Group #29580143 — subgroups

Subgroups — #29580143

Order 15616.

430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 167 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	#21445936	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	#21443702	C₂	1	4	yes	maximal
4	C₆₁×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	C₂²	1	2	yes	—
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	—	2	1	no	—
8	#16839517	—	2	1	no	—
9	#16839517	—	2	2	no	—
10	C₂×#125×C₆₁	—	2	1	no	—
11	C₆₁×#389	—	2	2	no	—
12	C₆₁×#389	C₄	1	4	yes	—
13	C₆₁×#384	C₂²	1	2	yes	—
14	C₆₁×#382	—	2	1	no	—
15	C₄³×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×#92418	—	2	2	no	—
17	C₂²×#92418	—	2	2	no	—
18	C₄⋊#92418	—	2	2	no	—
19	C₄⋊#92418	—	2	4	no	—
20	C₄⋊#92418	Q₈	1	4	yes	—
21	C₄×#92418	—	4	2	no	—
22	#125×C₆₁	—	4	2	no	—
23	#125×C₆₁	—	4	2	no	—
24	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	D₄	1	2	yes	—
31	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#95×C₆₁	—	2	4	no	—
34	C₂²×C₄×C₆₁	—	2	4	no	—
35	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₆₁	—	4	4	no	—
41	C₄⋊C₄×C₆₁	—	4	8	no	—
42	C₄⋊C₄×C₆₁	—	4	2	no	—
43	C₄⋊C₄×C₆₁	—	4	2	no	—
44	C₄⋊C₄×C₆₁	—	4	4	no	—
45	C₄²×C₆₁	—	2	4	no	—
46	C₄²×C₆₁	—	2	2	no	—
47	C₄²×C₆₁	—	4	2	no	—
48	C₄²×C₆₁	D₈	1	2	yes	—
49	C₄²×C₆₁	SD₁₆	1	2	yes	—
50	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
51	C₄²×C₆₁	#41	1	2	yes	—
52	C₂×#92418	—	2	4	no	—
53	C₂×#92418	—	4	2	no	—
54	C₂×#92418	—	4	2	no	—
55	C₂×#92418	—	4	2	no	—
56	C₂×#92418	—	4	4	no	—
57	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	—	2	2	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	2	no	—
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	4	4	no	—
65	C₂×C₄×C₆₁	—	4	4	no	—
66	C₂×C₄×C₆₁	—	4	4	no	—
67	C₂×C₄×C₆₁	#102	1	4	yes	—
68	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
70	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#135	1	4	yes	—
75	C₂×C₄×C₆₁	#137	1	2	yes	—
76	#92418	—	4	4	no	—
77	#92418	—	8	2	no	—
78	#9899	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	#391	1	1	yes	characteristic
80	C₂²×C₆₁	#413	1	2	yes	—
81	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
82	C₂²×C₆₁	#416	1	2	yes	—
83	C₂²×C₆₁	#419	1	1	yes	characteristic
84	C₄×C₆₁	—	2	4	no	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	4	2	no	—
87	C₄×C₆₁	—	8	4	no	—
88	C₄×C₆₁	#474	1	2	yes	—
89	C₄×C₆₁	#475	1	2	yes	—
90	C₄×C₆₁	#488	1	2	yes	—
91	C₄×C₆₁	#499	1	2	yes	—
