Group #46820655 — subgroups

Subgroups — C₂×#31770899 #46820655

Order 34432.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#22575452	C₂	1	1	yes	maximal, characteristic
2	C₂×#22575418	C₂	1	1	yes	maximal, characteristic
3	#31770899	C₂	1	8	yes	maximal, direct product factor
4	C₂×#22575386	C₂	1	1	yes	maximal, characteristic
5	C₂×#22575382	C₂	1	1	yes	maximal, characteristic
6	C₂×#22575378	C₂	1	1	yes	maximal, characteristic
7	C₂×#22575376	C₂	1	1	yes	maximal, characteristic
8	C₂×#107×C₂₆₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₃₈	—	2	1	no	—
10	C₂²×Dic₅₃₈	C₂²	1	1	yes	characteristic
11	C₂²×#14524130	C₂²	1	1	yes	characteristic
12	#22575452	—	2	2	no	—
13	#22575452	C₂²	1	4	yes	—
14	#22575418	—	2	2	no	—
15	#22575418	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
17	C₂×#16864381	—	2	1	no	—
18	C₂×#16864381	—	2	1	no	—
19	C₂×C₄×Dic₂₆₉	—	2	1	no	—
20	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
21	#22575386	C₂²	1	4	yes	—
22	#22575382	C₂²	1	4	yes	—
23	#22575378	C₂²	1	4	yes	—
24	#22575376	C₂²	1	4	yes	—
25	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
28	#107×C₂₆₉	C₂²	1	4	yes	—
29	C₂×Dic₅₃₈	—	2	4	no	—
30	C₂×Dic₅₃₈	—	2	1	no	—
31	C₂×Dic₅₃₈	—	2	2	no	—
32	C₂×Dic₅₃₈	—	4	4	no	—
33	C₂×Dic₅₃₈	Q₈	1	4	yes	complemented
34	C₂×Dic₅₃₈	C₂³	1	2	yes	—
35	C₂²×Dic₂₆₉	—	2	1	no	—
36	C₂²×Dic₂₆₉	—	2	1	no	—
37	C₂²×Dic₂₆₉	D₄	1	1	yes	characteristic
38	C₂²×Dic₂₆₉	D₄	1	1	yes	characteristic
39	C₂×#14524130	—	2	2	no	—
40	C₂×#14524130	C₂³	1	2	yes	—
41	C₄⋊Dic₂₆₉	—	2	1	no	—
42	C₄⋊Dic₂₆₉	C₂³	1	2	yes	—
43	#16864381	—	2	4	no	—
44	#16864381	—	2	4	no	—
45	C₄×Dic₂₆₉	—	2	4	no	—
46	C₄×Dic₂₆₉	D₄	1	2	yes	—
47	C₄×Dic₂₆₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₆₉	—	2	1	no	—
49	C₂²×C₄×C₂₆₉	—	2	1	no	—
50	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₂₆₉	—	2	2	no	—
52	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₂₆₉	—	2	1	no	—
54	C₄⋊C₄×C₂₆₉	—	2	1	no	—
55	C₄⋊C₄×C₂₆₉	C₂³	1	2	yes	—
56	C₄⋊C₄×C₂₆₉	C₂³	1	2	yes	—
57	Dic₅₃₈	—	2	4	no	—
58	Dic₅₃₈	—	2	2	no	—
59	Dic₅₃₈	—	4	8	no	—
60	Dic₅₃₈	C₂×Q₈	1	4	yes	complemented
61	C₂×Dic₂₆₉	—	2	4	no	—
62	C₂×Dic₂₆₉	—	2	2	no	—
63	C₂×Dic₂₆₉	—	2	2	no	—
64	C₂×Dic₂₆₉	—	2	2	no	—
65	C₂×Dic₂₆₉	—	4	2	no	—
66	C₂×Dic₂₆₉	Q₁₆	1	4	yes	—
67	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
68	C₂×Dic₂₆₉	C₂×D₄	1	2	yes	—
69	#14524130	—	2	2	no	—
70	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₂₆₉	—	2	2	no	—
72	C₂×C₄×C₂₆₉	—	2	2	no	—
73	C₂×C₄×C₂₆₉	—	4	2	no	—
74	C₂×C₄×C₂₆₉	—	4	2	no	—
75	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₂₆₉	#41	1	2	yes	—
78	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
79	C₈×C₂₆₉	—	2	2	no	—
80	C₂²×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
81	C₂²×C₂₆₉	C₂×Q₁₆	1	2	yes	—
82	C₂²×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₂₆₉	—	4	2	no	—
85	C₄×C₂₆₉	—	4	2	no	—
86	C₄×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
87	C₄×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
