Group #29835332 — subgroups

Subgroups — C₂×#21448066 #29835332

Order 15616.

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×#16840142	C₂	1	1	yes	maximal, characteristic
2	#21448066	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16839565	C₂	1	4	yes	maximal
4	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#499	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	—	2	1	no	—
7	C₂²×Dic₂₄₄	C₂²	1	2	yes	—
8	C₂×#14519777	—	2	1	no	—
9	C₂²×#11758185	C₂²	1	2	yes	—
10	#16840142	—	2	2	no	—
11	#16840142	C₂²	1	4	yes	—
12	#16839565	C₂²	1	16	yes	—
13	#16839538	C₂²	1	4	yes	—
14	C₂×#125×C₆₁	—	2	1	no	—
15	C₆₁×#499	—	2	2	no	—
16	C₆₁×#499	C₂²	1	4	yes	—
17	C₂×#107×C₆₁	C₂²	1	2	yes	—
18	C₂×#106×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₂₂	—	2	2	no	—
21	C₂×Dic₂₄₄	—	2	8	no	—
22	C₂×Dic₂₄₄	—	2	4	no	—
23	C₂×Dic₂₄₄	—	2	2	no	—
24	C₂×Dic₂₄₄	—	4	4	no	—
25	C₂×Dic₂₄₄	C₂³	1	4	yes	—
26	#14519777	—	2	4	no	—
27	#14519777	—	4	2	no	—
28	C₂×C₄⋊Dic₆₁	—	4	2	no	—
29	C₂×#11758185	—	2	4	no	—
30	C₂×#11758185	C₂³	1	4	yes	—
31	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	#125×C₆₁	—	2	4	no	—
34	#125×C₆₁	—	4	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
37	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
38	#107×C₆₁	—	2	2	no	—
39	#107×C₆₁	C₂³	1	4	yes	—
40	#106×C₆₁	—	2	4	no	—
41	C₄×C₈×C₆₁	D₄	1	2	yes	—
42	C₄×C₈×C₆₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₆₁	—	4	2	no	—
44	C₂²×C₄×C₆₁	D₈	1	2	yes	—
45	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₆₁	—	4	2	no	—
47	C₂×Dic₁₂₂	—	2	4	no	—
48	C₂×Dic₁₂₂	—	4	4	no	—
49	C₂×Dic₁₂₂	—	4	4	no	—
50	C₂×Dic₁₂₂	—	4	2	no	—
51	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
52	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₆₁	#41	1	4	yes	—
55	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₆₁	—	2	4	no	—
57	C₄⋊C₄×C₆₁	—	4	2	no	—
58	C₄⋊C₄×C₆₁	—	4	8	no	—
59	C₄²×C₆₁	D₈	1	2	yes	—
60	C₄²×C₆₁	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₆₁	—	4	8	no	—
62	Dic₂₄₄	—	2	8	no	—
63	Dic₂₄₄	—	4	4	no	—
64	Dic₂₄₄	—	4	8	no	—
65	#11758185	—	2	4	no	—
66	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	4	no	—
68	C₂×C₄×C₆₁	—	4	4	no	—
69	C₂×C₄×C₆₁	—	8	4	no	—
70	C₂×C₄×C₆₁	#124	1	2	yes	—
71	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
72	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₆₁	—	4	4	no	—
76	C₂×Dic₆₁	—	8	4	no	—
77	Dic₁₂₂	—	4	4	no	—
78	Dic₁₂₂	—	4	4	no	—
79	Dic₁₂₂	—	8	4	no	—
80	C₈×C₆₁	#124	1	2	yes	—
81	C₈×C₆₁	C₂×SD₁₆	1	4	yes	—
82	C₈×C₆₁	C₂×#41	1	2	yes	—
83	C₂×#2245	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	#486	1	2	yes	—
85	C₂²×C₆₁	#510	1	4	yes	—
86	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₆₁	—	2	4	no	—
88	C₄×C₆₁	—	8	4	no	—
89	C₄×C₆₁	#486	1	2	yes	—
90	C₄×C₆₁	C₂×#124	1	1	yes	characteristic
91	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
92	Dic₆₁	—	8	4	no	—
93	C₂×#499	D₆₁	1	1	yes	characteristic
94	C₂×#494	—	61	1	no	—
95	#2245	—	61	8	no	—
96	C₂×#358	—	61	4	no	—
97	C₂×#346	—	61	1	no	—
98	C₂×C₆₁	#2245	1	4	yes	complemented
99	C₂×C₆₁	C₂×#486	1	1	yes	characteristic
100	C₂×C₆₁	C₂×#510	1	2	yes	—
101	C₂²×Q₁₆	—	61	2	no	—
102	C₂²×Q₁₆	—	122	1	no	—
103	C₂×C₄⋊Q₈	—	122	1	no	—
104	C₂×#125	—	2	1	no	—
105	C₂²×C₁₆	—	61	2	no	—
106	#499	—	2	2	no	—
107	#499	C₂×D₆₁	1	4	yes	—
108	#494	—	61	4	no	—
109	#494	—	122	2	no	—
110	C₂×#107	C₂×D₆₁	1	2	yes	—
111	C₂×#106	—	2	1	no	—
112	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
113	#358	—	61	16	no	—
114	#346	—	61	4	no	—
115	C₆₁	C₂×#2245	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	122	2	no	—
117	C₂×Q₁₆	—	61	4	no	—
118	C₂×Q₁₆	—	122	8	no	—
119	C₂×Q₁₆	—	122	2	no	—
120	C₂×Q₁₆	—	122	4	no	—
121	C₂×Q₁₆	—	244	4	no	—
122	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
124	C₄⋊Q₈	—	122	4	no	—
125	C₄⋊Q₈	—	244	2	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	—	244	2	no	—
131	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
132	C₂×C₁₆	—	61	4	no	—
133	C₂×C₁₆	—	122	4	no	—
134	#107	—	2	2	no	—
135	#107	C₂²×D₆₁	1	4	yes	—
136	#106	—	2	4	no	—
137	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
138	C₄×C₈	C₂²×D₆₁	1	2	yes	—
139	C₂×Q₈	—	122	4	no	—
140	C₂×Q₈	—	244	4	no	—
141	C₂×Q₈	—	244	4	no	—
142	C₂×Q₈	—	244	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	244	2	no	—
145	C₂²×C₄	#11758199	1	2	yes	—
146	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
147	Q₁₆	—	122	8	no	—
148	Q₁₆	—	244	4	no	—
149	Q₁₆	—	244	8	no	—
150	C₂×C₈	#11758201	1	4	yes	—
151	C₂×C₈	#11758226	1	4	yes	—
152	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
154	C₂×C₈	C₂³×D₆₁	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	—	244	8	no	—
159	C₄²	#11758199	1	2	yes	—
160	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
161	C₁₆	—	122	4	no	—
162	C₂³	C₂×#11758199	1	1	yes	characteristic
163	Q₈	—	488	4	no	—
164	Q₈	—	244	4	no	—
165	Q₈	—	244	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	488	4	no	—
170	C₂×C₄	—	244	4	no	—
171	C₂×C₄	#14519766	1	2	yes	—
172	C₂×C₄	C₂×#11758199	1	4	yes	—
173	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
174	C₂×C₄	C₂×#11758201	1	2	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
176	C₈	#14519766	1	2	yes	—
177	C₈	C₂×#11758201	1	4	yes	—
178	C₈	C₂×#11758226	1	2	yes	—
179	C₂²	#16840110	1	2	yes	—
180	C₂²	#16840202	1	4	yes	—
181	C₂²	C₂²×#11758199	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	488	4	no	—
185	C₄	#16840110	1	2	yes	—
186	C₄	C₂×#14519766	1	1	yes	characteristic
187	C₄	C₂²×#11758201	1	1	yes	characteristic
188	C₂	#21448066	1	4	yes	direct product factor
189	C₂	C₂×#16840110	1	1	yes	characteristic
190	C₂	C₂×#16840202	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840142

