Group #29514601 — subgroups

Subgroups — C₆₁×#11168 #29514601

Order 15616.

802 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 222 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	C₂²×#101×C₆₁	C₂	1	2	yes	maximal
2	C₆₁×#2310	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#344	C₂	1	4	yes	maximal
4	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₆₁	C₂²	1	4	yes	—
6	C₂×#117×C₆₁	C₂²	1	2	yes	—
7	C₂×#101×C₆₁	—	2	4	no	—
8	C₂×#101×C₆₁	—	2	8	no	—
9	C₆₁×#391	—	2	4	no	—
10	C₆₁×#381	C₄	1	4	yes	—
11	C₆₁×#344	—	2	8	no	—
12	C₂²×Q₈×C₆₁	—	2	4	no	—
13	C₂²×Q₈×C₆₁	—	2	4	no	—
14	C₂²×Q₈×C₆₁	—	2	4	no	—
15	C₂²×Q₈×C₆₁	D₄	1	4	yes	—
16	C₂³×C₄×C₆₁	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₆₁	—	2	4	no	—
19	C₂×M₁₆×C₆₁	—	2	4	no	—
20	C₂×M₁₆×C₆₁	—	2	4	no	—
21	C₂×M₁₆×C₆₁	—	2	4	no	—
22	C₂×M₁₆×C₆₁	—	2	4	no	—
23	C₂×M₁₆×C₆₁	D₄	1	8	yes	—
24	C₂²×C₈×C₆₁	—	2	4	no	—
25	#117×C₆₁	—	2	4	no	—
26	#117×C₆₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₆₁	—	2	2	no	—
30	#101×C₆₁	—	2	16	no	—
31	#95×C₆₁	—	2	8	no	—
32	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×Q₈×C₆₁	—	2	4	no	—
34	C₂×Q₈×C₆₁	—	2	8	no	—
35	C₂×Q₈×C₆₁	—	2	8	no	—
36	C₂×Q₈×C₆₁	—	2	4	no	—
37	C₂×Q₈×C₆₁	—	4	4	no	—
38	C₂×Q₈×C₆₁	—	4	4	no	—
39	C₂×Q₈×C₆₁	—	4	4	no	—
40	C₂×Q₈×C₆₁	—	4	4	no	—
41	C₂²×C₄×C₆₁	—	2	4	no	—
42	C₂²×C₄×C₆₁	—	2	8	no	—
43	C₂²×C₄×C₆₁	—	2	4	no	—
44	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₆₁	#41	1	4	yes	—
48	C₂²×C₄×C₆₁	#41	1	4	yes	—
49	M₁₆×C₆₁	—	2	8	no	—
50	M₁₆×C₆₁	—	2	8	no	—
51	M₁₆×C₆₁	—	4	4	no	—
52	M₁₆×C₆₁	—	4	4	no	—
53	C₂×C₈×C₆₁	—	2	8	no	—
54	C₂×C₈×C₆₁	—	4	4	no	—
55	C₂×C₈×C₆₁	—	4	4	no	—
56	C₄⋊C₄×C₆₁	—	4	4	no	—
57	C₂²⋊C₄×C₆₁	—	2	8	no	—
58	C₄²×C₆₁	—	4	2	no	—
59	C₄²×C₆₁	—	4	2	no	—
60	C₂³×C₆₁	—	4	2	no	—
61	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₆₁	#117	1	2	yes	—
63	C₂³×C₆₁	C₄×D₄	1	4	yes	—
64	Q₈×C₆₁	—	4	8	no	—
65	Q₈×C₆₁	—	4	8	no	—
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	—	2	8	no	—
68	C₂×C₄×C₆₁	—	4	8	no	—
69	C₂×C₄×C₆₁	—	4	8	no	—
70	C₂×C₄×C₆₁	—	4	4	no	—
71	C₂×C₄×C₆₁	#120	1	2	yes	—
72	C₂×C₄×C₆₁	#121	1	4	yes	—
73	C₂×C₄×C₆₁	#121	1	4	yes	—
74	C₂×C₄×C₆₁	#123	1	4	yes	—
75	C₂×C₄×C₆₁	#123	1	4	yes	—
76	C₂×C₄×C₆₁	#123	1	2	yes	—
77	C₂×C₄×C₆₁	#124	1	4	yes	—
78	C₂×C₄×C₆₁	#126	1	4	yes	—
79	C₈×C₆₁	—	4	8	no	—
80	#11168	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₆₁	—	2	2	no	—
82	C₂²×C₆₁	—	2	4	no	—
83	C₂²×C₆₁	—	4	4	no	—
84	C₂²×C₆₁	#381	1	1	yes	characteristic
85	C₂²×C₆₁	#386	1	2	yes	—
86	C₂²×C₆₁	#388	1	4	yes	—
87	C₄×C₆₁	—	4	4	no	—
88	C₄×C₆₁	—	8	4	no	—
89	C₄×C₆₁	#394	1	4	yes	—
