Group #37573591 — subgroups

Subgroups — #37573591

Order 24448.

510 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 99 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×#15690272	C₂	1	2	yes	maximal, complemented
2	C₂×#20259243	C₂	1	2	yes	maximal, complemented
3	#27132565	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₆₄	C₂²	1	1	yes	characteristic
7	C₂×#15690272	—	2	4	no	—
8	C₂×#15690272	—	2	4	no	—
9	C₂×#15690272	—	2	4	no	—
10	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
13	#20259243	—	2	4	no	—
14	C₂²×D₄×C₁₉₁	—	2	2	no	—
15	C₂²×D₄×C₁₉₁	C₂²	1	2	yes	—
16	#127×C₁₉₁	—	2	4	no	—
17	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₂³×D₁₉₁	—	2	1	no	—
19	C₂×D₇₆₄	—	2	2	no	—
20	C₂×D₇₆₄	—	2	2	no	—
21	C₂×D₇₆₄	—	2	1	no	—
22	C₂×D₇₆₄	D₄	1	2	yes	complemented
23	C₂²×Dic₁₉₁	—	2	1	no	—
24	#15690272	—	2	16	no	—
25	C₂×C₁₉₁⋊C₈	—	2	2	no	—
26	C₂×C₁₉₁⋊C₈	—	2	2	no	—
27	C₂×C₁₉₁⋊C₈	D₄	1	4	yes	complemented
28	C₄⋊Dic₁₉₁	—	2	2	no	—
29	(C₂×D₁₉₁)⋊C₄	—	4	4	no	—
30	C₂⁴×C₁₉₁	—	2	2	no	—
31	C₂×D₄×C₁₉₁	—	2	8	no	—
32	C₂×D₄×C₁₉₁	—	2	2	no	—
33	C₂×D₄×C₁₉₁	—	2	2	no	—
34	C₂×D₄×C₁₉₁	—	2	2	no	—
35	C₂×D₄×C₁₉₁	—	2	2	no	—
36	C₂×D₄×C₁₉₁	—	2	2	no	—
37	C₂×D₄×C₁₉₁	—	4	4	no	—
38	C₂×D₄×C₁₉₁	—	4	4	no	—
39	C₂×D₄×C₁₉₁	D₄	1	4	yes	—
40	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
41	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
42	C₄²×C₁₉₁	—	2	2	no	—
43	C₂²×D₁₉₁	—	2	2	no	—
44	C₂²×D₁₉₁	—	4	1	no	—
45	C₂²×D₁₉₁	—	4	1	no	—
46	C₂²×D₁₉₁	—	4	2	no	—
47	C₂²×D₁₉₁	—	4	2	no	—
48	D₇₆₄	—	2	4	no	—
49	D₇₆₄	—	4	2	no	—
50	C₂×Dic₁₉₁	—	4	1	no	—
51	C₂×Dic₁₉₁	—	4	2	no	—
52	C₁₉₁⋊C₈	—	2	4	no	—
53	C₂³×C₁₉₁	—	2	4	no	—
54	C₂³×C₁₉₁	—	4	2	no	—
55	C₂³×C₁₉₁	—	4	2	no	—
56	C₂³×C₁₉₁	—	4	4	no	—
57	C₂³×C₁₉₁	—	4	4	no	—
58	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
59	D₄×C₁₉₁	—	2	4	no	—
60	D₄×C₁₉₁	—	2	4	no	—
61	D₄×C₁₉₁	—	4	16	no	—
62	D₄×C₁₉₁	—	4	2	no	—
63	D₄×C₁₉₁	—	4	2	no	—
64	C₂×C₄×C₁₉₁	—	2	2	no	—
65	C₂×C₄×C₁₉₁	—	2	2	no	—
66	C₂×C₄×C₁₉₁	D₈	1	4	yes	complemented
67	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
71	C₂²×C₁₉₁	—	4	4	no	—
72	C₂²×C₁₉₁	—	4	8	no	—
73	C₂²×C₁₉₁	—	8	4	no	—
74	C₂²×C₁₉₁	—	8	2	no	—
75	C₂²×C₁₉₁	—	8	2	no	—
76	C₂²×C₁₉₁	#120	1	1	yes	characteristic
77	C₂²×C₁₉₁	#121	1	1	yes	characteristic
78	C₂²×C₁₉₁	#127	1	1	yes	characteristic
79	C₂²×C₁₉₁	C₂×D₈	1	2	yes	—
80	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×D₁₉₁	—	4	2	no	—
82	C₂×D₁₉₁	—	4	4	no	—
83	C₂×D₁₉₁	—	8	2	no	—
84	C₂×D₁₉₁	—	8	1	no	—
85	C₂×D₁₉₁	—	8	1	no	—
86	C₄×C₁₉₁	—	2	4	no	—
87	C₄×C₁₉₁	#120	1	2	yes	—
88	C₄×C₁₉₁	#121	1	2	yes	—
89	Dic₁₉₁	—	8	1	no	—
90	C₂×C₁₉₁	—	8	4	no	—
91	C₂×C₁₉₁	#392	1	1	yes	characteristic
92	C₂×C₁₉₁	#447	1	2	yes	—
93	C₂×C₁₉₁	#459	1	2	yes	—
94	C₂×C₁₉₁	#493	1	1	yes	characteristic
