Group #29570939 — subgroups

Subgroups — #29570939

Order 15616.

444 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 151 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	#21445568	C₂	1	2	yes	maximal
2	C₂×#16839504	C₂	1	2	yes	maximal
3	#21443700	C₂	1	2	yes	maximal
4	C₄×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519688	C₂²	1	4	yes	—
6	#16839517	—	2	2	no	—
7	#16839517	—	2	2	no	—
8	#16839504	C₄	1	8	yes	complemented
9	C₂×#125×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#389	C₄	1	2	yes	—
11	C₆₁×#387	—	2	2	no	—
12	C₆₁×#382	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
14	C₄×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	—	2	4	no	—
17	#14519688	—	2	4	no	—
18	#14519688	C₂×C₄	1	8	yes	—
19	#125×C₆₁	—	2	2	no	—
20	#125×C₆₁	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₁	—	2	2	no	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₆₁	D₄	1	2	yes	—
29	C₂×C₄²×C₆₁	D₄	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	#95×C₆₁	—	2	2	no	—
32	#95×C₆₁	—	2	2	no	—
33	#95×C₆₁	—	2	2	no	—
34	C₂²×C₄×C₆₁	—	2	2	no	—
35	C₂²×C₄×C₆₁	—	2	2	no	—
36	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₆₁	—	2	2	no	—
42	C₄⋊C₄×C₆₁	—	2	8	no	—
43	C₄⋊C₄×C₆₁	—	4	4	no	—
44	C₄⋊C₄×C₆₁	—	4	2	no	—
45	C₄⋊C₄×C₆₁	C₄²	1	4	yes	—
46	C₄²×C₆₁	—	2	2	no	—
47	C₄²×C₆₁	—	4	1	no	—
48	C₄²×C₆₁	—	4	2	no	—
49	C₄²×C₆₁	—	4	2	no	—
50	C₄²×C₆₁	—	4	4	no	—
51	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
52	C₄²×C₆₁	#41	1	2	yes	—
53	C₂×#92418	—	2	8	no	—
54	C₂×#92418	—	4	4	no	—
55	C₂×#92418	—	4	4	no	—
56	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	—	2	1	no	—
58	C₂×C₄×C₆₁	—	2	1	no	—
59	C₂×C₄×C₆₁	—	2	2	no	—
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	4	no	—
63	C₂×C₄×C₆₁	—	2	1	no	—
64	C₂×C₄×C₆₁	—	2	2	no	—
65	C₂×C₄×C₆₁	—	2	2	no	—
66	C₂×C₄×C₆₁	—	2	2	no	—
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	4	2	no	—
69	C₂×C₄×C₆₁	—	4	2	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	#101	1	2	yes	—
74	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
77	#92418	—	4	8	no	—
78	#8225	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
81	C₂²×C₆₁	#413	1	1	yes	characteristic
82	C₂²×C₆₁	#421	1	2	yes	—
83	C₂²×C₆₁	#421	1	2	yes	—
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	—	2	1	no	—
86	C₄×C₆₁	—	2	4	no	—
87	C₄×C₆₁	—	2	2	no	—
88	C₄×C₆₁	—	4	1	no	—
89	C₄×C₆₁	—	4	4	no	—
90	C₄×C₆₁	—	4	2	no	—
91	C₄×C₆₁	—	8	2	no	—
92	C₄×#125	D₆₁	1	1	yes	characteristic
93	#1745	—	61	2	no	—
94	C₂×#330	—	61	2	no	—
95	#1295	—	61	2	no	—
96	C₂×C₆₁	#1534	1	1	yes	characteristic
97	C₂×C₆₁	#1535	1	2	yes	—
98	C₂×C₆₁	#1537	1	1	yes	characteristic
99	C₂×C₆₁	C₄×#101	1	1	yes	characteristic
100	C₂×C₆₁	#1759	1	2	yes	—
101	C₂×#125	C₂×D₆₁	1	1	yes	characteristic
102	C₂×#97	—	61	4	no	—
103	#389	Dic₆₁	1	2	yes	—
104	#387	—	2	2	no	—
105	#382	C₂×D₆₁	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	2	no	—
107	C₄×C₄⋊C₄	Dic₆₁	1	2	yes	—
108	C₄³	C₂×D₆₁	1	1	yes	characteristic
109	#336	—	122	2	no	—
110	#336	—	122	2	no	—
111	#330	—	61	8	no	—
112	C₆₁	#8225	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	122	4	no	—
114	#125	—	2	2	no	—
115	#125	C₄×D₆₁	1	4	yes	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
125	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
126	#97	—	61	8	no	—
127	#97	—	122	4	no	—
128	#95	—	2	2	no	—
129	#95	—	2	2	no	—
130	#95	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
135	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
136	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
138	C₂×C₈	—	122	8	no	—
139	C₂×C₈	—	244	4	no	—
140	C₂×C₈	—	244	4	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	8	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	C₄×Dic₆₁	1	4	yes	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	1	no	—
148	C₄²	—	4	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	4	no	—
151	C₄²	C₂×C₄×D₆₁	1	2	yes	—
152	C₄²	#11758222	1	2	yes	—
153	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#14519696	1	2	yes	—
171	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
172	C₂×C₄	#14519736	1	1	yes	characteristic
173	C₂×C₄	C₄×C₂²⋊D₆₁	1	4	yes	—
174	C₈	—	244	8	no	—
175	C₂²	C₄×C₂²⋊Dic₆₁	1	1	yes	characteristic
176	C₂²	C₂×#14519696	1	1	yes	characteristic
177	C₂²	#16839746	1	1	yes	characteristic
178	C₂²	#16839779	1	2	yes	—
179	C₂²	#16839779	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	4	no	—
187	C₄	—	8	2	no	—
188	C₂	#21444446	1	1	yes	characteristic
189	C₂	#21444450	1	2	yes	—
190	C₂	#21444457	1	1	yes	characteristic
191	C₂	C₄×#14519696	1	1	yes	characteristic
192	C₂	#21445629	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445568

