Group #37582978 — subgroups

Subgroups — #37582978

Order 24448.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#20259354	C₂	1	4	yes	maximal, complemented
2	#27133214	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#27133056	C₂	1	4	yes	maximal, complemented
5	#27133042	C₂	1	1	yes	maximal, characteristic, complemented
6	#27133037	C₂	1	1	yes	maximal, characteristic, complemented
7	#27133036	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#15690270	C₂	1	1	yes	maximal, characteristic, complemented
9	#494×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15690288	C₂²	1	2	yes	—
11	C₂×Q₈×D₁₉₁	C₂²	1	2	yes	—
12	#20259354	—	2	16	no	—
13	C₂×#15690277	C₂²	1	4	yes	—
14	C₂×#15690276	C₂²	1	4	yes	—
15	C₂×#15690274	C₂²	1	4	yes	—
16	C₂×#15690270	C₂²	1	2	yes	—
17	#20259322	C₂²	1	1	yes	characteristic
18	#20259321	—	2	2	no	—
19	#20259320	C₂²	1	2	yes	—
20	#20259319	C₂²	1	1	yes	characteristic
21	#20259311	C₂²	1	2	yes	—
22	#20259297	—	2	2	no	—
23	#20259284	—	2	2	no	—
24	C₄⋊C₄×D₁₉₁	—	2	2	no	—
25	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
26	#20259232	C₂²	1	2	yes	—
27	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₁₉₁	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
32	#15690288	—	2	8	no	—
33	C₂×Dic₃₈₂	—	2	2	no	—
34	C₂×Dic₃₈₂	C₂³	1	2	yes	—
35	Q₈×D₁₉₁	—	2	8	no	—
36	C₂×D₇₆₄	—	2	2	no	—
37	C₂×D₇₆₄	—	2	2	no	—
38	C₂×D₇₆₄	C₂³	1	2	yes	—
39	C₂×C₄×D₁₉₁	—	2	2	no	—
40	C₂×C₄×D₁₉₁	—	2	2	no	—
41	C₂×C₄×D₁₉₁	D₄	1	2	yes	—
42	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
43	#15690277	—	2	8	no	—
44	#15690276	—	2	8	no	—
45	#15690274	—	2	8	no	—
46	#15690270	D₄	1	8	yes	complemented
47	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₁₉₁	—	2	2	no	—
49	Dic₁₉₁⋊C₄	—	2	4	no	—
50	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
51	C₄×Dic₁₉₁	D₄	1	2	yes	—
52	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₁₉₁	C₂³	1	2	yes	—
54	C₂×Q₈×C₁₉₁	C₂³	1	2	yes	—
55	Q₁₆×C₁₉₁	—	2	8	no	—
56	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
57	C₄⋊C₄×C₁₉₁	—	2	2	no	—
58	C₄⋊C₄×C₁₉₁	—	2	2	no	—
59	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
60	C₂²×D₁₉₁	—	2	2	no	—
61	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
62	Dic₃₈₂	—	2	4	no	—
63	Dic₃₈₂	—	4	4	no	—
64	D₇₆₄	—	2	4	no	—
65	D₇₆₄	—	4	4	no	—
66	D₇₆₄	—	4	4	no	—
67	C₂×Dic₁₉₁	—	2	2	no	—
68	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
69	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
70	C₄×D₁₉₁	—	2	4	no	—
71	C₄×D₁₉₁	—	4	4	no	—
72	C₄×D₁₉₁	—	4	4	no	—
73	C₄×D₁₉₁	C₂×D₄	1	4	yes	—
74	C₁₉₁⋊C₈	C₂×D₄	1	4	yes	—
75	Q₈×C₁₉₁	—	2	4	no	—
76	Q₈×C₁₉₁	—	2	4	no	—
77	C₂×C₄×C₁₉₁	—	2	2	no	—
78	C₂×C₄×C₁₉₁	—	2	2	no	—
79	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
81	C₈×C₁₉₁	C₂×D₄	1	4	yes	—
82	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₁₉₁	—	2	2	no	—
84	C₂×D₁₉₁	—	4	2	no	—
85	C₂×D₁₉₁	—	4	4	no	—
86	C₂×D₁₉₁	#127	1	2	yes	—
