Group #35559723 — subgroups

Subgroups — C₂×#24909710 #35559723

Order 21248.

582 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 203 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×#19128081	C₂	1	1	yes	maximal, characteristic
2	#24909710	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19127504	C₂	1	4	yes	maximal
4	C₂×#19127477	C₂	1	1	yes	maximal, characteristic
5	C₂×#499×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	—	2	1	no	—
7	C₂²×Dic₃₃₂	C₂²	1	2	yes	—
8	C₂×#14581792	—	2	1	no	—
9	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
10	#19128081	—	2	2	no	—
11	#19128081	C₂²	1	4	yes	—
12	#19127504	C₂²	1	16	yes	—
13	#19127477	C₂²	1	4	yes	—
14	C₂×#125×C₈₃	—	2	1	no	—
15	#499×C₈₃	—	2	2	no	—
16	#499×C₈₃	C₂²	1	4	yes	—
17	C₂×#107×C₈₃	C₂²	1	2	yes	—
18	C₂×#106×C₈₃	—	2	1	no	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₆₆	—	2	2	no	—
21	C₂×Dic₃₃₂	—	2	2	no	—
22	C₂×Dic₃₃₂	—	2	4	no	—
23	C₂×Dic₃₃₂	—	2	8	no	—
24	C₂×Dic₃₃₂	—	4	4	no	—
25	C₂×Dic₃₃₂	C₂³	1	4	yes	—
26	#14581792	—	2	4	no	—
27	#14581792	—	4	2	no	—
28	C₂×C₄⋊Dic₈₃	—	4	2	no	—
29	C₂×C₈₃⋊C₁₆	—	2	4	no	—
30	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
31	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	#125×C₈₃	—	2	4	no	—
34	#125×C₈₃	—	4	2	no	—
35	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₃	—	4	2	no	—
37	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
38	#107×C₈₃	—	2	2	no	—
39	#107×C₈₃	C₂³	1	4	yes	—
40	#106×C₈₃	—	2	4	no	—
41	C₄×C₈×C₈₃	D₄	1	2	yes	—
42	C₄×C₈×C₈₃	C₂³	1	2	yes	—
43	C₂×Dic₁₆₆	—	2	4	no	—
44	C₂×Dic₁₆₆	—	4	4	no	—
45	C₂×Dic₁₆₆	—	4	4	no	—
46	C₂×Dic₁₆₆	—	4	2	no	—
47	C₂²×Dic₈₃	—	4	2	no	—
48	Dic₃₃₂	—	2	8	no	—
49	Dic₃₃₂	—	4	4	no	—
50	Dic₃₃₂	—	4	8	no	—
51	C₄⋊Dic₈₃	—	4	8	no	—
52	C₈₃⋊C₁₆	—	2	4	no	—
53	C₂²×C₄×C₈₃	—	4	2	no	—
54	C₂²×C₄×C₈₃	D₈	1	2	yes	—
55	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
57	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₈₃	#41	1	4	yes	—
60	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₈₃	—	2	4	no	—
62	C₄⋊C₄×C₈₃	—	4	2	no	—
63	C₄⋊C₄×C₈₃	—	4	8	no	—
64	C₄²×C₈₃	D₈	1	2	yes	—
65	C₄²×C₈₃	C₂×D₄	1	2	yes	—
66	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	4	no	—
68	C₂×C₄×C₈₃	—	4	4	no	—
69	C₂×C₄×C₈₃	—	8	4	no	—
70	C₂×C₄×C₈₃	#124	1	2	yes	—
71	C₂×C₄×C₈₃	C₂×D₈	1	4	yes	—
72	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₈₃	—	4	4	no	—
76	C₂×Dic₈₃	—	8	4	no	—
77	Dic₁₆₆	—	4	4	no	—
78	Dic₁₆₆	—	4	4	no	—
79	Dic₁₆₆	—	8	4	no	—
80	C₈×C₈₃	#124	1	2	yes	—
81	C₈×C₈₃	C₂×SD₁₆	1	4	yes	—
82	C₈×C₈₃	C₂×#41	1	2	yes	—
83	C₂²×C₈₃	#486	1	2	yes	—
84	C₂²×C₈₃	#510	1	4	yes	—
85	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
86	C₄×C₈₃	—	2	4	no	—
87	C₄×C₈₃	—	8	4	no	—
88	C₄×C₈₃	#486	1	2	yes	—
89	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
90	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
91	Dic₈₃	—	8	4	no	—