88	C₄×C₂₆₉	C₂×#41	1	1	yes	characteristic
89	Dic₂₆₉	—	2	4	no	—
90	Dic₂₆₉	—	2	2	no	—
91	Dic₂₆₉	—	4	2	no	—
92	Dic₂₆₉	C₂×Q₁₆	1	4	yes	—
93	C₂×C₂₆₉	#477	1	4	yes	complemented
94	C₂×C₂₆₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₂₆₉	C₂²×Q₁₆	1	1	yes	characteristic
96	C₂×C₂₆₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂₆₉	C₂×#477	1	1	yes	characteristic, complemented, Sylow
98	C₂×#477	—	269	1	no	maximal, Sylow
99	C₂×C₄⋊Q₈	—	269	1	no	—
100	C₂×C₄×Q₈	—	269	1	no	—
101	#477	—	269	8	no	—
102	C₂×#107	—	269	1	no	—
103	C₂×#107	D₂₆₉	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	269	1	no	—
105	C₂×#103	—	269	1	no	—
106	C₂×#103	—	269	1	no	—
107	C₂²×Q₈	—	269	1	no	—
108	C₂²×Q₈	—	538	1	no	—
109	C₂²×C₈	—	269	1	no	—
110	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
111	C₄⋊Q₈	—	269	4	no	—
112	C₄⋊Q₈	—	538	2	no	—
113	C₄×Q₈	—	269	4	no	—
114	C₄×Q₈	—	538	2	no	—
115	C₂×C₄⋊C₄	—	269	1	no	—
116	C₂×C₄⋊C₄	—	538	1	no	—
117	C₂×C₄⋊C₄	—	538	1	no	—
118	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
120	C₂×C₄²	—	269	1	no	—
121	C₂×C₄²	—	538	1	no	—
122	#107	—	269	4	no	—
123	#107	C₂×D₂₆₉	1	4	yes	—
124	C₄⋊C₈	—	269	4	no	—
125	#103	—	269	4	no	—
126	#103	—	269	4	no	—
127	C₂×Q₈	—	269	4	no	—
128	C₂×Q₈	—	269	2	no	—
129	C₂×Q₈	—	538	4	no	—
130	C₂×Q₈	—	538	1	no	—
131	C₂×Q₈	—	538	2	no	—
132	C₂×Q₈	—	1076	4	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	269	1	no	—
136	C₂²×C₄	—	269	1	no	—
137	C₂²×C₄	—	538	1	no	—
138	C₂²×C₄	—	538	1	no	—
139	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	269	2	no	—
142	C₂×C₈	—	538	2	no	—
143	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	269	2	no	—
147	C₄⋊C₄	—	538	4	no	—
148	C₄⋊C₄	—	538	4	no	—
149	C₄⋊C₄	—	538	1	no	—
150	C₄⋊C₄	C₂²×D₂₆₉	1	2	yes	—
151	C₄⋊C₄	C₂²×D₂₆₉	1	2	yes	—
152	C₄²	—	269	2	no	—
153	C₄²	—	269	2	no	—
154	C₄²	—	538	4	no	—
155	C₂³	D₄×D₂₆₉	1	1	yes	characteristic
156	Q₈	—	269	4	no	—
157	Q₈	—	538	4	no	—
158	Q₈	—	538	2	no	—
159	Q₈	—	1076	8	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	269	4	no	—
165	C₂×C₄	—	269	2	no	—
166	C₂×C₄	—	269	2	no	—
167	C₂×C₄	—	538	4	no	—
168	C₂×C₄	—	538	2	no	—
169	C₂×C₄	—	538	2	no	—
170	C₂×C₄	—	538	2	no	—
171	C₂×C₄	—	1076	2	no	—
172	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
173	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
174	C₂×C₄	#16864402	1	2	yes	—
175	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	538	2	no	—
178	C₂²	#22575431	1	2	yes	—
179	C₂²	Q₁₆×D₂₆₉	1	2	yes	—
180	C₂²	#22575477	1	2	yes	—
181	C₂²	C₂×D₄×D₂₆₉	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	269	4	no	—
185	C₄	—	538	4	no	—
186	C₄	—	538	2	no	—
187	C₄	—	1076	2	no	—
188	C₄	#22575431	1	2	yes	—
189	C₄	C₂×Q₈×D₂₆₉	1	1	yes	characteristic
190	C₄	C₂×#16864402	1	1	yes	characteristic
191	C₂	#31770899	1	4	yes	direct product factor
192	C₂	C₂×#22575431	1	1	yes	characteristic
193	C₂	C₂×Q₁₆×D₂₆₉	1	1	yes	characteristic
194	C₂	C₂×#22575477	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575452