95	C₂×C₁₉₁	#496	1	1	yes	characteristic
96	D₁₉₁	—	8	2	no	—
97	C₁₉₁	#2011	1	1	yes	characteristic, complemented, Sylow
98	#2011	—	191	1	no	maximal, Sylow
99	C₂²×D₈	—	191	2	no	—
100	C₂×#127	D₁₉₁	1	1	yes	characteristic, complemented
101	C₂×#102	—	191	2	no	—
102	#390	—	191	1	no	—
103	#336	—	191	1	no	—
104	C₂²×D₄	—	2	2	no	—
105	C₂²×D₄	—	191	1	no	—
106	C₂²×D₄	C₂×D₁₉₁	1	2	yes	—
107	C₂×D₈	—	382	4	no	—
108	C₂×D₈	—	382	4	no	—
109	C₂×D₈	—	382	4	no	—
110	C₂²×C₈	—	191	2	no	—
111	#127	—	2	4	no	—
112	C₂×C₄⋊C₄	—	191	1	no	—
113	C₂×C₂²⋊C₄	—	382	2	no	—
114	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
115	#102	—	382	4	no	—
116	C₂⁴	—	2	2	no	—
117	C₂⁴	—	382	1	no	—
118	C₂×D₄	—	2	8	no	—
119	C₂×D₄	—	2	2	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	—	4	4	no	—
125	C₂×D₄	—	4	4	no	—
126	C₂×D₄	—	382	2	no	—
127	C₂×D₄	—	382	2	no	—
128	C₂×D₄	—	382	1	no	—
129	C₂×D₄	—	191	2	no	—
130	C₂×D₄	C₂²⋊D₁₉₁	1	4	yes	—
131	C₂²×C₄	—	382	1	no	—
132	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
133	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
134	D₈	—	382	16	no	—
135	C₂×C₈	—	382	2	no	—
136	C₂×C₈	—	382	2	no	—
137	C₂×C₈	—	191	4	no	—
138	C₄⋊C₄	—	382	2	no	—
139	C₂²⋊C₄	—	764	4	no	—
140	C₄²	—	2	2	no	—
141	C₂³	—	2	4	no	—
142	C₂³	—	4	4	no	—
143	C₂³	—	4	4	no	—
144	C₂³	—	4	2	no	—
145	C₂³	—	4	2	no	—
146	C₂³	—	382	2	no	—
147	C₂³	—	764	2	no	—
148	C₂³	—	764	2	no	—
149	C₂³	—	764	1	no	—
150	C₂³	—	764	1	no	—
151	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
152	D₄	—	2	4	no	—
153	D₄	—	2	4	no	—
154	D₄	—	4	16	no	—
155	D₄	—	4	2	no	—
156	D₄	—	4	2	no	—
157	D₄	—	382	4	no	—
158	D₄	—	764	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	764	2	no	—
162	C₂×C₄	—	764	1	no	—
163	C₂×C₄	#15690272	1	4	yes	complemented
164	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
165	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
167	C₂×C₄	#15690286	1	1	yes	characteristic
168	C₈	—	382	4	no	—
169	C₂²	—	4	8	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	8	2	no	—
172	C₂²	—	8	2	no	—
173	C₂²	—	8	4	no	—
174	C₂²	—	1528	1	no	—
175	C₂²	—	1528	1	no	—
176	C₂²	—	1528	2	no	—
177	C₂²	—	764	2	no	—
178	C₂²	—	764	4	no	—
179	C₂²	#20259291	1	1	yes	characteristic
180	C₂²	#20259293	1	1	yes	characteristic
181	C₂²	#20259318	1	1	yes	characteristic
182	C₂²	C₂×#15690272	1	2	yes	—
183	C₂²	#20259348	1	2	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	1528	1	no	—
186	C₄	#20259290	1	2	yes	—
187	C₄	#20259294	1	2	yes	—
188	C₂	—	8	4	no	—
189	C₂	—	1528	2	no	—
190	C₂	#27132571	1	1	yes	characteristic
191	C₂	#27132797	1	2	yes	—
192	C₂	#27132855	1	2	yes	—
193	C₂	#27133039	1	1	yes	characteristic
194	C₂	#27133050	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15690272