90	C₄×C₆₁	#397	1	4	yes	—
91	C₂²×#101	C₂×C₆₁	1	2	yes	—
92	#2310	C₂×C₆₁	1	1	yes	characteristic
93	C₂×#344	C₂×C₆₁	1	4	yes	—
94	C₂×C₆₁	—	2	2	no	—
95	C₂×C₆₁	—	2	4	no	—
96	C₂×C₆₁	#1440	1	1	yes	characteristic
97	C₂×C₆₁	#1830	1	2	yes	—
98	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
99	C₂²×M₁₆	C₂²×C₆₁	1	4	yes	—
100	C₂×#117	C₂²×C₆₁	1	2	yes	—
101	C₂×#101	—	2	4	no	—
102	C₂×#101	—	2	8	no	—
103	#391	—	2	4	no	—
104	#381	C₄×C₆₁	1	4	yes	—
105	#344	—	2	8	no	—
106	C₆₁	#11168	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×Q₈	—	2	4	no	—
108	C₂²×Q₈	—	2	4	no	—
109	C₂²×Q₈	—	2	4	no	—
110	C₂²×Q₈	D₄×C₆₁	1	4	yes	—
111	C₂³×C₄	C₂×C₄×C₆₁	1	2	yes	—
112	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
113	C₂×M₁₆	—	2	4	no	—
114	C₂×M₁₆	—	2	4	no	—
115	C₂×M₁₆	—	2	4	no	—
116	C₂×M₁₆	—	2	4	no	—
117	C₂×M₁₆	—	2	4	no	—
118	C₂×M₁₆	D₄×C₆₁	1	8	yes	—
119	C₂²×C₈	—	2	4	no	—
120	#117	—	2	4	no	—
121	#117	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
124	C₂×C₄²	—	2	2	no	—
125	#101	—	2	16	no	—
126	#95	—	2	8	no	—
127	C₂⁴	C₂²×C₄×C₆₁	1	1	yes	characteristic
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	8	no	—
130	C₂×Q₈	—	2	8	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	8	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
140	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
141	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
142	C₂²×C₄	#41×C₆₁	1	4	yes	—
143	C₂²×C₄	#41×C₆₁	1	4	yes	—
144	M₁₆	—	2	8	no	—
145	M₁₆	—	2	8	no	—
146	M₁₆	—	4	4	no	—
147	M₁₆	—	4	4	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	—	4	4	no	—
150	C₂×C₈	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	2	8	no	—
153	C₄²	—	4	2	no	—
154	C₄²	—	4	2	no	—
155	C₂³	—	4	2	no	—
156	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
157	C₂³	#117×C₆₁	1	2	yes	—
158	C₂³	C₄×D₄×C₆₁	1	4	yes	—
159	Q₈	—	4	8	no	—
160	Q₈	—	4	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	#120×C₆₁	1	2	yes	—
167	C₂×C₄	#121×C₆₁	1	4	yes	—
168	C₂×C₄	#121×C₆₁	1	4	yes	—
169	C₂×C₄	#123×C₆₁	1	4	yes	—
170	C₂×C₄	#123×C₆₁	1	4	yes	—
171	C₂×C₄	#123×C₆₁	1	2	yes	—
172	C₂×C₄	#124×C₆₁	1	4	yes	—
173	C₂×C₄	#126×C₆₁	1	4	yes	—
174	C₈	—	4	8	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	4	4	no	—
178	C₂²	C₆₁×#381	1	1	yes	characteristic
179	C₂²	C₆₁×#386	1	2	yes	—
180	C₂²	C₆₁×#388	1	4	yes	—
181	C₄	—	4	4	no	—
182	C₄	—	8	4	no	—
183	C₄	C₆₁×#394	1	4	yes	—
184	C₄	C₆₁×#397	1	4	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	2	4	no	—
187	C₂	C₆₁×#1440	1	1	yes	characteristic
188	C₂	C₆₁×#1830	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#101×C₆₁

