Group #44551277 — subgroups

Subgroups — #44551277

Order 32128.

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×#16860800	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548650	C₂	1	1	yes	maximal, characteristic
3	#30631697	C₂	1	1	yes	maximal, characteristic
4	#30631696	C₂	1	1	yes	maximal, characteristic
5	C₂×#22548633	C₂	1	1	yes	maximal, characteristic
6	#30631645	C₂	1	1	yes	maximal, characteristic
7	C₂×#101×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
9	C₂×#16860800	—	2	1	no	—
10	C₂×#16860800	—	2	1	no	—
11	C₂×#16860800	—	2	2	no	—
12	C₂×#16860800	C₄	1	4	yes	complemented
13	C₂×#16860794	C₂²	1	1	yes	characteristic
14	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
15	#22548674	C₂²	1	1	yes	characteristic
16	#22548650	—	2	2	no	—
17	#22548640	—	2	2	no	—
18	#22548639	—	2	2	no	—
19	#22548633	—	2	2	no	—
20	#22548625	—	2	1	no	—
21	C₈×Dic₂₅₁	—	2	1	no	—
22	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
25	#101×C₂₅₁	—	2	2	no	—
26	C₂×Dic₅₀₂	—	2	1	no	—
27	C₂×Dic₅₀₂	—	2	1	no	—
28	C₂×Dic₅₀₂	—	2	1	no	—
29	C₂×Dic₅₀₂	—	2	1	no	—
30	C₂×Dic₅₀₂	—	2	1	no	—
31	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
32	C₂²×Dic₂₅₁	—	2	1	no	—
33	#16860800	—	2	4	no	—
34	#16860800	—	2	4	no	—
35	#16860794	—	2	1	no	—
36	#16860794	—	2	1	no	—
37	C₂×C₂₅₁⋊C₈	—	2	1	no	—
38	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₂₅₁⋊C₈	D₄	1	2	yes	—
42	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
43	C₂²⋊Dic₂₅₁	—	2	1	no	—
44	C₄×Dic₂₅₁	—	2	1	no	—
45	C₂×Q₈×C₂₅₁	—	2	1	no	—
46	C₂×Q₈×C₂₅₁	—	2	1	no	—
47	C₂×Q₈×C₂₅₁	—	2	1	no	—
48	C₂×Q₈×C₂₅₁	—	2	1	no	—
49	C₂×Q₈×C₂₅₁	—	2	1	no	—
50	C₂×Q₈×C₂₅₁	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₂₅₁	—	2	1	no	—
52	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
53	M₁₆×C₂₅₁	—	2	1	no	—
54	M₁₆×C₂₅₁	—	2	1	no	—
55	M₁₆×C₂₅₁	—	2	1	no	—
56	M₁₆×C₂₅₁	D₄	1	2	yes	—
57	C₂×C₈×C₂₅₁	—	2	1	no	—
58	C₂×C₈×C₂₅₁	—	2	1	no	—
59	Dic₅₀₂	—	2	2	no	—
60	Dic₅₀₂	—	2	2	no	—
61	Dic₅₀₂	—	4	1	no	—
62	Dic₅₀₂	—	4	1	no	—
63	C₂×Dic₂₅₁	—	2	2	no	—
64	C₂×Dic₂₅₁	—	2	2	no	—
65	C₂×Dic₂₅₁	—	4	1	no	—
66	C₂×Dic₂₅₁	—	4	1	no	—
67	C₂₅₁⋊C₈	—	4	1	no	—
68	C₂₅₁⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₂₅₁⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₂³×C₂₅₁	#41	1	1	yes	characteristic
71	Q₈×C₂₅₁	—	2	2	no	—
72	Q₈×C₂₅₁	—	2	2	no	—
73	Q₈×C₂₅₁	—	4	1	no	—
74	Q₈×C₂₅₁	—	4	1	no	—
75	C₂×C₄×C₂₅₁	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	4	1	no	—
77	C₂×C₄×C₂₅₁	—	4	1	no	—
78	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
84	C₈×C₂₅₁	—	2	2	no	—
85	C₈×C₂₅₁	—	4	1	no	—
86	C₂²×C₂₅₁	—	2	2	no	—
87	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₂₅₁	#121	1	1	yes	characteristic
89	C₂²×C₂₅₁	#124	1	1	yes	characteristic
90	C₄×C₂₅₁	—	4	2	no	—
91	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
93	C₄×C₂₅₁	#121	1	1	yes	characteristic
94	C₄×C₂₅₁	#127	1	1	yes	characteristic
95	Dic₂₅₁	—	4	2	no	—
