Group #46821339 — subgroups

Subgroups — #46821339

Order 34432.

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×#22575482	C₂	1	4	yes	maximal, complemented
2	#31771153	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770996	C₂	1	4	yes	maximal, complemented
5	#31770985	C₂	1	1	yes	maximal, characteristic, complemented
6	#31770984	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770977	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#16864385	C₂	1	1	yes	maximal, characteristic, complemented
9	#492×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16864401	C₂²	1	2	yes	—
11	C₂×Q₈×D₂₆₉	C₂²	1	2	yes	—
12	#22575482	—	2	16	no	—
13	C₂×Dic₁₀₇₆	C₂²	1	4	yes	—
14	C₂×#16864392	C₂²	1	4	yes	—
15	C₂×#16864390	C₂²	1	4	yes	—
16	C₂×#16864385	C₂²	1	2	yes	—
17	#22575453	C₂²	1	1	yes	characteristic
18	#22575453	C₂²	1	1	yes	characteristic
19	#22575452	—	2	2	no	—
20	#22575451	C₂²	1	2	yes	—
21	#22575440	C₂²	1	2	yes	—
22	#22575424	—	2	2	no	—
23	D₄×Dic₂₆₉	—	2	2	no	—
24	C₄⋊C₄×D₂₆₉	—	2	2	no	—
25	C₄²×D₂₆₉	C₂²	1	1	yes	characteristic
26	#22575363	C₂²	1	2	yes	—
27	C₄×#14524130	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₂₆₉	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
30	#127×C₂₆₉	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
32	#16864401	—	2	8	no	—
33	C₂×Dic₅₃₈	—	2	2	no	—
34	C₂×Dic₅₃₈	C₂³	1	2	yes	—
35	C₂×Dic₅₃₈	C₂³	1	2	yes	—
36	Q₈×D₂₆₉	—	2	8	no	—
37	C₂×C₂²⋊D₂₆₉	—	2	2	no	—
38	C₂²×Dic₂₆₉	—	2	2	no	—
39	C₂×C₄×D₂₆₉	—	2	2	no	—
40	C₂×C₄×D₂₆₉	D₄	1	2	yes	—
41	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
42	Dic₁₀₇₆	—	2	8	no	—
43	#16864392	—	2	8	no	—
44	#16864390	—	2	8	no	—
45	#16864385	D₄	1	8	yes	complemented
46	C₂×#14524130	C₂³	1	2	yes	—
47	C₄⋊Dic₂₆₉	—	2	2	no	—
48	C₄⋊Dic₂₆₉	—	2	2	no	—
49	#16864381	—	2	4	no	—
50	C₂²⋊Dic₂₆₉	—	2	4	no	—
51	C₄×Dic₂₆₉	D₄	1	2	yes	—
52	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₂₆₉	C₂³	1	2	yes	—
54	C₂×D₄×C₂₆₉	—	2	2	no	—
55	C₂×D₄×C₂₆₉	C₂³	1	2	yes	—
56	SD₁₆×C₂₆₉	—	2	8	no	—
57	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₆₉	—	2	2	no	—
59	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
60	C₂²×D₂₆₉	C₂×D₄	1	1	yes	characteristic
61	Dic₅₃₈	—	2	4	no	—
62	Dic₅₃₈	—	2	4	no	—
63	Dic₅₃₈	—	4	4	no	—
64	C₂²⋊D₂₆₉	—	4	4	no	—
65	C₂×Dic₂₆₉	—	2	2	no	—
66	C₂×Dic₂₆₉	—	2	2	no	—
67	C₂×Dic₂₆₉	—	4	4	no	—
68	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
69	C₂×Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
70	C₄×D₂₆₉	—	2	4	no	—
71	C₄×D₂₆₉	—	4	4	no	—
72	C₄×D₂₆₉	C₂×D₄	1	4	yes	—
73	#14524130	C₂×D₄	1	4	yes	—
74	C₂³×C₂₆₉	—	2	2	no	—
75	Q₈×C₂₆₉	—	2	4	no	—
76	D₄×C₂₆₉	—	2	4	no	—
77	D₄×C₂₆₉	—	4	4	no	—
78	C₂×C₄×C₂₆₉	—	2	2	no	—
79	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
81	C₈×C₂₆₉	C₂×D₄	1	4	yes	—
82	C₂²×C₂₆₉	—	4	2	no	—
83	C₂²×C₂₆₉	—	4	4	no	—
84	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₂₆₉	—	2	2	no	—
86	C₂×D₂₆₉	#127	1	2	yes	—
87	C₄×C₂₆₉	—	4	2	no	—
88	C₄×C₂₆₉	#137	1	4	yes	complemented