C₂

yes

maximal, complemented

C₂×#20259243

C₂

yes

maximal, complemented

#27132565

C₂

yes

maximal, characteristic, complemented

#27132418

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₆₄

C₂²

yes

—

—

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×(C₂×D₁₉₁)⋊C₄

—

#20259243

—

C₂²×D₄×C₁₉₁

—

C₂²×D₄×C₁₉₁

C₂²

yes

—

#127×C₁₉₁

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂³×D₁₉₁

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

D₄

yes

complemented

C₂²×Dic₁₉₁

—

#15690272

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

D₄

yes

complemented

C₄⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

—

C₂⁴×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

D₈

yes

complemented

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

C₂×D₈

yes

—

C₂²×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#120

yes

—

C₄×C₁₉₁

#121

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#392

yes

characteristic

C₂×C₁₉₁

#447

yes

—

C₂×C₁₉₁

#459

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

D₁₉₁

—

C₁₉₁

#2011

yes

characteristic, complemented, Sylow

#2011

—

191

maximal, Sylow

C₂²×D₈

—

191

—

100

C₂×#127

D₁₉₁

yes

characteristic, complemented

101

C₂×#102

—

191

—

102

#390

—

191

—

103

#336

—

191

—

104

C₂²×D₄

—

105

C₂²×D₄

—

191

—

106

C₂²×D₄

C₂×D₁₉₁

yes

—

107

C₂×D₈

—

382

—

108

C₂×D₈

—

382

—

109

C₂×D₈

—

382

—

110

C₂²×C₈

—

191

—

111

#127

—

112

C₂×C₄⋊C₄

—

191

—

113

C₂×C₂²⋊C₄

—

382

—

114

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

115

#102

—

382

—

116

C₂⁴

—

117

C₂⁴

—

382

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

382

—

127

C₂×D₄

—

382

—

128

C₂×D₄

—

382

—

129

C₂×D₄

—

191

—

130

C₂×D₄

C₂²⋊D₁₉₁

yes

—

131

C₂²×C₄

—

382

—

132

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

133

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

134

D₈

—

382

—

135

C₂×C₈

—

382

—

136

C₂×C₈

—

382

—

137

C₂×C₈

—

191

—

138

C₄⋊C₄

—

382

—

139

C₂²⋊C₄

—

764

—

140

C₄²

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

382

—

147

C₂³

—

764

—

148

C₂³

—

764

—

149

C₂³

—

764

—

150

C₂³

—

764

—

151

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

152

D₄

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

382

—

158

D₄

—

764

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

764

—

162

C₂×C₄

—

764

—

163

C₂×C₄

#15690272

yes

complemented

164

C₂×C₄

D₄×D₁₉₁

yes

characteristic

165

C₂×C₄

D₄×D₁₉₁

yes

—

166

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

167

C₂×C₄

#15690286

yes

characteristic

168

C₈

—

382

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

1528

—

175

C₂²

—

1528

—

176

C₂²

—

1528

—

177

C₂²

—

764

—

178

C₂²

—

764

—

179

C₂²

#20259291

yes

characteristic

180

C₂²

#20259293

yes

characteristic

181

C₂²

#20259318

yes

characteristic

182

C₂²

C₂×#15690272

yes

—

183

C₂²

#20259348

yes

—

184

C₄

—

185

C₄

—

1528

—

186

C₄

#20259290

yes

—

187

C₄

#20259294

yes

—

188

C₂

—

189

C₂

—

1528

—

190

C₂

#27132571

yes

characteristic

191

C₂

#27132797

yes

—

192

C₂

#27132855

yes

—

193

C₂

#27133039

yes

characteristic

194

C₂

#27133050

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×#127×C₁₉₁	C₂
socle	C₂³×C₁₉₁	C₂×D₄
hypercenter	C₂×#127	D₁₉₁
hypocenter	C₁₉₁	#2011
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×#127×C₁₉₁

C₂

socle

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂²×#15690272	C₂	semidirect
2. C₂×#20259243	C₂	semidirect
3. #27132565	C₂	semidirect
4. #27132418	C₂	semidirect
5. C₂×#127×C₁₉₁	C₂	semidirect
22. C₂×D₇₆₄	D₄	semidirect
27. C₂×C₁₉₁⋊C₈	D₄	semidirect
66. C₂×C₄×C₁₉₁	D₈	semidirect
97. C₁₉₁	#2011	semidirect
100. C₂×#127	D₁₉₁	semidirect
163. C₂×C₄	#15690272	semidirect

H ≅ G/N

type

1. C₂²×#15690272

C₂

semidirect

2. C₂×#20259243

C₂

semidirect

3. #27132565

C₂

semidirect

4. #27132418

C₂

semidirect

5. C₂×#127×C₁₉₁

C₂

semidirect

22. C₂×D₇₆₄

D₄

semidirect

27. C₂×C₁₉₁⋊C₈

D₄

semidirect

66. C₂×C₄×C₁₉₁

D₈

semidirect

97. C₁₉₁

#2011

semidirect

100. C₂×#127

D₁₉₁

semidirect

163. C₂×C₄

#15690272

semidirect

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

subgroup	index
C₂²×#15690272	2
C₂×#20259243	2
#27132565	2
#27132418	2
C₂×#127×C₁₉₁	2
#2011	191

subgroup

index

C₂×#127×C₁₉₁

191

Subgroups — #37573591

Conjugacy classes of subgroups

Complements

Maximal subgroups