C₂

yes

maximal, characteristic

#21448066

C₂

yes

maximal, direct product factor

C₂×#16839565

C₂

yes

maximal

C₂×#16839538

C₂

yes

maximal, characteristic

C₂×C₆₁×#499

C₂

yes

maximal, characteristic

C₂²×Dic₂₄₄

—

C₂²×Dic₂₄₄

C₂²

yes

—

C₂×#14519777

—

C₂²×#11758185

C₂²

yes

—

#16840142

—

#16840142

C₂²

yes

—

#16839565

C₂²

yes

—

#16839538

C₂²

yes

—

C₂×#125×C₆₁

—

C₆₁×#499

—

C₆₁×#499

C₂²

yes

—

C₂×#107×C₆₁

C₂²

yes

—

C₂×#106×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×Dic₁₂₂

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

C₂³

yes

—

#14519777

—

#14519777

—

C₂×C₄⋊Dic₆₁

—

C₂×#11758185

—

C₂×#11758185

C₂³

yes

—

C₂²×C₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#125×C₆₁

—

#125×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

#107×C₆₁

—

#107×C₆₁

C₂³

yes

—

#106×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₈

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₂×C₈×C₆₁

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₈

yes

—

C₄²×C₆₁

C₂×D₄

yes

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

#11758185

—

C₂³×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#124

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂²×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