89	C₄×C₂₆₉	C₂²×D₄	1	2	yes	—
90	Dic₂₆₉	—	2	2	no	—
91	Dic₂₆₉	—	4	2	no	—
92	Dic₂₆₉	—	4	2	no	—
93	Dic₂₆₉	#127	1	2	yes	—
94	C₂×C₂₆₉	—	8	2	no	—
95	C₂×C₂₆₉	C₂×#127	1	1	yes	characteristic
96	C₂×C₂₆₉	C₂×#137	1	2	yes	—
97	D₂₆₉	—	2	2	no	—
98	C₂₆₉	#3152	1	1	yes	characteristic, complemented, Sylow
99	#3152	—	269	1	no	maximal, Sylow
100	C₂×#137	—	269	4	no	—
101	#532	—	269	1	no	—
102	C₂×C₄⋊Q₈	—	269	1	no	—
103	#497	—	269	4	no	—
104	#494	—	269	1	no	—
105	#494	—	269	1	no	—
106	#492	—	269	1	no	—
107	#492	D₂₆₉	1	1	yes	characteristic, complemented
108	C₄×M₁₆	—	269	1	no	—
109	C₂×#41	—	269	2	no	—
110	C₂²×Q₈	—	269	2	no	—
111	#137	—	538	16	no	—
112	C₂×Q₁₆	—	269	4	no	—
113	C₂×Q₁₆	—	269	4	no	—
114	C₂×SD₁₆	—	269	4	no	—
115	C₂×SD₁₆	C₂×D₂₆₉	1	4	yes	—
116	C₂×M₁₆	—	269	2	no	—
117	C₄⋊Q₈	—	269	2	no	—
118	C₄⋊Q₈	—	269	1	no	—
119	C₄⋊Q₈	—	269	1	no	—
120	C₄⋊Q₈	—	538	2	no	—
121	C₄⋊Q₈	C₂×D₂₆₉	1	1	yes	characteristic
122	#127	C₂×D₂₆₉	1	1	yes	characteristic
123	#124	—	269	2	no	—
124	#121	—	538	2	no	—
125	C₄×D₄	—	538	2	no	—
126	C₂×C₄⋊C₄	—	538	2	no	—
127	C₂×C₄²	—	269	1	no	—
128	#97	—	269	2	no	—
129	C₄×C₈	—	269	1	no	—
130	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
131	#41	—	538	8	no	—
132	C₂×Q₈	—	269	2	no	—
133	C₂×Q₈	—	269	2	no	—
134	C₂×Q₈	—	538	2	no	—
135	C₂×Q₈	—	538	8	no	—
136	C₂×Q₈	C₂²×D₂₆₉	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	538	2	no	—
139	C₂×D₄	C₂²×D₂₆₉	1	2	yes	—
140	C₂²×C₄	—	269	1	no	—
141	C₂²×C₄	—	269	2	no	—
142	C₂²×C₄	—	538	2	no	—
143	C₂²×C₄	—	538	2	no	—
144	Q₁₆	—	538	8	no	—
145	Q₁₆	—	538	8	no	—
146	SD₁₆	—	2	8	no	—
147	SD₁₆	—	538	8	no	—
148	M₁₆	—	269	8	no	—
149	C₂×C₈	—	269	2	no	—
150	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	538	4	no	—
153	C₄⋊C₄	—	538	2	no	—
154	C₄⋊C₄	—	538	2	no	—
155	C₂²⋊C₄	—	538	4	no	—
156	C₄²	—	269	2	no	—
157	C₄²	—	269	1	no	—
158	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
159	C₂³	—	2	2	no	—
160	C₂³	—	269	1	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	538	4	no	—
163	Q₈	—	538	4	no	—
164	Q₈	—	1076	4	no	—
165	D₄	—	2	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	1076	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	269	1	no	—
170	C₂×C₄	—	269	4	no	—
171	C₂×C₄	—	269	2	no	—
172	C₂×C₄	—	538	4	no	—
173	C₂×C₄	—	538	2	no	—
174	C₂×C₄	—	538	2	no	—
175	C₂×C₄	—	1076	4	no	—
176	C₂×C₄	—	1076	4	no	—
177	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
178	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
179	C₈	—	269	4	no	—
180	C₈	D₄×D₂₆₉	1	4	yes	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	269	2	no	—
184	C₂²	—	538	2	no	—
185	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	269	2	no	—
188	C₄	—	538	2	no	—
189	C₄	—	1076	2	no	—
190	C₄	—	1076	2	no	—
191	C₄	#22575482	1	4	yes	complemented
192	C₄	C₂×D₄×D₂₆₉	1	2	yes	—
193	C₂	—	8	2	no	—
194	C₂	—	538	2	no	—
195	C₂	#127×D₂₆₉	1	1	yes	characteristic
196	C₂	C₂×#22575482	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575482