92	C₂×#2245	—	83	1	no	maximal, Sylow
93	C₂×C₈₃	#2245	1	4	yes	complemented
94	C₂×C₈₃	C₂×#486	1	1	yes	characteristic
95	C₂×C₈₃	C₂×#510	1	2	yes	—
96	C₂×#499	D₈₃	1	1	yes	characteristic
97	C₂×#494	—	83	1	no	—
98	#2245	—	83	8	no	—
99	C₂×#358	—	83	4	no	—
100	C₂×#346	—	83	1	no	—
101	C₈₃	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
102	C₂²×Q₁₆	—	83	2	no	—
103	C₂²×Q₁₆	—	166	1	no	—
104	C₂×C₄⋊Q₈	—	166	1	no	—
105	C₂×#125	—	2	1	no	—
106	C₂²×C₁₆	—	83	2	no	—
107	#499	—	2	2	no	—
108	#499	C₂×D₈₃	1	4	yes	—
109	#494	—	83	4	no	—
110	#494	—	166	2	no	—
111	C₂×#107	C₂×D₈₃	1	2	yes	—
112	C₂×#106	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
114	#358	—	83	16	no	—
115	#346	—	83	4	no	—
116	C₂²×Q₈	—	166	2	no	—
117	C₂×Q₁₆	—	332	4	no	—
118	C₂×Q₁₆	—	83	4	no	—
119	C₂×Q₁₆	—	166	2	no	—
120	C₂×Q₁₆	—	166	4	no	—
121	C₂×Q₁₆	—	166	8	no	—
122	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	332	2	no	—
125	C₄⋊Q₈	—	166	4	no	—
126	#125	—	2	4	no	—
127	#125	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	2	no	—
130	C₂×C₄⋊C₄	—	332	2	no	—
131	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
132	C₂×C₁₆	—	83	4	no	—
133	C₂×C₁₆	—	166	4	no	—
134	#107	—	2	2	no	—
135	#107	C₂²×D₈₃	1	4	yes	—
136	#106	—	2	4	no	—
137	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
138	C₄×C₈	C₂²×D₈₃	1	2	yes	—
139	C₂×Q₈	—	332	2	no	—
140	C₂×Q₈	—	332	4	no	—
141	C₂×Q₈	—	332	4	no	—
142	C₂×Q₈	—	166	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	332	2	no	—
145	C₂²×C₄	#13050575	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
147	Q₁₆	—	332	4	no	—
148	Q₁₆	—	332	8	no	—
149	Q₁₆	—	166	8	no	—
150	C₂×C₈	#13050577	1	4	yes	—
151	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
153	C₂×C₈	#13050591	1	4	yes	—
154	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	8	no	—
158	C₄⋊C₄	—	332	8	no	—
159	C₄²	#13050575	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
161	C₁₆	—	166	4	no	—
162	C₂³	C₂×#13050575	1	1	yes	characteristic
163	Q₈	—	332	4	no	—
164	Q₈	—	332	4	no	—
165	Q₈	—	664	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	332	4	no	—
170	C₂×C₄	—	664	4	no	—
171	C₂×C₄	#14581781	1	2	yes	—
172	C₂×C₄	C₂×#13050575	1	4	yes	—
173	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
174	C₂×C₄	C₂×#13050577	1	2	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
176	C₈	#14581781	1	2	yes	—
177	C₈	C₂×#13050577	1	4	yes	—
178	C₈	C₂×#13050591	1	2	yes	—
179	C₂²	#19128049	1	2	yes	—
180	C₂²	#19128141	1	4	yes	—
181	C₂²	C₂²×#13050575	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	664	4	no	—
185	C₄	#19128049	1	2	yes	—
186	C₄	C₂×#14581781	1	1	yes	characteristic
187	C₄	C₂²×#13050577	1	1	yes	characteristic
188	C₂	#24909710	1	4	yes	direct product factor
189	C₂	C₂×#19128049	1	1	yes	characteristic
190	C₂	C₂×#19128141	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128081

