Group #29579852 — subgroups

Subgroups — #29579852

Order 15616.

710 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 199 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	#21446626	C₂	1	4	yes	maximal, complemented
2	#21445976	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16839483	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₆₁×#493	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519709	—	2	2	no	—
6	C₂×#14519698	—	2	8	no	—
7	C₂×C₄×#92418	C₂²	1	2	yes	—
8	#16839632	C₂²	1	2	yes	—
9	#16839483	—	2	2	no	—
10	C₂²×D₈×C₆₁	—	2	2	no	—
11	C₂×#127×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#493	—	2	2	no	—
13	C₆₁×#493	C₄	1	4	yes	complemented
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×#92418	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₆₁	—	2	4	no	—
17	C₂×C₄⋊Dic₆₁	—	2	2	no	—
18	#14519709	—	4	4	no	—
19	#14519698	—	4	16	no	—
20	C₄×#92418	—	2	2	no	—
21	C₄×#92418	—	2	2	no	—
22	C₂²×D₄×C₆₁	—	2	2	no	—
23	C₂²×D₄×C₆₁	—	2	4	no	—
24	C₂×D₈×C₆₁	—	2	8	no	—
25	C₂×D₈×C₆₁	—	4	4	no	—
26	C₂×D₈×C₆₁	—	4	4	no	—
27	C₂²×C₈×C₆₁	D₄	1	2	yes	—
28	#127×C₆₁	—	2	2	no	—
29	#127×C₆₁	—	2	4	no	—
30	#127×C₆₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₆₁	D₄	1	2	yes	—
34	C₂⁴×C₆₁	—	4	2	no	—
35	C₂×D₄×C₆₁	—	2	4	no	—
36	C₂×D₄×C₆₁	—	2	8	no	—
37	C₂×D₄×C₆₁	—	4	4	no	—
38	C₂×D₄×C₆₁	—	4	2	no	—
39	C₂×D₄×C₆₁	—	4	4	no	—
40	C₂×D₄×C₆₁	—	4	4	no	—
41	C₂×D₄×C₆₁	—	4	8	no	—
42	C₂×D₄×C₆₁	—	4	8	no	—
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
45	C₂²×Dic₆₁	—	4	2	no	—
46	D₈×C₆₁	—	4	16	no	—
47	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
48	C₂×C₈×C₆₁	D₈	1	4	yes	—
49	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
50	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₆₁	#41	1	2	yes	—
53	C₄⋊Dic₆₁	—	4	4	no	—
54	C₄⋊Dic₆₁	—	4	8	no	—
55	C₂×#92418	—	2	4	no	—
56	C₂×#92418	—	2	8	no	—
57	C₂³×C₆₁	—	4	4	no	—
58	C₂³×C₆₁	—	8	2	no	—
59	C₂³×C₆₁	—	8	2	no	—
60	C₂³×C₆₁	—	8	4	no	—
61	C₂³×C₆₁	—	8	4	no	—
62	C₂³×C₆₁	#127	1	1	yes	characteristic
63	D₄×C₆₁	—	4	8	no	—
64	D₄×C₆₁	—	4	16	no	—
65	D₄×C₆₁	—	8	8	no	—
66	D₄×C₆₁	—	8	4	no	—
67	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₆₁	#121	1	2	yes	—
70	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	8	yes	—
74	C₂×Dic₆₁	—	8	2	no	—
75	C₂×Dic₆₁	—	8	4	no	—
76	C₈×C₆₁	#102	1	8	yes	complemented
77	#92418	—	4	4	no	—
78	#9837	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	—	8	4	no	—
80	C₂²×C₆₁	—	8	8	no	—
81	C₂²×C₆₁	—	16	2	no	—
82	C₂²×C₆₁	—	16	2	no	—
83	C₂²×C₆₁	—	16	4	no	—
84	C₂²×C₆₁	#390	1	1	yes	characteristic
85	C₂²×C₆₁	#492	1	1	yes	characteristic
86	C₂²×C₆₁	#493	1	1	yes	characteristic
87	C₂²×C₆₁	#496	1	4	yes	—
88	C₄×C₆₁	C₂×#102	1	2	yes	—
89	C₄×C₆₁	#442	1	4	yes	—
90	C₄×C₆₁	#469	1	4	yes	—
91	C₄×C₆₁	#486	1	2	yes	—
92	Dic₆₁	—	16	2	no	—
93	C₂×#493	D₆₁	1	1	yes	characteristic
94	#1957	—	61	4	no	—
95	#1844	—	61	1	no	—