C₂

yes

maximal, complemented

#31771153

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₂₆₉

C₂

yes

maximal, characteristic, complemented

#31770996

C₂

yes

maximal, complemented

#31770985

C₂

yes

maximal, characteristic, complemented

#31770984

C₂

yes

maximal, characteristic, complemented

#31770977

C₂

yes

maximal, characteristic, complemented

C₄×#16864385

C₂

yes

maximal, characteristic, complemented

#492×C₂₆₉

C₂

yes

maximal, characteristic, complemented

C₂×#16864401

C₂²

yes

—

C₂×Q₈×D₂₆₉

C₂²

yes

—

#22575482

—

C₂×Dic₁₀₇₆

C₂²

yes

—

C₂×#16864392

C₂²

yes

—

C₂×#16864390

C₂²

yes

—

C₂×#16864385

C₂²

yes

—

#22575453

C₂²

yes

characteristic

#22575453

C₂²

yes

characteristic

#22575452

—

#22575451

C₂²

yes

—

#22575440

C₂²

yes

—

#22575424

—

D₄×Dic₂₆₉

—

C₄⋊C₄×D₂₆₉

—

C₄²×D₂₆₉

C₂²

yes

characteristic

#22575363

C₂²

yes

—

C₄×#14524130

C₂²

yes

characteristic

C₂×SD₁₆×C₂₆₉

C₂²

yes

—

C₄⋊Q₈×C₂₆₉

C₂²

yes

characteristic

#127×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₂²

yes

characteristic

#16864401

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂³

yes

—

C₂×Dic₅₃₈

C₂³

yes

—

Q₈×D₂₆₉

—

C₂×C₂²⋊D₂₆₉

—

C₂²×Dic₂₆₉

—

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

D₄

yes

—

C₂×C₄×D₂₆₉

C₂³

yes

characteristic

Dic₁₀₇₆

—

#16864392

—

#16864390

—

#16864385

D₄

yes

complemented

C₂×#14524130

C₂³

yes

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

#16864381

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

D₄

yes

—

C₄×Dic₂₆₉

C₂³

yes

characteristic

C₂×Q₈×C₂₆₉

C₂³

yes

—

C₂×D₄×C₂₆₉

—

C₂×D₄×C₂₆₉

C₂³

yes

—

SD₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

C₂³

yes

characteristic

C₂²×D₂₆₉

C₂×D₄

yes

characteristic

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂²⋊D₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂×D₄