C₂

yes

maximal

C₆₁×#2310

C₂

yes

maximal, characteristic

C₂×C₆₁×#344

C₂

yes

maximal

C₂³×Q₈×C₆₁

C₂²

yes

characteristic

C₂²×M₁₆×C₆₁

C₂²

yes

—

C₂×#117×C₆₁

C₂²

yes

—

C₂×#101×C₆₁

—

C₂×#101×C₆₁

—

C₆₁×#391

—

C₆₁×#381

C₄

yes

—

C₆₁×#344

—

C₂²×Q₈×C₆₁

—

C₂²×Q₈×C₆₁

—

C₂²×Q₈×C₆₁

—

C₂²×Q₈×C₆₁

D₄

yes

—

C₂³×C₄×C₆₁

C₂×C₄

yes

—

C₂³×C₄×C₆₁

C₂³

yes

characteristic

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

—

#117×C₆₁

—

#117×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

—

#101×C₆₁

—

#95×C₆₁

—

C₂⁴×C₆₁

C₂²×C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×C₄×C₆₁

#41

yes

—

C₂²×C₄×C₆₁

#41

yes

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₆₁

#117

yes

—

C₂³×C₆₁

C₄×D₄

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#120

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#121

yes

—

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#123

yes

—

C₂×C₄×C₆₁

#124

yes

—

C₂×C₄×C₆₁

#126

yes

—

C₈×C₆₁

—

#11168

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#381

yes

characteristic

C₂²×C₆₁

#386

yes

—

C₂²×C₆₁

#388

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#394

yes

—

C₄×C₆₁

#397

yes

—

C₂²×#101

C₂×C₆₁

yes

—

#2310

C₂×C₆₁

yes

characteristic

C₂×#344

C₂×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1440

yes

characteristic

C₂×C₆₁

#1830

yes

—

C₂³×Q₈

C₂²×C₆₁

yes

characteristic

C₂²×M₁₆

C₂²×C₆₁

yes

—

100

C₂×#117

C₂²×C₆₁

yes

—

101

C₂×#101

—

102

C₂×#101

—

103

#391

—

104

#381

C₄×C₆₁

yes

—

105

#344

—

106

C₆₁

#11168

yes

characteristic, direct product factor, Sylow

107

C₂²×Q₈

—

108

C₂²×Q₈

—

109

C₂²×Q₈

—

110

C₂²×Q₈

D₄×C₆₁

yes

—

111

C₂³×C₄

C₂×C₄×C₆₁

yes

—

112

C₂³×C₄

C₂³×C₆₁

yes

characteristic

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

D₄×C₆₁

yes

—

119

C₂²×C₈

—

120

#117

—

121

#117

—

122

C₂×C₄⋊C₄

—

123

C₂×C₂²⋊C₄

C₂×C₄×C₆₁

yes

—

124

C₂×C₄²

—

125

#101

—

126

#95

—

127

C₂⁴

C₂²×C₄×C₆₁

yes

characteristic

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²⋊C₄×C₆₁

yes

—

140

C₂²×C₄

C₂×D₄×C₆₁

yes

—

141

C₂²×C₄

C₂×D₄×C₆₁

yes

—

142

C₂²×C₄

#41×C₆₁

yes

—

143

C₂²×C₄

#41×C₆₁

yes

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₄⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₂³

—

156

C₂³

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

157

C₂³

#117×C₆₁

yes

—

158

C₂³

C₄×D₄×C₆₁

yes

—

159

Q₈

—

160

Q₈

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#120×C₆₁

yes

—

167

C₂×C₄

#121×C₆₁

yes

—

168

C₂×C₄

#121×C₆₁

yes

—

169

C₂×C₄

#123×C₆₁

yes

—

170

C₂×C₄

#123×C₆₁

yes

—

171

C₂×C₄

#123×C₆₁

yes

—

172

C₂×C₄

#124×C₆₁

yes

—

173

C₂×C₄

#126×C₆₁

yes

—

174

C₈

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

C₆₁×#381

yes

characteristic

179

C₂²

C₆₁×#386

yes

—

180

C₂²

C₆₁×#388

yes

—

181

C₄

—

182

C₄

—

183

C₄

C₆₁×#394

yes

—

184

C₄

C₆₁×#397

yes

—

185

C₂

—

186

C₂

—

187

C₂

C₆₁×#1440

yes

characteristic

188

C₂

C₆₁×#1830

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₁	#381
commutator	C₂⁴	C₂²×C₄×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

#381

commutator

C₂⁴

C₂²×C₄×C₆₁

Frattini, cosocle

C₂³×C₄

C₂³×C₆₁

prime radical

C₂⁴×C₆₁

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

N	H ≅ G/N	type
80. #11168	C₆₁	direct (N × H)
106. C₆₁	#11168	direct (N × H)

H ≅ G/N

type

80. #11168

C₆₁

direct (N × H)

106. C₆₁

#11168

direct (N × H)

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

subgroup	index
C₂²×#101×C₆₁	2
C₆₁×#2310	2
C₂×C₆₁×#344	2
#11168	61

subgroup

index

C₂²×#101×C₆₁

C₆₁×#2310

C₂×C₆₁×#344

#11168

Subgroups — C₆₁×#11168 #29514601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#11168 #29514601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11168 #29514601