96	C₂×#322	—	61	1	no	—
97	C₂×C₆₁	—	16	4	no	—
98	C₂×C₆₁	#1715	1	2	yes	—
99	C₂×C₆₁	#1721	1	2	yes	—
100	C₂×C₆₁	#1957	1	1	yes	characteristic
101	C₂×C₆₁	#1972	1	2	yes	—
102	C₂²×D₈	—	2	2	no	—
103	C₂×#127	C₂×D₆₁	1	2	yes	—
104	#493	—	2	2	no	—
105	#493	Dic₆₁	1	4	yes	complemented
106	C₂×#106	—	122	2	no	—
107	C₂×#102	—	122	8	no	—
108	C₂×C₄×C₈	—	61	2	no	—
109	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
110	#384	—	61	2	no	—
111	#322	—	122	2	no	—
112	C₆₁	#9837	1	1	yes	characteristic, complemented, Sylow
113	C₂²×D₄	—	2	4	no	—
114	C₂²×D₄	—	2	2	no	—
115	C₂×D₈	—	2	8	no	—
116	C₂×D₈	—	4	4	no	—
117	C₂×D₈	—	4	4	no	—
118	C₂²×C₈	—	61	4	no	—
119	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
120	#127	—	2	4	no	—
121	#127	—	2	2	no	—
122	#127	C₂×Dic₆₁	1	4	yes	—
123	C₂×C₄⋊C₄	—	122	4	no	—
124	C₂×C₄⋊C₄	—	122	2	no	—
125	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
126	#106	—	244	4	no	—
127	#102	—	244	16	no	—
128	C₄×C₈	—	122	2	no	—
129	C₄×C₈	—	122	2	no	—
130	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
131	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
132	C₂⁴	—	4	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	4	no	—
138	C₂×D₄	—	4	4	no	—
139	C₂×D₄	—	4	8	no	—
140	C₂×D₄	—	4	8	no	—
141	C₂²×C₄	—	244	2	no	—
142	C₂²×C₄	D₄×D₆₁	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
144	D₈	—	4	16	no	—
145	C₂×C₈	—	122	4	no	—
146	C₂×C₈	—	122	8	no	—
147	C₂×C₈	#11758199	1	4	yes	—
148	C₂×C₈	#11758200	1	4	yes	—
149	C₂×C₈	C₂²⋊Dic₆₁	1	4	yes	—
150	C₄⋊C₄	—	244	4	no	—
151	C₄⋊C₄	—	244	8	no	—
152	C₄²	#11758224	1	2	yes	—
153	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
154	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
155	C₂³	—	4	4	no	—
156	C₂³	—	8	2	no	—
157	C₂³	—	8	2	no	—
158	C₂³	—	8	4	no	—
159	C₂³	—	8	4	no	—
160	C₂³	#14519773	1	1	yes	characteristic
161	D₄	—	4	8	no	—
162	D₄	—	4	16	no	—
163	D₄	—	8	8	no	—
164	D₄	—	8	4	no	—
165	C₂×C₄	—	488	2	no	—
166	C₂×C₄	—	488	4	no	—
167	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
168	C₂×C₄	D₄×Dic₆₁	1	2	yes	—
169	C₂×C₄	#14519748	1	2	yes	—
170	C₂×C₄	#14519764	1	1	yes	characteristic
171	C₂×C₄	C₂×#11758199	1	2	yes	—
172	C₂×C₄	C₂×#11758200	1	2	yes	—
173	C₂×C₄	#14519801	1	8	yes	—
174	C₈	—	244	4	no	—
175	C₈	#14519698	1	8	yes	complemented
176	C₂²	—	8	8	no	—
177	C₂²	—	8	4	no	—
178	C₂²	—	16	2	no	—
179	C₂²	—	16	2	no	—
180	C₂²	—	16	4	no	—
181	C₂²	#16839665	1	1	yes	characteristic
182	C₂²	#16840134	1	1	yes	characteristic
183	C₂²	#16840138	1	1	yes	characteristic
184	C₂²	#16840150	1	4	yes	—
185	C₄	—	976	2	no	—
186	C₄	C₂×#14519698	1	2	yes	—
187	C₄	#16839874	1	4	yes	—
188	C₄	#16840012	1	4	yes	—
189	C₄	#16840108	1	2	yes	—
190	C₂	—	16	4	no	—
191	C₂	#21445445	1	2	yes	—
192	C₂	#21445486	1	2	yes	—
193	C₂	#21446626	1	1	yes	characteristic
194	C₂	#21446696	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21446626