92	#2859	D₆₁	1	1	yes	characteristic
93	#1831	—	61	1	no	—
94	C₄×C₄⋊C₈	—	61	1	no	—
95	#1296	—	61	4	no	—
96	C₂×C₆₁	#1540	1	2	yes	—
97	C₂×C₆₁	#1554	1	2	yes	—
98	C₂×C₆₁	#1988	1	1	yes	characteristic
99	C₂×C₆₁	#1991	1	1	yes	characteristic
100	C₂×C₆₁	#1998	1	1	yes	characteristic
101	C₂×#125	—	2	1	no	—
102	C₂×C₄⋊C₈	—	61	2	no	—
103	C₂×C₄⋊C₈	—	61	2	no	—
104	C₂×C₄×C₈	—	122	1	no	—
105	#389	—	2	2	no	—
106	#389	Dic₆₁	1	4	yes	—
107	#384	C₂×D₆₁	1	2	yes	—
108	#382	—	2	1	no	—
109	C₄³	C₂×D₆₁	1	1	yes	characteristic
110	#336	—	122	2	no	—
111	#336	—	122	1	no	—
112	C₆₁	#9899	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	122	2	no	—
114	C₂²×C₈	—	122	2	no	—
115	#125	—	4	2	no	—
116	#125	—	4	2	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
121	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
122	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
126	C₄⋊C₈	—	61	4	no	—
127	C₄⋊C₈	—	122	2	no	—
128	C₄⋊C₈	—	122	4	no	—
129	C₄×C₈	—	244	2	no	—
130	#95	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
133	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
134	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
137	C₂×C₈	—	122	4	no	—
138	C₂×C₈	—	244	2	no	—
139	C₂×C₈	—	244	4	no	—
140	C₂×C₈	—	244	2	no	—
141	C₂×C₈	—	244	2	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	#11758199	1	2	yes	—
151	C₄²	#11758200	1	2	yes	—
152	C₄²	Q₈×D₆₁	1	2	yes	—
153	C₄²	#11758226	1	2	yes	—
154	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	#14519698	1	4	yes	—
165	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
166	C₂×C₄	#14519753	1	2	yes	—
167	C₂×C₄	#14519766	1	1	yes	characteristic
168	C₂×C₄	#14519775	1	1	yes	characteristic
169	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
170	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
171	C₂×C₄	#14519797	1	4	yes	—
172	C₂×C₄	#14519808	1	2	yes	—
173	C₈	—	488	2	no	—
174	C₈	—	244	4	no	—
175	C₂²	#16839669	1	1	yes	characteristic
176	C₂²	#16839746	1	2	yes	—
177	C₂²	C₂×#14519698	1	1	yes	characteristic
178	C₂²	#16839758	1	2	yes	—
179	C₂²	#16839771	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	4	no	—
184	C₄	#16840039	1	2	yes	—
185	C₄	#16840044	1	2	yes	—
186	C₄	#16840119	1	2	yes	—
187	C₄	#16840161	1	2	yes	—
188	C₂	#21444468	1	2	yes	—
189	C₂	#21444529	1	2	yes	—
190	C₂	#21446772	1	1	yes	characteristic
191	C₂	#21446785	1	1	yes	characteristic
192	C₂	#21446813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21445936