C₂

yes

maximal, characteristic

C₂×#22575418

C₂

yes

maximal, characteristic

#31770899

C₂

yes

maximal, direct product factor

C₂×#22575386

C₂

yes

maximal, characteristic

C₂×#22575382

C₂

yes

maximal, characteristic

C₂×#22575378

C₂

yes

maximal, characteristic

C₂×#22575376

C₂

yes

maximal, characteristic

C₂×#107×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₃₈

—

C₂²×Dic₅₃₈

C₂²

yes

characteristic

C₂²×#14524130

C₂²

yes

characteristic

#22575452

—

#22575452

C₂²

yes

—

#22575418

—

#22575418

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

characteristic

C₂×#16864381

—

C₂×#16864381

—

C₂×C₄×Dic₂₆₉

—

C₂×C₄×Dic₂₆₉

C₂²

yes

characteristic

#22575386

C₂²

yes

—

#22575382

C₂²

yes

—

#22575378

C₂²

yes

—

#22575376

C₂²

yes

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

#107×C₂₆₉

C₂²

yes

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

Q₈

yes

complemented

C₂×Dic₅₃₈

C₂³

yes

—

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

D₄

yes

characteristic

C₂²×Dic₂₆₉

D₄

yes

characteristic

C₂×#14524130

—

C₂×#14524130

C₂³

yes

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂³

yes

—

#16864381

—

#16864381

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

D₄

yes

—

C₄×Dic₂₆₉

C₂³

yes

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

C₂³

yes

—

C₄⋊C₄×C₂₆₉

C₂³

yes

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

—

Dic₅₃₈

C₂×Q₈

yes

complemented

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

Q₁₆

yes

—

C₂×Dic₂₆₉

C₂×D₄

yes

—

C₂×Dic₂₆₉

C₂×D₄

yes

—

#14524130

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

—

C₂×C₄×C₂₆₉

#41

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

—

C₂²×C₂₆₉

C₂²⋊Q₈

yes

—

C₂²×C₂₆₉

C₂×Q₁₆

yes

—

C₂²×C₂₆₉

AGL(1,ℤ/8)