C₂

yes

maximal, characteristic

#24909710

C₂

yes

maximal, direct product factor

C₂×#19127504

C₂

yes

maximal

C₂×#19127477

C₂

yes

maximal, characteristic

C₂×#499×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

—

C₂²×Dic₃₃₂

C₂²

yes

—

C₂×#14581792

—

C₂²×C₈₃⋊C₁₆

C₂²

yes

—

#19128081

—

#19128081

C₂²

yes

—

#19127504

C₂²

yes

—

#19127477

C₂²

yes

—

C₂×#125×C₈₃

—

#499×C₈₃

—

#499×C₈₃

C₂²

yes

—

C₂×#107×C₈₃

C₂²

yes

—

C₂×#106×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

C₂³

yes

—

#14581792

—

#14581792

—

C₂×C₄⋊Dic₈₃

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

C₂³

yes

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

#125×C₈₃

—

#125×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

characteristic

#107×C₈₃

—

#107×C₈₃

C₂³

yes

—

#106×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

C₂³

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₈

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₂×C₈×C₈₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

D₈

yes

—

C₄²×C₈₃

C₂×D₄

yes

—

C₂³×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#124

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂²×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

#124

yes

—

C₈×C₈₃

C₂×SD₁₆

yes

—

C₈×C₈₃

C₂×#41

yes

—

C₂²×C₈₃

#486

yes

—

C₂²×C₈₃

#510

yes

—

C₂²×C₈₃

C₂²×D₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#486

yes

—

C₄×C₈₃

C₂×#124

yes

characteristic

C₄×C₈₃

C₂²×SD₁₆

yes

characteristic

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×#499

D₈₃

yes

—

—

—

100

C₂×#346

—

101

C₈₃

C₂×#2245

yes

characteristic, complemented, Sylow

102

C₂²×Q₁₆

—

103

C₂²×Q₁₆

—

166

—

104

C₂×C₄⋊Q₈

—

166

—

105

C₂×#125

—

106

C₂²×C₁₆

—

107

#499

—

108

#499

C₂×D₈₃

yes

—

109

#494

—

110

#494

—

166

—

111

C₂×#107

C₂×D₈₃

yes

—

112

C₂×#106

—

113

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

114

#358

—

115

#346

—

116

C₂²×Q₈

—

166

—

117

C₂×Q₁₆

—

332

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

166

—

120

C₂×Q₁₆

—

166

—

121

C₂×Q₁₆

—

166

—

122

C₂²×C₈

C₂²⋊D₈₃

yes

characteristic

123

C₂²×C₈

C₂²×D₈₃

yes

characteristic

124

C₄⋊Q₈

—

332

—

125

C₄⋊Q₈

—

166

—

126

#125

—

127

#125

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

332

—

131

C₂×C₄²

C₂²⋊D₈₃

yes

characteristic

132

C₂×C₁₆

—

133

C₂×C₁₆

—

166

—

134

#107

—

135

#107

C₂²×D₈₃

yes

—

136

#106

—

137

C₄×C₈

C₂²⋊D₈₃

yes

—

138

C₄×C₈

C₂²×D₈₃

yes

—

139

C₂×Q₈

—

332

—

140

C₂×Q₈

—

332

—

141

C₂×Q₈

—

332

—

142

C₂×Q₈

—

166

—

143

C₂²×C₄

—

144

C₂²×C₄

—

332

—

145

C₂²×C₄

#13050575

yes

—

146

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

147

Q₁₆

—

332

—

148

Q₁₆

—

332

—

149

Q₁₆

—

166

—

150

C₂×C₈

#13050577

yes

—

151

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

152

C₂×C₈

C₂×C₂²⋊D₈₃

yes

—

153

C₂×C₈

#13050591

yes

—

154

C₂×C₈

C₂³×D₈₃

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

332

—

159

C₄²

#13050575

yes

—

160

C₄²

C₂×C₂²⋊D₈₃

yes

—

161

C₁₆

—

166

—

162

C₂³

C₂×#13050575

yes

characteristic

163

Q₈

—

332

—

164

Q₈

—

332

—

165

Q₈

—

664

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

332

—

170

C₂×C₄

—

664

—

171

C₂×C₄

#14581781

yes

—

172

C₂×C₄

C₂×#13050575

yes

—

173

C₂×C₄

C₂×#13050575

yes

characteristic

174

C₂×C₄

C₂×#13050577

yes

—

175

C₂×C₄

C₂²×C₂²⋊D₈₃

yes

characteristic

176

C₈

#14581781

yes

—

177

C₈

C₂×#13050577

yes

—

178

C₈

C₂×#13050591

yes

—

179

C₂²

#19128049

yes

—

180

C₂²

#19128141

yes

—

181

C₂²

C₂²×#13050575

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

664

—

185

C₄

#19128049

yes

—

186

C₄

C₂×#14581781

yes

characteristic

187

C₄

C₂²×#13050577

yes

characteristic

188

C₂

#24909710

yes

direct product factor

189

C₂

C₂×#19128049

yes

characteristic

190

C₂

C₂×#19128141

yes

—

	subgroup	G/H
center	C₂³	C₂×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#499×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	C₂×#499	D₈₃
hypocenter	C₈₃	C₂×#2245
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#13050575

commutator, cosocle

C₂×C₈×C₈₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₃

Fitting

C₂×#499×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₈

D₈₃

C₈₃

G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
2. #24909710	C₂	direct (N × H)
93. C₂×C₈₃	#2245	semidirect
101. C₈₃	C₂×#2245	semidirect
188. C₂	#24909710	direct (N × H)

H ≅ G/N

type

2. #24909710

C₂

direct (N × H)

93. C₂×C₈₃

#2245

semidirect

101. C₈₃

C₂×#2245

semidirect

188. C₂

#24909710

direct (N × H)

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

subgroup	index
C₂×#19128081	2
#24909710	2
C₂×#19127504	2
C₂×#19127477	2
C₂×#499×C₈₃	2
C₂×#2245	83

subgroup

index

C₂×#499×C₈₃

Subgroups — C₂×#24909710 #35559723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#24909710 #35559723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909710 #35559723