C₂

yes

maximal

C₂×#16839504

C₂

yes

maximal

#21443700

C₂

yes

maximal

C₄×#125×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519688

C₂²

yes

—

#16839517

—

#16839517

—

#16839504

C₄

yes

complemented

C₂×#125×C₆₁

C₂²

yes

characteristic

C₆₁×#389

C₄

yes

—

C₆₁×#387

—

C₆₁×#382

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₆₁

—

C₄×C₄⋊C₄×C₆₁

C₄

yes

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

#14519688

—

#14519688

C₂×C₄

yes

—

#125×C₆₁

—

#125×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#95×C₆₁

—

#95×C₆₁

—

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₄²

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

—

C₄²×C₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#101

yes

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

#117

yes

characteristic

C₂×C₄×C₆₁

C₄×D₄

yes

—

#92418

—

#8225

—

maximal, Sylow

C₂²×C₆₁

C₄×C₂²⋊C₄

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#421

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×#125

D₆₁

yes

—

—

—

C₂×C₆₁

#1534

yes

characteristic

C₂×C₆₁

#1535

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

100

C₂×C₆₁

#1759

yes

—

101

C₂×#125

C₂×D₆₁

yes

characteristic

102

C₂×#97

—

103

#389

Dic₆₁

yes

—

104

#387

—

105

#382

C₂×D₆₁

yes

characteristic

106

C₄×C₄⋊C₄

—

107

C₄×C₄⋊C₄

Dic₆₁

yes

—

108

C₄³

C₂×D₆₁

yes

characteristic

109

#336

—

122

—

110

#336

—

122

—

111

#330

—

112

C₆₁

#8225

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

122

—

114

#125

—

115

#125

C₄×D₆₁

yes

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

119

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

122

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

123

C₂×C₄²

C₂²⋊D₆₁

yes

—

124

C₂×C₄²

C₂²⋊D₆₁

yes

—

125

C₂×C₄²

C₂²×D₆₁

yes

characteristic

126

#97

—

127

#97

—

122

—

128

#95

—

129

#95

—

130

#95

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

134

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

135

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

136

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

137

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

—

138

C₂×C₈

—

122

—

139

C₂×C₈

—

244

—

140

C₂×C₈

—

244

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₄×Dic₆₁

yes

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂×C₄×D₆₁

yes

—

152

C₄²

#11758222

yes

—

153

C₂³

C₂×C₂²⋊Dic₆₁

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#14519696

yes

—

171

C₂×C₄

C₂×C₄×Dic₆₁

yes

characteristic

172

C₂×C₄

#14519736

yes

characteristic

173

C₂×C₄

C₄×C₂²⋊D₆₁

yes

—

174

C₈

—

244

—

175

C₂²

C₄×C₂²⋊Dic₆₁

yes

characteristic

176

C₂²

C₂×#14519696

yes

characteristic

177

C₂²

#16839746

yes

characteristic

178

C₂²

#16839779

yes

—

179

C₂²

#16839779

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

#21444446

yes

characteristic

189

C₂

#21444450

yes

—

190

C₂

#21444457

yes

characteristic

191

C₂

C₄×#14519696

yes

characteristic

192

C₂

#21445629

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#125×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#125	D₆₁
hypocenter	C₆₁	#8225
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₁

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₄×#125×C₆₁

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
8. #16839504	C₄	semidirect
112. C₆₁	#8225	semidirect

H ≅ G/N

type

8. #16839504

C₄

semidirect

112. C₆₁

#8225

semidirect

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

subgroup	index
#21445568	2
C₂×#16839504	2
#21443700	2
C₄×#125×C₆₁	2
#8225	61

subgroup

index

C₄×#125×C₆₁

Subgroups — #29570939

Conjugacy classes of subgroups

Complements

Maximal subgroups