87	C₄×C₁₉₁	—	4	2	no	—
88	C₄×C₁₉₁	—	4	2	no	—
89	C₄×C₁₉₁	#137	1	4	yes	complemented
90	C₄×C₁₉₁	C₂²×D₄	1	2	yes	—
91	Dic₁₉₁	—	2	2	no	—
92	Dic₁₉₁	—	4	2	no	—
93	Dic₁₉₁	#127	1	2	yes	—
94	C₂×C₁₉₁	C₂×#127	1	1	yes	characteristic
95	C₂×C₁₉₁	C₂×#137	1	2	yes	—
96	D₁₉₁	—	2	2	no	—
97	D₁₉₁	—	8	2	no	—
98	C₁₉₁	#3152	1	1	yes	characteristic, complemented, Sylow
99	#3152	—	191	1	no	maximal, Sylow
100	C₂×#137	—	191	4	no	—
101	#532	—	191	1	no	—
102	C₂×C₄⋊Q₈	—	191	1	no	—
103	#497	—	191	4	no	—
104	#494	—	191	1	no	—
105	#494	D₁₉₁	1	1	yes	characteristic, complemented
106	#492	—	191	1	no	—
107	#492	—	191	1	no	—
108	C₄×M₁₆	—	191	1	no	—
109	C₂×#41	—	191	2	no	—
110	C₂²×Q₈	—	191	2	no	—
111	#137	—	382	16	no	—
112	C₂×Q₁₆	—	191	4	no	—
113	C₂×Q₁₆	C₂×D₁₉₁	1	4	yes	—
114	C₂×SD₁₆	—	191	4	no	—
115	C₂×SD₁₆	—	191	4	no	—
116	C₂×M₁₆	—	191	2	no	—
117	C₄⋊Q₈	—	382	2	no	—
118	C₄⋊Q₈	—	191	2	no	—
119	C₄⋊Q₈	—	191	1	no	—
120	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
122	#127	—	191	1	no	—
123	#124	—	191	2	no	—
124	#121	—	382	2	no	—
125	C₄×D₄	—	382	2	no	—
126	C₂×C₄⋊C₄	—	382	2	no	—
127	C₂×C₄²	—	191	1	no	—
128	#97	—	191	2	no	—
129	C₄×C₈	—	191	1	no	—
130	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
131	#41	—	382	8	no	—
132	C₂×Q₈	—	382	2	no	—
133	C₂×Q₈	—	382	8	no	—
134	C₂×Q₈	—	191	2	no	—
135	C₂×Q₈	C₂²×D₁₉₁	1	2	yes	—
136	C₂×Q₈	C₂²×D₁₉₁	1	2	yes	—
137	C₂×D₄	—	382	2	no	—
138	C₂×D₄	—	382	2	no	—
139	C₂×D₄	—	191	2	no	—
140	C₂²×C₄	—	382	2	no	—
141	C₂²×C₄	—	382	2	no	—
142	C₂²×C₄	—	191	2	no	—
143	C₂²×C₄	—	191	1	no	—
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	382	8	no	—
146	SD₁₆	—	382	8	no	—
147	SD₁₆	—	382	8	no	—
148	M₁₆	—	191	8	no	—
149	C₂×C₈	—	191	2	no	—
150	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	382	4	no	—
154	C₄⋊C₄	—	382	2	no	—
155	C₂²⋊C₄	—	382	4	no	—
156	C₄²	—	191	2	no	—
157	C₄²	—	191	1	no	—
158	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
159	C₂³	—	382	2	no	—
160	C₂³	—	191	1	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	382	4	no	—
164	Q₈	—	764	4	no	—
165	D₄	—	382	4	no	—
166	D₄	—	764	4	no	—
167	D₄	—	764	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	382	4	no	—
171	C₂×C₄	—	382	2	no	—
172	C₂×C₄	—	191	2	no	—
173	C₂×C₄	—	191	1	no	—
174	C₂×C₄	—	191	4	no	—
175	C₂×C₄	—	764	4	no	—
176	C₂×C₄	—	764	4	no	—
177	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
178	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
179	C₈	—	191	4	no	—
180	C₈	D₄×D₁₉₁	1	4	yes	—
181	C₂²	—	382	2	no	—
182	C₂²	—	191	2	no	—
183	C₂²	—	764	2	no	—
184	C₂²	—	764	4	no	—
185	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	382	2	no	—
189	C₄	—	191	2	no	—
190	C₄	—	764	2	no	—
191	C₄	#20259354	1	4	yes	complemented
192	C₄	C₂×D₄×D₁₉₁	1	2	yes	—
193	C₂	—	382	2	no	—
194	C₂	—	1528	2	no	—
195	C₂	#127×D₁₉₁	1	1	yes	characteristic
196	C₂	C₂×#20259354	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259354