yes

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂²⋊Q₈

yes

—

C₄×C₂₆₉

C₂²×Q₈

yes

characteristic

C₄×C₂₆₉

C₂×#41

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

C₂×Q₁₆

yes

—

C₂×C₂₆₉

#477

yes

complemented

C₂×C₂₆₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₂₆₉

C₂²×Q₁₆

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

C₂×#477

—

269

maximal, Sylow

C₂×C₄⋊Q₈

—

269

—

100

C₂×C₄×Q₈

—

269

—

101

#477

—

269

—

102

C₂×#107

—

269

—

103

C₂×#107

D₂₆₉

yes

characteristic

104

C₂×C₄⋊C₈

—

269

—

105

C₂×#103

—

269

—

106

C₂×#103

—

269

—

107

C₂²×Q₈

—

269

—

108

C₂²×Q₈

—

538

—

109

C₂²×C₈

—

269

—

110

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

111

C₄⋊Q₈

—

269

—

112

C₄⋊Q₈

—

538

—

113

C₄×Q₈

—

269

—

114

C₄×Q₈

—

538

—

115

C₂×C₄⋊C₄

—

269

—

116

C₂×C₄⋊C₄

—

538

—

117

C₂×C₄⋊C₄

—

538

—

118

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

120

C₂×C₄²

—

269

—

121

C₂×C₄²

—

538

—

122

#107

—

269

—

123

#107

C₂×D₂₆₉

yes

—

124

C₄⋊C₈

—

269

—

125

#103

—

269

—

126

#103

—

269

—

127

C₂×Q₈

—

269

—

128

C₂×Q₈

—

269

—

129

C₂×Q₈

—

538

—

130

C₂×Q₈

—

538

—

131

C₂×Q₈

—

538

—

132

C₂×Q₈

—

1076

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

269

—

136

C₂²×C₄

—

269

—

137

C₂²×C₄

—

538

—

138

C₂²×C₄

—

538

—

139

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

269

—

142

C₂×C₈

—

538

—

143

C₂×C₈

C₂²×D₂₆₉

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

269

—

147

C₄⋊C₄

—

538

—

148

C₄⋊C₄

—

538

—

149

C₄⋊C₄

—

538

—

150

C₄⋊C₄

C₂²×D₂₆₉

yes

—

151

C₄⋊C₄

C₂²×D₂₆₉

yes

—

152

C₄²

—

269

—

153

C₄²

—

269

—

154

C₄²

—

538

—

155

C₂³

D₄×D₂₆₉

yes

characteristic

156

Q₈

—

269

—

157

Q₈

—

538

—

158

Q₈

—

538

—

159

Q₈

—

1076

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

269

—

165

C₂×C₄

—

269

—

166

C₂×C₄

—

269

—

167

C₂×C₄

—

538

—

168

C₂×C₄

—

538

—

169

C₂×C₄

—

538

—

170

C₂×C₄

—

538

—

171

C₂×C₄

—

1076

—

172

C₂×C₄

D₄×D₂₆₉

yes

characteristic

173

C₂×C₄

Q₈×D₂₆₉

yes

—

174

C₂×C₄

#16864402

yes

—

175

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

176

C₈

—

177

C₈

—

538

—

178

C₂²

#22575431

yes

—

179

C₂²

Q₁₆×D₂₆₉

yes

—

180

C₂²

#22575477

yes

—

181

C₂²

C₂×D₄×D₂₆₉

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

269

—

185

C₄

—

538

—

186

C₄

—

538

—

187

C₄

—

1076

—

188

C₄

#22575431

yes

—

189

C₄

C₂×Q₈×D₂₆₉

yes

characteristic

190

C₄

C₂×#16864402

yes

characteristic

191

C₂

#31770899

yes

direct product factor

192

C₂

C₂×#22575431

yes

characteristic

193

C₂

C₂×Q₁₆×D₂₆₉

yes

characteristic

194

C₂

C₂×#22575477

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	C₂×#107×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
hypercenter	C₂×#107	D₂₆₉
hypocenter	C₂₆₉	C₂×#477
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₂₆₉

commutator, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

C₂×#107×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #31770899	C₂	direct (N × H)
33. C₂×Dic₅₃₈	Q₈	semidirect
60. Dic₅₃₈	C₂×Q₈	semidirect
93. C₂×C₂₆₉	#477	semidirect
97. C₂₆₉	C₂×#477	semidirect
191. C₂	#31770899	direct (N × H)

H ≅ G/N

type

3. #31770899

C₂

direct (N × H)

33. C₂×Dic₅₃₈

Q₈

semidirect

60. Dic₅₃₈

C₂×Q₈

semidirect

93. C₂×C₂₆₉

#477

semidirect

97. C₂₆₉

C₂×#477

semidirect

191. C₂

#31770899

direct (N × H)

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

subgroup	index
C₂×#22575452	2
C₂×#22575418	2
#31770899	2
C₂×#22575386	2
C₂×#22575382	2
C₂×#22575378	2
C₂×#22575376	2
C₂×#107×C₂₆₉	2
C₂×#477	269

subgroup

index

C₂×#107×C₂₆₉

269

Subgroups — C₂×#31770899 #46820655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#31770899 #46820655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31770899 #46820655