Group #46820373 — subgroups

Subgroups — #46820373

Order 34432.

480 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22575486	C₂	1	2	yes	maximal
2	C₂×#22575453	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#16864402	C₂	1	1	yes	maximal, characteristic
4	#31770816	C₂	1	4	yes	maximal, complemented
5	#31770811	C₂	1	4	yes	maximal, complemented
6	#31770615	C₂	1	2	yes	maximal, complemented
7	#423×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16864402	C₂²	1	1	yes	characteristic
9	C₂²×Dic₅₃₈	C₂²	1	2	yes	—
10	#22575486	—	2	8	no	—
11	C₂×Dic₁₀₇₆	C₂²	1	4	yes	—
12	C₂×#16864391	C₂²	1	4	yes	—
13	#22575453	—	2	1	no	—
14	#22575453	—	2	1	no	—
15	#22575453	C₂²	1	4	yes	—
16	C₄×Dic₅₃₈	C₂²	1	2	yes	—
17	C₄×D₁₀₇₆	C₂²	1	2	yes	—
18	C₄×C₂²⋊D₂₆₉	—	2	2	no	—
19	#22575412	C₂²	1	1	yes	characteristic
20	#22575410	—	2	1	no	—
21	C₂×C₄⋊Dic₂₆₉	—	2	2	no	—
22	C₄²×D₂₆₉	—	2	1	no	—
23	#22575378	C₂²	1	8	yes	—
24	C₂×M₁₆×C₂₆₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₂₆₉	C₂²	1	4	yes	—
27	#16864402	—	2	2	no	—
28	#16864402	D₄	1	4	yes	—
29	C₂×Dic₅₃₈	—	2	2	no	—
30	C₂×Dic₅₃₈	—	2	4	no	—
31	C₂×Dic₅₃₈	—	2	4	no	—
32	C₂×Dic₅₃₈	C₂³	1	4	yes	—
33	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic
34	C₂×D₁₀₇₆	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊D₂₆₉	—	2	1	no	—
36	C₂²×Dic₂₆₉	—	2	2	no	—
37	C₂×C₄×D₂₆₉	—	2	1	no	—
38	C₂×C₄×D₂₆₉	—	2	2	no	—
39	Dic₁₀₇₆	—	2	8	no	—
40	#16864391	—	2	8	no	—
41	C₄⋊Dic₂₆₉	—	2	4	no	—
42	C₄⋊Dic₂₆₉	—	2	4	no	—
43	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
44	#16864381	—	2	2	no	—
45	C₂²⋊Dic₂₆₉	—	2	1	no	—
46	#16864379	—	2	2	no	—
47	C₄×Dic₂₆₉	—	2	1	no	—
48	C₄×Dic₂₆₉	—	2	2	no	—
49	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
50	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
51	M₁₆×C₂₆₉	—	2	4	no	—
52	C₂×C₈×C₂₆₉	C₂³	1	4	yes	—
53	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
54	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
55	C₄²×C₂₆₉	C₂³	1	2	yes	—
56	C₂²×D₂₆₉	—	2	1	no	—
57	Dic₅₃₈	—	2	1	no	—
58	Dic₅₃₈	—	2	4	no	—
59	Dic₅₃₈	—	2	4	no	—
60	Dic₅₃₈	—	4	4	no	—
61	Dic₅₃₈	C₂×D₄	1	2	yes	—
62	D₁₀₇₆	—	2	1	no	—
63	D₁₀₇₆	C₂×D₄	1	2	yes	—
64	C₂²⋊D₂₆₉	—	2	4	no	—
65	C₂×Dic₂₆₉	—	2	1	no	—
66	C₂×Dic₂₆₉	—	2	2	no	—
67	C₂×Dic₂₆₉	—	2	4	no	—
68	C₂×Dic₂₆₉	—	4	4	no	—
69	C₄×D₂₆₉	—	2	4	no	—
70	C₄×D₂₆₉	—	4	4	no	—
71	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₂₆₉	—	2	4	no	—
73	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₂₆₉	#41	1	2	yes	—
78	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
79	C₈×C₂₆₉	—	2	4	no	—
80	C₂²×C₂₆₉	—	2	2	no	—
81	C₂²×C₂₆₉	#121	1	2	yes	—
82	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₂₆₉	—	2	2	no	—
84	C₂×D₂₆₉	—	4	2	no	—
85	C₄×C₂₆₉	—	2	2	no	—
86	C₄×C₂₆₉	#121	1	2	yes	—
87	C₄×C₂₆₉	#137	1	4	yes	complemented
88	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
90	Dic₂₆₉	—	2	2	no	—
91	Dic₂₆₉	—	4	4	no	—
92	Dic₂₆₉	—	4	2	no	—
93	C₂×C₂₆₉	—	2	2	no	—