C₂

yes

maximal, complemented

#27133214

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₉₁

C₂

yes

maximal, characteristic, complemented

#27133056

C₂

yes

maximal, complemented

#27133042

C₂

yes

maximal, characteristic, complemented

#27133037

C₂

yes

maximal, characteristic, complemented

#27133036

C₂

yes

maximal, characteristic, complemented

C₄×#15690270

C₂

yes

maximal, characteristic, complemented

#494×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690288

C₂²

yes

—

C₂×Q₈×D₁₉₁

C₂²

yes

—

#20259354

—

C₂×#15690277

C₂²

yes

—

C₂×#15690276

C₂²

yes

—

C₂×#15690274

C₂²

yes

—

C₂×#15690270

C₂²

yes

—

#20259322

C₂²

yes

characteristic

#20259321

—

#20259320

C₂²

yes

—

#20259319

C₂²

yes

characteristic

#20259311

C₂²

yes

—

#20259297

—

#20259284

—

C₄⋊C₄×D₁₉₁

—

C₄²×D₁₉₁

C₂²

yes

characteristic

#20259232

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉₁

C₂²

yes

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

#15690288

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

C₂³

yes

—

Q₈×D₁₉₁

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

C₂³

yes

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

D₄

yes

—

C₂×C₄×D₁₉₁

C₂³

yes

—

—

—

D₄

yes

complemented

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₉₁

—

Dic₁₉₁⋊C₄

—

(C₂×D₁₉₁)⋊C₄

—

C₄×Dic₁₉₁

D₄

yes

—

C₄×Dic₁₉₁

C₂³

yes

characteristic

C₂×Q₈×C₁₉₁

C₂³

yes

—

C₂×Q₈×C₁₉₁

C₂³

yes

—

Q₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

—

C₂²×D₁₉₁

C₂×D₄

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₂×Dic₁₉₁

C₂×D₄

yes

characteristic

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

C₂×D₄

yes

—

C₁₉₁⋊C₈

C₂×D₄

yes

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

C₂×D₄

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

#127

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#137

yes

complemented

C₄×C₁₉₁

C₂²×D₄

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

#127

yes

—

C₂×C₁₉₁

C₂×#127

yes

characteristic

C₂×C₁₉₁

C₂×#137

yes

—

D₁₉₁

—

D₁₉₁

—

C₁₉₁

#3152

yes

characteristic, complemented, Sylow

#3152

—

191

maximal, Sylow

100

C₂×#137

—

191

—

101

#532

—

191

—

102

C₂×C₄⋊Q₈

—

191

—

103

#497

—

191

—

104

#494

—

191

—

105

#494

D₁₉₁

yes

characteristic, complemented

106

#492

—

191

—

107

#492

—

191

—

108

C₄×M₁₆

—

191

—

109

C₂×#41

—

191

—

110

C₂²×Q₈

—

191

—

111

#137

—

382

—

112

C₂×Q₁₆

—

191

—

113

C₂×Q₁₆

C₂×D₁₉₁

yes

—

114

C₂×SD₁₆

—

191

—

115

C₂×SD₁₆

—

191

—

116

C₂×M₁₆

—

191

—

117

C₄⋊Q₈

—

382

—

118

C₄⋊Q₈

—

191

—

119

C₄⋊Q₈

—

191

—

120

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

121

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

122

#127

—

191

—

123

#124

—

191

—

124

#121

—

382

—

125

C₄×D₄

—

382

—

126

C₂×C₄⋊C₄

—

382

—

127

C₂×C₄²

—

191

—

128

#97

—

191

—

129

C₄×C₈

—

191

—

130