C₂

yes

maximal, characteristic

C₄×C₄⋊#92418

C₂

yes

maximal, characteristic

#21443702

C₂

yes

maximal

C₆₁×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

C₂²

yes

—

C₂×C₄⋊#92418

C₂²

yes

—

C₂×C₄×#92418

—

#16839517

—

#16839517

—

C₂×#125×C₆₁

—

C₆₁×#389

—

C₆₁×#389

C₄

yes

—

C₆₁×#384

C₂²

yes

—

C₆₁×#382

—

C₄³×C₆₁

C₂²

yes

characteristic

C₂²×#92418

—

C₂²×#92418

—

C₄⋊#92418

—

C₄⋊#92418

—

C₄⋊#92418

Q₈

yes

—

C₄×#92418

—

#125×C₆₁

—

#125×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₈

yes

—

C₄²×C₆₁

SD₁₆

yes

—

C₄²×C₆₁

C₂×Q₈

yes

—

C₄²×C₆₁

#41

yes

—

—

—

—

—

—

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

#102

yes

—

C₂×C₄×C₆₁

C₄×Q₈

yes

—

C₂×C₄×C₆₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₁

#124

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₁

#135

yes

—

C₂×C₄×C₆₁

#137

yes

—

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

C₂×#102

yes

characteristic

C₂²×C₆₁

#416

yes

—

C₂²×C₆₁

#419

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#474

yes

—

C₄×C₆₁

#475

yes

—

C₄×C₆₁

#488

yes

—

C₄×C₆₁

#499

yes

—

#2859

D₆₁

yes

characteristic

#1831

—

C₄×C₄⋊C₈

—

#1296

—

C₂×C₆₁

#1540

yes

—

C₂×C₆₁

#1554

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

100

C₂×C₆₁

#1998

yes

characteristic

101

C₂×#125

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

—

104

C₂×C₄×C₈

—

122

—

105

#389

—

106

#389

Dic₆₁

yes

—

107

#384

C₂×D₆₁

yes

—

108

#382

—

109

C₄³

C₂×D₆₁

yes

characteristic

110

#336

—

122

—

111

#336

—

122

—

112

C₆₁

#9899

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

122

—

114

C₂²×C₈

—

122

—

115

#125

—

116

#125

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂×Dic₆₁

yes

—

121

C₂×C₄²

C₂×Dic₆₁

yes

—

122

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

123

C₂×C₄²

C₂²⋊D₆₁

yes

—

124

C₂×C₄²

C₂²⋊D₆₁

yes

characteristic

125

C₂×C₄²

C₂²×D₆₁

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

122

—

128

C₄⋊C₈

—

122

—

129

C₄×C₈

—

244

—

130

#95

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

133

C₂²×C₄

C₂²⋊Dic₆₁

yes

—

134

C₂²×C₄

C₂²×Dic₆₁

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

136

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

137

C₂×C₈

—

122

—

138

C₂×C₈

—

244

—

139

C₂×C₈

—

244

—

140

C₂×C₈

—

244

—

141

C₂×C₈

—

244

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

#11758199

yes

—

151

C₄²

#11758200

yes

—

152

C₄²

Q₈×D₆₁

yes

—

153

C₄²

#11758226

yes

—

154

C₂³

C₂×C₂²⋊Dic₆₁

yes

characteristic

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

#14519698

yes

—

165

C₂×C₄

Q₈×Dic₆₁

yes

—

166

C₂×C₄

#14519753

yes

—

167

C₂×C₄

#14519766

yes

characteristic

168

C₂×C₄

#14519775

yes

characteristic

169

C₂×C₄

C₂×#11758199

yes

characteristic

170

C₂×C₄

C₂×#11758200

yes

characteristic

171

C₂×C₄

#14519797

yes

—

172

C₂×C₄

#14519808

yes

—

173

C₈

—

488

—

174

C₈

—

244

—

175

C₂²

#16839669

yes

characteristic

176

C₂²

#16839746

yes

—

177

C₂²

C₂×#14519698

yes

characteristic

178

C₂²

#16839758

yes

—

179

C₂²

#16839771

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#16840039

yes

—

185

C₄

#16840044

yes

—

186

C₄

#16840119

yes

—

187

C₄

#16840161

yes

—

188

C₂

#21444468

yes

—

189

C₂

#21444529

yes

—

190

C₂

#21446772

yes

characteristic

191

C₂

#21446785

yes

characteristic

192

C₂

#21446813

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2859	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#2859	D₆₁
hypocenter	C₆₁	#9899
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₆₁

commutator

C₂²×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#2859

C₂

socle, prime radical

C₂³×C₆₁

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
112. C₆₁	#9899	semidirect

H ≅ G/N

type

112. C₆₁

#9899

semidirect

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

subgroup	index
#21445936	2
C₄×C₄⋊#92418	2
#21443702	2
C₆₁×#2859	2
#9899	61

subgroup

index

#21445936

C₄×C₄⋊#92418

#21443702

C₆₁×#2859

#9899

Subgroups — #29580143

Conjugacy classes of subgroups

Complements

Maximal subgroups