94	C₂×C₂₆₉	C₂×#121	1	1	yes	characteristic
95	C₂×C₂₆₉	C₂×#137	1	2	yes	—
96	D₂₆₉	—	4	2	no	—
97	C₂₆₉	#3039	1	1	yes	characteristic, complemented, Sylow
98	#3039	—	269	1	no	maximal, Sylow
99	C₂×#137	—	269	2	no	—
100	C₂×C₄⋊Q₈	—	269	1	no	—
101	C₄×#41	—	269	1	no	—
102	#462	—	269	4	no	—
103	#461	—	269	4	no	—
104	#423	D₂₆₉	1	1	yes	characteristic, complemented
105	#419	—	269	2	no	—
106	C₂×#41	—	269	1	no	—
107	C₂²×Q₈	—	269	2	no	—
108	#137	—	538	8	no	—
109	C₂×Q₁₆	—	269	4	no	—
110	C₂×SD₁₆	—	269	4	no	—
111	C₂×M₁₆	C₂×D₂₆₉	1	2	yes	—
112	C₄⋊Q₈	—	269	4	no	—
113	C₄⋊Q₈	—	538	1	no	—
114	C₄⋊Q₈	—	538	1	no	—
115	C₄×Q₈	—	269	2	no	—
116	C₄×D₄	—	269	2	no	—
117	C₄×D₄	—	538	2	no	—
118	#117	—	269	1	no	—
119	#117	—	538	1	no	—
120	C₂×C₄⋊C₄	—	538	2	no	—
121	C₂×C₄²	—	538	1	no	—
122	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₂₆₉	1	4	yes	—
124	#103	—	269	8	no	—
125	#41	—	269	4	no	—
126	#41	—	538	2	no	—
127	C₂×Q₈	—	269	1	no	—
128	C₂×Q₈	—	269	4	no	—
129	C₂×Q₈	—	538	2	no	—
130	C₂×Q₈	—	538	4	no	—
131	C₂×Q₈	—	538	4	no	—
132	C₂×D₄	—	269	1	no	—
133	C₂×D₄	—	538	1	no	—
134	C₂²×C₄	—	538	1	no	—
135	C₂²×C₄	—	538	2	no	—
136	C₂²×C₄	—	538	2	no	—
137	C₂²×C₄	D₁₀₇₆	1	2	yes	—
138	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
139	Q₁₆	—	538	8	no	—
140	SD₁₆	—	538	8	no	—
141	M₁₆	—	2	4	no	—
142	C₂×C₈	C₂²×D₂₆₉	1	4	yes	—
143	C₄⋊C₄	—	269	2	no	—
144	C₄⋊C₄	—	538	4	no	—
145	C₄⋊C₄	—	538	4	no	—
146	C₄⋊C₄	—	538	2	no	—
147	C₂²⋊C₄	—	538	1	no	—
148	C₂²⋊C₄	—	538	2	no	—
149	C₄²	—	538	1	no	—
150	C₄²	—	538	2	no	—
151	C₄²	D₁₀₇₆	1	1	yes	characteristic
152	C₄²	D₁₀₇₆	1	1	yes	characteristic
153	C₄²	C₂²×D₂₆₉	1	2	yes	—
154	C₂³	—	538	1	no	—
155	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
156	Q₈	—	269	2	no	—
157	Q₈	—	538	1	no	—
158	Q₈	—	538	4	no	—
159	Q₈	—	538	4	no	—
160	Q₈	—	1076	4	no	—
161	D₄	—	269	2	no	—
162	D₄	—	538	4	no	—
163	D₄	—	538	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	538	1	no	—
166	C₂×C₄	—	538	4	no	—
167	C₂×C₄	—	538	2	no	—
168	C₂×C₄	—	538	4	no	—
169	C₂×C₄	—	1076	4	no	—
170	C₂×C₄	—	1076	4	no	—
171	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
172	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
173	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
174	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
175	C₂×C₄	#16864403	1	2	yes	—
176	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
177	C₈	—	2	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	538	2	no	—
180	C₂²	—	1076	2	no	—
181	C₂²	#22575428	1	2	yes	—
182	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	538	2	no	—
185	C₄	—	1076	2	no	—
186	C₄	—	1076	4	no	—
187	C₄	#22575428	1	2	yes	—
188	C₄	#22575486	1	4	yes	complemented
189	C₄	C₂×D₄×D₂₆₉	1	1	yes	characteristic
190	C₄	C₂×#16864403	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	1076	2	no	—
193	C₂	C₂×#22575428	1	1	yes	characteristic
194	C₂	C₂×#22575486	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575486

