	C₈	—	4	1	no	—
184	C₈	—	382	2	no	—
185	C₈	—	191	2	no	—
186	C₈	—	191	2	no	—
187	C₈	—	764	1	no	—
188	C₂²	—	2	2	no	—
189	C₂²	(C₂²⋊D₁₉₁)⋊C₄	1	1	yes	characteristic
190	C₂²	#20259294	1	1	yes	characteristic
191	C₂²	#20259308	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	764	2	no	—
195	C₄	C₂²⋊C₄×D₁₉₁	1	1	yes	characteristic
196	C₄	C₄×C₂²⋊D₁₉₁	1	1	yes	characteristic
197	C₄	#20259295	1	1	yes	characteristic
198	C₄	#20259318	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#27132563	1	1	yes	characteristic
202	C₂	#27132936	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15690277

C₂

yes

maximal, characteristic

C₂×#20259257

C₂

yes

maximal, characteristic

#27132671

C₂

yes

maximal, characteristic

C₂×#20259241

C₂

yes

maximal, characteristic

C₂×#20259240

C₂

yes

maximal, characteristic

#27132619

C₂

yes

maximal, characteristic

#419×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#15690271

C₂²

yes

characteristic

C₂×#15690271

C₂²

yes

characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

—

—

—

—

C₄×C₁₉₁⋊C₈

—

C₂²×Q₈×C₁₉₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₁

C₂²

yes

characteristic

#117×C₁₉₁

C₂²

yes

characteristic

#103×C₁₉₁

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

C₂×C₄

yes

—

C₂²×Dic₁₉₁

—

—

—

—

—

—

D₄

yes

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

C₂²⋊C₄

yes

—

C₁₉₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₉₁

#41

yes

characteristic

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₄×D₄

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₁

C₄×D₄

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#390

yes

characteristic

100

C₂×C₁₉₁

#473

yes

—

101

C₁₉₁

#1979

yes

characteristic, complemented, Sylow

102

#1979

—

191

maximal, Sylow

103

C₂²×Q₁₆

—

191

—

104

C₂×#108

—

191

—

105

#419

—

191

—

106

#419

D₁₉₁

yes

characteristic

107

C₂×#101

—

191

—

108

C₂×#101

—

191

—

109

#405

—

191

—

110

C₂²×Q₈

—

191

—

111

C₂²×Q₈

C₂×D₁₉₁

yes

characteristic

112

C₂×Q₁₆

—

382

—

113

C₂×Q₁₆

—

382

—

114

C₂×Q₁₆

—

382

—

115

C₂×Q₁₆

—

191

—

116

C₂×M₁₆

—

191

—

117

C₂×M₁₆

—

191

—

118

C₂×M₁₆

C₂×D₁₉₁

yes

characteristic

119

C₂²×C₈

—

191

—

120

#117

C₂×D₁₉₁

yes

characteristic

121

#108

—

382

—

122

#103

—

123

#103

—

382

—

124

#101

—

382

—

125

#101

—

382

—

126

#97

—

382

—

127

C₄×C₈

—

382

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

382

—

134

C₂×Q₈

—

382

—

135

C₂×Q₈

—

382

—

136

C₂×Q₈

—

382

—

137

C₂×Q₈

—

382

—

138

C₂×Q₈

—

191

—

139

C₂×Q₈

C₄×D₁₉₁

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

382

—

142

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

143

Q₁₆

—

382

—

144

Q₁₆

—

382

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

382

—

148

M₁₆

—

382

—

149

M₁₆

—

382

—

150

M₁₆

—

191

—

151

C₂×C₈

—

152

C₂×C₈

—

382

—

153

C₂×C₈

—

382

—

154

C₂×C₈

—

191

—

155

C₂×C₈

—

191

—

156

C₂×C₈

—

191

—

157

C₂×C₈

—

191

—

158

C₄⋊C₄

C₂²⋊D₁₉₁

yes

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₂³

#15690286

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

382

—

167

Q₈

—

382

—

168

Q₈

—

764

—

169

Q₈

—

764

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

382

—

175

C₂×C₄

—

764

—

176

C₂×C₄

—

764

—

177

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

178

C₂×C₄

D₄×D₁₉₁

yes

characteristic

179

C₂×C₄

D₄×D₁₉₁

yes

characteristic

180

C₂×C₄

D₄×D₁₉₁

yes

characteristic

181

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

182

C₂×C₄

#15690287

yes

characteristic

183

C₈

—

184

C₈

—

382

—

185

C₈

—

191

—

186

C₈

—

191

—

187

C₈

—

764

—

188

C₂²

—

189

C₂²

(C₂²⋊D₁₉₁)⋊C₄

yes

characteristic

190

C₂²

#20259294

yes

characteristic

191

C₂²

#20259308

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

764

—

195

C₄

C₂²⋊C₄×D₁₉₁

yes

characteristic

196

C₄

C₄×C₂²⋊D₁₉₁

yes

characteristic

197

C₄

#20259295

yes

characteristic

198

C₄

#20259318

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#27132563

yes

characteristic

202

C₂

#27132936

yes

—

	subgroup	G/H
center	C₂²	#20259294
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#419×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#121
prime radical	C₂³×C₁₉₁	#41
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#419	D₁₉₁
hypocenter	C₁₉₁	#1979
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#20259294

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#419×C₁₉₁

C₂

C₂²×C₁₉₁

C₂³×C₁₉₁

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
12. C₂×#15690277	C₄	semidirect
101. C₁₉₁	#1979	semidirect

H ≅ G/N

type

12. C₂×#15690277

C₄

semidirect

101. C₁₉₁

#1979

semidirect

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

subgroup	index
C₂²×#15690277	2
C₂×#20259257	2
#27132671	2
C₂×#20259241	2
C₂×#20259240	2
#27132619	2
#419×C₁₉₁	2
#1979	191

subgroup

index

#419×C₁₉₁

191

Subgroups — #37573434

Conjugacy classes of subgroups

Complements

Maximal subgroups