yes

—

C₂×Dic₂₆₉

C₂×D₄

yes

characteristic

C₄×D₂₆₉

—

C₄×D₂₆₉

—

C₄×D₂₆₉

C₂×D₄

yes

—

#14524130

C₂×D₄

yes

—

C₂³×C₂₆₉

—

Q₈×C₂₆₉

—

D₄×C₂₆₉

—

D₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

C₂×D₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

#127

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#137

yes

complemented

C₄×C₂₆₉

C₂²×D₄

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

#127

yes

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂×#127

yes

characteristic

C₂×C₂₆₉

C₂×#137

yes

—

D₂₆₉

—

C₂₆₉

#3152

yes

characteristic, complemented, Sylow

#3152

—

269

maximal, Sylow

100

C₂×#137

—

269

—

101

#532

—

269

—

102

C₂×C₄⋊Q₈

—

269

—

103

#497

—

269

—

104

#494

—

269

—

105

#494

—

269

—

106

#492

—

269

—

107

#492

D₂₆₉

yes

characteristic, complemented

108

C₄×M₁₆

—

269

—

109

C₂×#41

—

269

—

110

C₂²×Q₈

—

269

—

111

#137

—

538

—

112

C₂×Q₁₆

—

269

—

113

C₂×Q₁₆

—

269

—

114

C₂×SD₁₆

—

269

—

115

C₂×SD₁₆

C₂×D₂₆₉

yes

—

116

C₂×M₁₆

—

269

—

117

C₄⋊Q₈

—

269

—

118

C₄⋊Q₈

—

269

—

119

C₄⋊Q₈

—

269

—

120

C₄⋊Q₈

—

538

—

121

C₄⋊Q₈

C₂×D₂₆₉

yes

characteristic

122

#127

C₂×D₂₆₉

yes

characteristic

123

#124

—

269

—

124

#121

—

538

—

125

C₄×D₄

—

538

—

126

C₂×C₄⋊C₄

—

538

—

127

C₂×C₄²

—

269

—

128

#97

—

269

—

129

C₄×C₈

—

269

—

130

C₄×C₈

C₂×D₂₆₉

yes

characteristic

131

#41

—

538

—

132

C₂×Q₈

—

269

—

133

C₂×Q₈

—

269

—

134

C₂×Q₈

—

538

—

135

C₂×Q₈

—

538

—

136

C₂×Q₈

C₂²×D₂₆₉

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

538

—

139

C₂×D₄

C₂²×D₂₆₉

yes

—

140

C₂²×C₄

—

269

—

141

C₂²×C₄

—

269

—

142

C₂²×C₄

—

538

—

143

C₂²×C₄

—

538

—

144

Q₁₆

—

538

—

145

Q₁₆

—

538

—

146

SD₁₆

—

147

SD₁₆

—

538

—

148

M₁₆

—

269

—

149

C₂×C₈

—

269

—

150

C₂×C₈

C₂²×D₂₆₉

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

538

—

153

C₄⋊C₄

—

538

—

154

C₄⋊C₄

—

538

—

155

C₂²⋊C₄

—

538

—

156

C₄²

—

269

—

157

C₄²

—

269

—

158

C₄²

C₂²×D₂₆₉

yes

characteristic

159

C₂³

—

160

C₂³

—

269

—

161

Q₈

—

162

Q₈

—

538

—

163

Q₈

—

538

—

164

Q₈

—

1076

—

165

D₄

—

166

D₄

—

167

D₄

—

1076

—

168

C₂×C₄

—

169

C₂×C₄

—

269

—

170

C₂×C₄

—

269

—

171

C₂×C₄

—

269

—

172

C₂×C₄

—

538

—

173

C₂×C₄

—

538

—

174

C₂×C₄

—

538

—

175

C₂×C₄

—

1076

—

176

C₂×C₄

—

1076

—

177

C₂×C₄

D₄×D₂₆₉

yes

—

178

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

179

C₈

—

269

—

180

C₈

D₄×D₂₆₉

yes

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

269

—

184

C₂²

—

538

—

185

C₂²

C₂×D₄×D₂₆₉

yes

characteristic

186

C₄

—

187

C₄

—

269

—

188

C₄

—

538

—

189

C₄

—

1076

—

190

C₄

—

1076

—

191

C₄

#22575482

yes

complemented

192

C₄

C₂×D₄×D₂₆₉

yes

—

193

C₂

—

194

C₂

—

538

—

195

C₂

#127×D₂₆₉

yes

characteristic

196

C₂

C₂×#22575482

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#492×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
prime radical	#31771153	C₂
hypercenter	#492	D₂₆₉
hypocenter	C₂₆₉	#3152
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

C₂×D₄×D₂₆₉

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#492×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂²×D₄

C₂

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₂×#22575482	C₂	semidirect
3. C₄⋊Q₈×D₂₆₉	C₂	semidirect
4. #31770996	C₂	semidirect
5. #31770985	C₂	semidirect
6. #31770984	C₂	semidirect
7. #31770977	C₂	semidirect
8. C₄×#16864385	C₂	semidirect
9. #492×C₂₆₉	C₂	semidirect
45. #16864385	D₄	semidirect
88. C₄×C₂₆₉	#137	semidirect
98. C₂₆₉	#3152	semidirect
107. #492	D₂₆₉	semidirect
191. C₄	#22575482	semidirect

H ≅ G/N

type

1. C₂×#22575482

C₂

semidirect

3. C₄⋊Q₈×D₂₆₉

C₂

semidirect

4. #31770996

C₂

semidirect

5. #31770985

C₂

semidirect

6. #31770984

C₂

semidirect

7. #31770977

C₂

semidirect

8. C₄×#16864385

C₂

semidirect

9. #492×C₂₆₉

C₂

semidirect

45. #16864385

D₄

semidirect

88. C₄×C₂₆₉

#137

semidirect

98. C₂₆₉

#3152

semidirect

107. #492

D₂₆₉

semidirect

191. C₄

#22575482

semidirect

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

subgroup	index
C₂×#22575482	2
#31771153	2
C₄⋊Q₈×D₂₆₉	2
#31770996	2
#31770985	2
#31770984	2
#31770977	2
C₄×#16864385	2
#492×C₂₆₉	2
#3152	269

subgroup

index

C₄⋊Q₈×D₂₆₉

#492×C₂₆₉

269

Subgroups — #46821339

Conjugacy classes of subgroups

Complements

Maximal subgroups