C₄×C₈

C₂×D₁₉₁

yes

characteristic

131

#41

—

382

—

132

C₂×Q₈

—

382

—

133

C₂×Q₈

—

382

—

134

C₂×Q₈

—

191

—

135

C₂×Q₈

C₂²×D₁₉₁

yes

—

136

C₂×Q₈

C₂²×D₁₉₁

yes

—

137

C₂×D₄

—

382

—

138

C₂×D₄

—

382

—

139

C₂×D₄

—

191

—

140

C₂²×C₄

—

382

—

141

C₂²×C₄

—

382

—

142

C₂²×C₄

—

191

—

143

C₂²×C₄

—

191

—

144

Q₁₆

—

145

Q₁₆

—

382

—

146

SD₁₆

—

382

—

147

SD₁₆

—

382

—

148

M₁₆

—

191

—

149

C₂×C₈

—

191

—

150

C₂×C₈

C₂²×D₁₉₁

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

382

—

154

C₄⋊C₄

—

382

—

155

C₂²⋊C₄

—

382

—

156

C₄²

—

191

—

157

C₄²

—

191

—

158

C₄²

C₂²×D₁₉₁

yes

characteristic

159

C₂³

—

382

—

160

C₂³

—

191

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

382

—

164

Q₈

—

764

—

165

D₄

—

382

—

166

D₄

—

764

—

167

D₄

—

764

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

382

—

171

C₂×C₄

—

382

—

172

C₂×C₄

—

191

—

173

C₂×C₄

—

191

—

174

C₂×C₄

—

191

—

175

C₂×C₄

—

764

—

176

C₂×C₄

—

764

—

177

C₂×C₄

D₄×D₁₉₁

yes

—

178

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

179

C₈

—

191

—

180

C₈

D₄×D₁₉₁

yes

—

181

C₂²

—

382

—

182

C₂²

—

191

—

183

C₂²

—

764

—

184

C₂²

—

764

—

185

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

382

—

189

C₄

—

191

—

190

C₄

—

764

—

191

C₄

#20259354

yes

complemented

192

C₄

C₂×D₄×D₁₉₁

yes

—

193

C₂

—

382

—

194

C₂

—

1528

—

195

C₂

#127×D₁₉₁

yes

characteristic

196

C₂

C₂×#20259354

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#494×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	#27133214	C₂
hypercenter	#494	D₁₉₁
hypocenter	C₁₉₁	#3152
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#494×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

C₂

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#20259354	C₂	semidirect
3. C₄⋊Q₈×D₁₉₁	C₂	semidirect
4. #27133056	C₂	semidirect
5. #27133042	C₂	semidirect
6. #27133037	C₂	semidirect
7. #27133036	C₂	semidirect
8. C₄×#15690270	C₂	semidirect
9. #494×C₁₉₁	C₂	semidirect
46. #15690270	D₄	semidirect
89. C₄×C₁₉₁	#137	semidirect
98. C₁₉₁	#3152	semidirect
105. #494	D₁₉₁	semidirect
191. C₄	#20259354	semidirect

H ≅ G/N

type

1. C₂×#20259354

C₂

semidirect

3. C₄⋊Q₈×D₁₉₁

C₂

semidirect

4. #27133056

C₂

semidirect

5. #27133042

C₂

semidirect

6. #27133037

C₂

semidirect

7. #27133036

C₂

semidirect

8. C₄×#15690270

C₂

semidirect

9. #494×C₁₉₁

C₂

semidirect

46. #15690270

D₄

semidirect

89. C₄×C₁₉₁

#137

semidirect

98. C₁₉₁

#3152

semidirect

105. #494

D₁₉₁

semidirect

191. C₄

#20259354

semidirect

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

subgroup	index
C₂×#20259354	2
#27133214	2
C₄⋊Q₈×D₁₉₁	2
#27133056	2
#27133042	2
#27133037	2
#27133036	2
C₄×#15690270	2
#494×C₁₉₁	2
#3152	191

subgroup

index

C₄⋊Q₈×D₁₉₁

#494×C₁₉₁

191

Subgroups — #37582978

Conjugacy classes of subgroups

Complements

Maximal subgroups