96	Dic₂₅₁	—	4	2	no	—
97	C₂×C₂₅₁	—	2	1	no	—
98	C₂×C₂₅₁	—	2	1	no	—
99	C₂×C₂₅₁	#390	1	1	yes	characteristic
100	C₂×C₂₅₁	#473	1	2	yes	—
101	C₂₅₁	#1979	1	1	yes	characteristic, complemented, Sylow
102	#1979	—	251	1	no	maximal, Sylow
103	C₂²×Q₁₆	—	251	1	no	—
104	C₂×#108	—	251	1	no	—
105	#419	—	251	1	no	—
106	#419	—	251	1	no	—
107	C₂×#101	—	251	1	no	—
108	C₂×#101	D₂₅₁	1	1	yes	characteristic
109	#405	—	251	1	no	—
110	C₂²×Q₈	—	251	1	no	—
111	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
112	C₂×Q₁₆	—	502	1	no	—
113	C₂×Q₁₆	—	502	1	no	—
114	C₂×Q₁₆	—	502	2	no	—
115	C₂×Q₁₆	—	251	4	no	—
116	C₂×M₁₆	—	251	1	no	—
117	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
118	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
119	C₂²×C₈	—	251	1	no	—
120	#117	—	251	1	no	—
121	#108	—	502	2	no	—
122	#103	—	502	2	no	—
123	#103	—	502	2	no	—
124	#101	—	2	2	no	—
125	#101	—	502	2	no	—
126	#97	—	502	1	no	—
127	C₄×C₈	—	502	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	—	502	1	no	—
134	C₂×Q₈	—	502	1	no	—
135	C₂×Q₈	—	502	1	no	—
136	C₂×Q₈	—	502	1	no	—
137	C₂×Q₈	—	502	1	no	—
138	C₂×Q₈	—	251	2	no	—
139	C₂×Q₈	C₄×D₂₅₁	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	502	1	no	—
142	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
143	Q₁₆	—	502	4	no	—
144	Q₁₆	—	502	4	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	2	1	no	—
148	M₁₆	—	502	1	no	—
149	M₁₆	—	502	1	no	—
150	M₁₆	C₂²⋊D₂₅₁	1	2	yes	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	502	1	no	—
154	C₂×C₈	—	251	2	no	—
155	C₂×C₈	—	251	1	no	—
156	C₂×C₈	—	251	1	no	—
157	C₂×C₈	—	251	2	no	—
158	C₄⋊C₄	—	251	2	no	—
159	C₂²⋊C₄	—	502	1	no	—
160	C₄²	—	502	1	no	—
161	C₂³	#16860810	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	—	1004	1	no	—
167	Q₈	—	1004	1	no	—
168	Q₈	—	502	2	no	—
169	Q₈	—	502	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	1004	1	no	—
174	C₂×C₄	—	1004	1	no	—
175	C₂×C₄	—	502	2	no	—
176	C₂×C₄	—	502	2	no	—
177	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
178	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
179	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
180	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
182	C₂×C₄	#16860809	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	4	1	no	—
185	C₈	—	1004	1	no	—
186	C₈	—	251	2	no	—
187	C₈	—	251	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	C₄×C₂²⋊D₂₅₁	1	1	yes	characteristic
190	C₂²	#22548688	1	1	yes	characteristic
191	C₂²	#22548701	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	1004	2	no	—
194	C₄	—	1004	2	no	—
195	C₄	C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
196	C₄	(C₂²⋊D₂₅₁)⋊C₄	1	1	yes	characteristic
197	C₄	#22548687	1	1	yes	characteristic
198	C₄	#22548711	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#30631587	1	1	yes	characteristic
202	C₂	#30631961	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#16860800