#124

yes

—

C₈×C₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

C₂×#41

yes

—

C₂×#2245

—

maximal, Sylow

C₂²×C₆₁

#486

yes

—

C₂²×C₆₁

#510

yes

—

C₂²×C₆₁

C₂²×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#486

yes

—

C₄×C₆₁

C₂×#124

yes

characteristic

C₄×C₆₁

C₂²×SD₁₆

yes

characteristic

Dic₆₁

—

C₂×#499

D₆₁

yes

—

—

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

100

C₂×C₆₁

C₂×#510

yes

—

101

C₂²×Q₁₆

—

102

C₂²×Q₁₆

—

122

—

103

C₂×C₄⋊Q₈

—

122

—

104

C₂×#125

—

105

C₂²×C₁₆

—

106

#499

—

107

#499

C₂×D₆₁

yes

—

108

#494

—

109

#494

—

122

—

110

C₂×#107

C₂×D₆₁

yes

—

111

C₂×#106

—

112

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

113

#358

—

114

#346

—

115

C₆₁

C₂×#2245

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

—

122

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

122

—

119

C₂×Q₁₆

—

122

—

120

C₂×Q₁₆

—

122

—

121

C₂×Q₁₆

—

244

—

122

C₂²×C₈

C₂²⋊D₆₁

yes

characteristic

123

C₂²×C₈

C₂²×D₆₁

yes

characteristic

124

C₄⋊Q₈

—

122

—

125

C₄⋊Q₈

—

244

—

126

#125

—

127

#125

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

244

—

131

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

132

C₂×C₁₆

—

133

C₂×C₁₆

—

122

—

134

#107

—

135

#107

C₂²×D₆₁

yes

—

136

#106

—

137

C₄×C₈

C₂²⋊D₆₁

yes

—

138

C₄×C₈

C₂²×D₆₁

yes

—

139

C₂×Q₈

—

122

—

140

C₂×Q₈

—

244

—

141

C₂×Q₈

—

244

—

142

C₂×Q₈

—

244

—

143

C₂²×C₄

—

144

C₂²×C₄

—

244

—

145

C₂²×C₄

#11758199

yes

—

146

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

147

Q₁₆

—

122

—

148

Q₁₆

—

244

—

149

Q₁₆

—

244

—

150

C₂×C₈

#11758201

yes

—

151

C₂×C₈

#11758226

yes

—

152

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

153

C₂×C₈

C₂×C₂²⋊D₆₁

yes

—

154

C₂×C₈

C₂³×D₆₁

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

244

—

159

C₄²

#11758199

yes

—

160

C₄²

C₂×C₂²⋊D₆₁

yes

—

161

C₁₆

—

122

—

162

C₂³

C₂×#11758199

yes

characteristic

163

Q₈

—

488

—

164

Q₈

—

244

—

165

Q₈

—

244

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

488

—

170

C₂×C₄

—

244

—

171

C₂×C₄

#14519766

yes

—

172

C₂×C₄

C₂×#11758199

yes

—

173

C₂×C₄

C₂×#11758199

yes

characteristic

174

C₂×C₄

C₂×#11758201

yes

—

175

C₂×C₄

C₂²×C₂²⋊D₆₁

yes

characteristic

176

C₈

#14519766

yes

—

177

C₈

C₂×#11758201

yes

—

178

C₈

C₂×#11758226

yes

—

179

C₂²

#16840110

yes

—

180

C₂²

#16840202

yes

—

181

C₂²

C₂²×#11758199

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

488

—

185

C₄

#16840110

yes

—

186

C₄

C₂×#14519766

yes

characteristic

187

C₄

C₂²×#11758201

yes

characteristic

188

C₂

#21448066

yes

direct product factor

189

C₂

C₂×#16840110

yes

characteristic

190

C₂

C₂×#16840202

yes

—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#499	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₂×#499	D₆₁
hypocenter	C₆₁	C₂×#2245
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂³

C₂×#11758199

commutator, cosocle

C₂×C₈×C₆₁

C₂⁴

Frattini

C₂×C₈

C₂³×D₆₁

Fitting

C₂×C₆₁×#499

C₂

socle, prime radical

C₂³×C₆₁

C₂×D₈

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

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

N	H ≅ G/N	type
2. #21448066	C₂	direct (N × H)
98. C₂×C₆₁	#2245	semidirect
115. C₆₁	C₂×#2245	semidirect
188. C₂	#21448066	direct (N × H)

H ≅ G/N

type

2. #21448066

C₂

direct (N × H)

98. C₂×C₆₁

#2245

semidirect

115. C₆₁

C₂×#2245

semidirect

188. C₂

#21448066

direct (N × H)

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

subgroup	index
C₂×#16840142	2
#21448066	2
C₂×#16839565	2
C₂×#16839538	2
C₂×C₆₁×#499	2
C₂×#2245	61

subgroup

index

C₂×C₆₁×#499

Subgroups — C₂×#21448066 #29835332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#21448066 #29835332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21448066 #29835332