C₂

yes

maximal

C₂×#22575453

C₂

yes

maximal, characteristic, complemented

C₄×#16864402

C₂

yes

maximal, characteristic

#31770816

C₂

yes

maximal, complemented

#31770811

C₂

yes

maximal, complemented

#31770615

C₂

yes

maximal, complemented

#423×C₂₆₉

C₂

yes

maximal, characteristic, complemented

C₂×#16864402

C₂²

yes

characteristic

C₂²×Dic₅₃₈

C₂²

yes

—

#22575486

—

C₂×Dic₁₀₇₆

C₂²

yes

—

C₂×#16864391

C₂²

yes

—

#22575453

—

#22575453

—

#22575453

C₂²

yes

—

C₄×Dic₅₃₈

C₂²

yes

—

C₄×D₁₀₇₆

C₂²

yes

—

C₄×C₂²⋊D₂₆₉

—

#22575412

C₂²

yes

characteristic

#22575410

—

C₂×C₄⋊Dic₂₆₉

—

C₄²×D₂₆₉

—

#22575378

C₂²

yes

—

C₂×M₁₆×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₆₉

C₂²

yes

—

#16864402

—

#16864402

D₄

yes

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₂³

yes

—

C₂×Dic₅₃₈

C₂³

yes

characteristic

C₂×D₁₀₇₆

C₂³

yes

characteristic

C₂×C₂²⋊D₂₆₉

—

C₂²×Dic₂₆₉

—

C₂×C₄×D₂₆₉

—

C₂×C₄×D₂₆₉

—

Dic₁₀₇₆

—

#16864391

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂³

yes

—

#16864381

—

C₂²⋊Dic₂₆₉

—

#16864379

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

M₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

C₂³

yes

—

C₂²×D₂₆₉

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

C₂×D₄

yes

—

D₁₀₇₆

—

D₁₀₇₆

C₂×D₄

yes

—

C₂²⋊D₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₄×D₂₆₉

—

C₄×D₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

#41

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#121

yes

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

#121

yes

—

C₄×C₂₆₉

#137

yes

complemented

C₄×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

C₂×#41

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂×#121

yes

characteristic

C₂×C₂₆₉

C₂×#137

yes

—

D₂₆₉

—

C₂₆₉

#3039

yes

characteristic, complemented, Sylow

#3039

—

269

maximal, Sylow

C₂×#137

—

269

—

100

C₂×C₄⋊Q₈

—

269

—

101

C₄×#41

—

269

—

102

#462

—

269

—

103

#461

—

269

—

104

#423

D₂₆₉

yes

characteristic, complemented

105

#419

—

269

—

106

C₂×#41

—

269

—

107

C₂²×Q₈

—

269

—

108

#137

—

538

—

109

C₂×Q₁₆

—

269

—

110

C₂×SD₁₆

—

269

—

111

C₂×M₁₆

C₂×D₂₆₉

yes

—

112

C₄⋊Q₈

—

269

—

113

C₄⋊Q₈

—

538

—

114

C₄⋊Q₈

—

538

—

115

C₄×Q₈

—

269

—

116

C₄×D₄

—

269

—

117

C₄×D₄

—

538

—

118

#117

—

269

—

119

#117

—

538

—

120

C₂×C₄⋊C₄

—

538

—

121

C₂×C₄²

—

538

—

122

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

123

C₄⋊C₈

C₂×D₂₆₉

yes

—

124

#103

—

269

—

125

#41

—

269

—

126

#41

—

538

—

127

C₂×Q₈

—

269

—

128

C₂×Q₈

—

269

—

129

C₂×Q₈

—

538

—

130

C₂×Q₈

—

538

—

131

C₂×Q₈

—

538

—

132

C₂×D₄

—

269

—

133

C₂×D₄

—

538

—

134

C₂²×C₄

—

538

—

135

C₂²×C₄

—

538

—

136

C₂²×C₄