C₂

yes

maximal, characteristic

C₂×#22548650

C₂

yes

maximal, characteristic

#30631697

C₂

yes

maximal, characteristic

#30631696

C₂

yes

maximal, characteristic

C₂×#22548633

C₂

yes

maximal, characteristic

#30631645

C₂

yes

maximal, characteristic

C₂×#101×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#16860794

C₂²

yes

characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

#22548674

C₂²

yes

—

—

—

—

—

C₈×Dic₂₅₁

—

C₂²×Q₈×C₂₅₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₅₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₅₁

C₂²

yes

characteristic

#101×C₂₅₁

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

C₂×C₄

yes

—

C₂²×Dic₂₅₁

—

—

—

—

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

C₂×C₂₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₅₁

D₄

yes

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

D₄

yes

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

C₂²⋊C₄

yes

—

C₂₅₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₂₅₁

#41

yes

characteristic

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

characteristic

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₄×D₄

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₅₁

C₄×D₄

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#390

yes

characteristic

100

C₂×C₂₅₁

#473

yes

—

101

C₂₅₁

#1979

yes

characteristic, complemented, Sylow

102

#1979

—

251

maximal, Sylow

103

C₂²×Q₁₆

—

251

—

104

C₂×#108

—

251

—

105

#419

—

251

—

106

#419

—

251

—

107

C₂×#101

—

251

—

108

C₂×#101

D₂₅₁

yes

characteristic

109

#405

—

251

—

110

C₂²×Q₈

—

251

—

111

C₂²×Q₈

C₂×D₂₅₁

yes

characteristic

112

C₂×Q₁₆

—

502

—

113

C₂×Q₁₆

—

502

—

114

C₂×Q₁₆

—

502

—

115

C₂×Q₁₆

—

251

—

116

C₂×M₁₆

—

251

—

117

C₂×M₁₆

C₂×D₂₅₁

yes

characteristic

118

C₂×M₁₆

C₂×D₂₅₁

yes

characteristic

119

C₂²×C₈

—

251

—

120

#117

—

251

—

121

#108

—

502

—

122

#103

—

502

—

123

#103

—

502

—

124

#101

—

125

#101

—

502

—

126

#97

—

502

—

127

C₄×C₈

—

502

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

502

—

134

C₂×Q₈

—

502

—

135

C₂×Q₈

—

502

—

136

C₂×Q₈

—

502

—

137

C₂×Q₈

—

502

—

138

C₂×Q₈

—

251

—

139

C₂×Q₈

C₄×D₂₅₁

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

502

—

142

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

143

Q₁₆

—

502

—

144

Q₁₆

—

502

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

502

—

149

M₁₆

—

502

—

150

M₁₆

C₂²⋊D₂₅₁

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

502

—

154

C₂×C₈

—

251

—

155

C₂×C₈

—

251

—

156

C₂×C₈

—

251

—

157

C₂×C₈

—

251

—

158

C₄⋊C₄

—

251

—

159

C₂²⋊C₄

—

502

—

160

C₄²

—

502

—

161

C₂³

#16860810

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

1004

—

167

Q₈

—

1004

—

168

Q₈

—

502

—

169

Q₈

—

502

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

1004

—

174

C₂×C₄

—

1004

—

175

C₂×C₄

—

502

—

176

C₂×C₄

—

502

—

177

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

178

C₂×C₄

D₄×D₂₅₁

yes

characteristic

179

C₂×C₄

D₄×D₂₅₁

yes

characteristic

180

C₂×C₄

D₄×D₂₅₁

yes

characteristic

181

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

182

C₂×C₄

#16860809

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

1004

—

186

C₈

—

251

—

187

C₈

—

251

—

188

C₂²

—

189

C₂²

C₄×C₂²⋊D₂₅₁

yes

characteristic

190

C₂²

#22548688

yes

characteristic

191

C₂²

#22548701

yes

characteristic

192

C₄

—

193

C₄

—

1004

—

194

C₄

—

1004

—

195

C₄

C₂²⋊C₄×D₂₅₁

yes

characteristic

196

C₄

(C₂²⋊D₂₅₁)⋊C₄

yes

characteristic

197

C₄

#22548687

yes

characteristic

198

C₄

#22548711

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#30631587

yes

characteristic

202

C₂

#30631961

yes

—

	subgroup	G/H
center	C₂²	#22548688
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂×#101×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#121
prime radical	C₂³×C₂₅₁	#41
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×#101	D₂₅₁
hypocenter	C₂₅₁	#1979
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#22548688

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₂×#101×C₂₅₁

C₂

C₂²×C₂₅₁

C₂³×C₂₅₁

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
12. C₂×#16860800	C₄	semidirect
101. C₂₅₁	#1979	semidirect

H ≅ G/N

type

12. C₂×#16860800

C₄

semidirect

101. C₂₅₁

#1979

semidirect

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

subgroup	index
C₂²×#16860800	2
C₂×#22548650	2
#30631697	2
#30631696	2
C₂×#22548633	2
#30631645	2
C₂×#101×C₂₅₁	2
#1979	251

subgroup

index

C₂×#101×C₂₅₁

251

Subgroups — #44551277

Conjugacy classes of subgroups

Complements

Maximal subgroups