C₂

yes

maximal, complemented

#21445976

C₂

yes

maximal, characteristic, complemented

C₂×#16839483

C₂

yes

maximal, characteristic, complemented

C₂×C₆₁×#493

C₂

yes

maximal, characteristic

C₂×#14519709

—

C₂×#14519698

—

C₂×C₄×#92418

C₂²

yes

—

#16839632

C₂²

yes

—

#16839483

—

C₂²×D₈×C₆₁

—

C₂×#127×C₆₁

C₂²

yes

—

C₆₁×#493

—

C₆₁×#493

C₄

yes

complemented

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×#92418

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

—

—

—

—

C₂²×D₄×C₆₁

—

C₂²×D₄×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂×D₈×C₆₁

—

C₂²×C₈×C₆₁

D₄

yes

—

#127×C₆₁

—

#127×C₆₁

—

#127×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂⁴×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

D₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄²×C₆₁

#41

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

#127

yes

characteristic

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

C₂×#102

yes

—

C₄×C₆₁

#442

yes

—

C₄×C₆₁

#469

yes

—

C₄×C₆₁

#486

yes

—

Dic₆₁

—

C₂×#493

D₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1715

yes

—

C₂×C₆₁

#1721

yes

—

100

C₂×C₆₁

#1957

yes

characteristic

101

C₂×C₆₁

#1972

yes

—

102

C₂²×D₈

—

103

C₂×#127

C₂×D₆₁

yes

—

104

#493

—

105

#493

Dic₆₁

yes

complemented

106

C₂×#106

—

122

—

107

C₂×#102

—

122

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

110

#384

—

111

#322

—

122

—

112

C₆₁

#9837

yes

characteristic, complemented, Sylow

113

C₂²×D₄

—

114

C₂²×D₄

—

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂²⋊D₆₁

yes

—

120

#127

—

121

#127

—

122

#127

C₂×Dic₆₁

yes

—

123

C₂×C₄⋊C₄

—

122

—

124

C₂×C₄⋊C₄

—

122

—

125

C₂×C₄²

C₂²×D₆₁

yes

characteristic

126

#106

—

244

—

127

#102

—

244

—

128

C₄×C₈

—

122

—

129

C₄×C₈

—

122

—

130

C₄×C₈

C₂×Dic₆₁

yes

—

131

C₄×C₈

C₂²⋊D₆₁

yes

—

132

C₂⁴

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂²×C₄

—

244

—

142

C₂²×C₄

D₄×D₆₁

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

144

D₈

—

145

C₂×C₈

—

122

—

146

C₂×C₈

—

122

—

147

C₂×C₈

#11758199

yes

—

148

C₂×C₈

#11758200

yes

—

149

C₂×C₈

C₂²⋊Dic₆₁

yes

—

150

C₄⋊C₄

—

244

—

151

C₄⋊C₄

—

244

—

152

C₄²

#11758224

yes

—

153

C₄²

C₂²×Dic₆₁

yes

characteristic

154

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

#14519773

yes

characteristic

161

D₄

—

162

D₄

—

163

D₄

—

164

D₄

—

165

C₂×C₄

—

488

—

166

C₂×C₄

—

488

—

167

C₂×C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

168

C₂×C₄

D₄×Dic₆₁

yes

—

169

C₂×C₄

#14519748

yes

—

170

C₂×C₄

#14519764

yes

characteristic

171

C₂×C₄

C₂×#11758199

yes

—

172

C₂×C₄

C₂×#11758200

yes

—

173

C₂×C₄

#14519801

yes

—

174

C₈

—

244

—

175

C₈

#14519698

yes

complemented

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#16839665

yes

characteristic

182

C₂²

#16840134

yes

characteristic

183

C₂²

#16840138

yes

characteristic

184

C₂²

#16840150

yes

—

185

C₄

—

976

—

186

C₄

C₂×#14519698

yes

—

187

C₄

#16839874

yes

—

188

C₄

#16840012

yes

—

189

C₄

#16840108

yes

—

190

C₂

—

191

C₂

#21445445

yes

—

192

C₂

#21445486

yes

—

193

C₂

#21446626

yes

characteristic

194

C₂

#21446696

yes

—

	subgroup	G/H
center	C₂³	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting, prime radical	C₂×C₆₁×#493	C₂
socle	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#493	D₆₁
hypocenter	C₆₁	#9837
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂³

#14519773

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting, prime radical

C₂×C₆₁×#493

C₂

socle

C₂³×C₆₁

#127

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #21446626	C₂	semidirect
2. #21445976	C₂	semidirect
3. C₂×#16839483	C₂	semidirect
13. C₆₁×#493	C₄	semidirect
76. C₈×C₆₁	#102	semidirect
105. #493	Dic₆₁	semidirect
112. C₆₁	#9837	semidirect
175. C₈	#14519698	semidirect

H ≅ G/N

type

1. #21446626

C₂

semidirect

2. #21445976

C₂

semidirect

3. C₂×#16839483

C₂

semidirect

13. C₆₁×#493

C₄

semidirect

76. C₈×C₆₁

#102

semidirect

105. #493

Dic₆₁

semidirect

112. C₆₁

#9837

semidirect

175. C₈

#14519698

semidirect

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

subgroup	index
#21446626	2
#21445976	2
C₂×#16839483	2
C₂×C₆₁×#493	2
#9837	61

subgroup

index

C₂×C₆₁×#493

Subgroups — #29579852

Conjugacy classes of subgroups

Complements

Maximal subgroups