—

538

—

137

C₂²×C₄

D₁₀₇₆

yes

—

138

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

139

Q₁₆

—

538

—

140

SD₁₆

—

538

—

141

M₁₆

—

142

C₂×C₈

C₂²×D₂₆₉

yes

—

143

C₄⋊C₄

—

269

—

144

C₄⋊C₄

—

538

—

145

C₄⋊C₄

—

538

—

146

C₄⋊C₄

—

538

—

147

C₂²⋊C₄

—

538

—

148

C₂²⋊C₄

—

538

—

149

C₄²

—

538

—

150

C₄²

—

538

—

151

C₄²

D₁₀₇₆

yes

characteristic

152

C₄²

D₁₀₇₆

yes

characteristic

153

C₄²

C₂²×D₂₆₉

yes

—

154

C₂³

—

538

—

155

C₂³

C₂×D₁₀₇₆

yes

characteristic

156

Q₈

—

269

—

157

Q₈

—

538

—

158

Q₈

—

538

—

159

Q₈

—

538

—

160

Q₈

—

1076

—

161

D₄

—

269

—

162

D₄

—

538

—

163

D₄

—

538

—

164

C₂×C₄

—

165

C₂×C₄

—

538

—

166

C₂×C₄

—

538

—

167

C₂×C₄

—

538

—

168

C₂×C₄

—

538

—

169

C₂×C₄

—

1076

—

170

C₂×C₄

—

1076

—

171

C₂×C₄

D₄×D₂₆₉

yes

—

172

C₂×C₄

C₂×D₁₀₇₆

yes

—

173

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

174

C₂×C₄

C₂×D₁₀₇₆

yes

—

175

C₂×C₄

#16864403

yes

—

176

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

177

C₈

—

178

C₂²

—

179

C₂²

—

538

—

180

C₂²

—

1076

—

181

C₂²

#22575428

yes

—

182

C₂²

C₂²×D₁₀₇₆

yes

characteristic

183

C₄

—

184

C₄

—

538

—

185

C₄

—

1076

—

186

C₄

—

1076

—

187

C₄

#22575428

yes

—

188

C₄

#22575486

yes

complemented

189

C₄

C₂×D₄×D₂₆₉

yes

characteristic

190

C₄

C₂×#16864403

yes

characteristic

191

C₂

—

192

C₂

—

1076

—

193

C₂

C₂×#22575428

yes

characteristic

194

C₂

C₂×#22575486

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₇₆
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	#423×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
prime radical	C₂×#16864402	C₂²
hypercenter	#423	D₂₆₉
hypocenter	C₂₆₉	#3039
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₁₀₇₆

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

#423×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂²×D₄

C₂²

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#22575453	C₂	semidirect
4. #31770816	C₂	semidirect
5. #31770811	C₂	semidirect
6. #31770615	C₂	semidirect
7. #423×C₂₆₉	C₂	semidirect
87. C₄×C₂₆₉	#137	semidirect
97. C₂₆₉	#3039	semidirect
104. #423	D₂₆₉	semidirect
188. C₄	#22575486	semidirect

H ≅ G/N

type

2. C₂×#22575453

C₂

semidirect

4. #31770816

C₂

semidirect

5. #31770811

C₂

semidirect

6. #31770615

C₂

semidirect

7. #423×C₂₆₉

C₂

semidirect

87. C₄×C₂₆₉

#137

semidirect

97. C₂₆₉

#3039

semidirect

104. #423

D₂₆₉

semidirect

188. C₄

#22575486

semidirect

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

subgroup	index
C₂×#22575486	2
C₂×#22575453	2
C₄×#16864402	2
#31770816	2
#31770811	2
#31770615	2
#423×C₂₆₉	2
#3039	269

subgroup

index

#423×C₂₆₉

269

Subgroups — #46820373

Conjugacy classes of subgroups

Complements

Maximal subgroups