Group #29516280 — subgroups

Subgroups — C₆₁×#12847 #29516280

Order 15616.

434 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 166 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×C₆₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2112	C₂	1	2	yes	maximal
3	C₆₁×#1377	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₆₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
6	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₆₁	C₂²	1	4	yes	—
8	C₆₁×#405	—	2	4	no	—
9	C₂×#97×C₆₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
11	C₆₁×C₂²⋊C₁₆	—	2	4	no	—
12	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₆₁	—	2	2	no	—
14	C₂×M₁₆×C₆₁	—	2	4	no	—
15	C₂²×C₈×C₆₁	—	2	4	no	—
16	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₆₁	D₄	1	4	yes	—
19	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
20	#117×C₆₁	—	2	2	no	—
21	#117×C₆₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
23	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
24	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
25	M₃₂×C₆₁	—	2	8	no	—
26	C₂×C₁₆×C₆₁	—	2	4	no	—
27	#97×C₆₁	—	2	4	no	—
28	C₄×C₈×C₆₁	—	2	4	no	—
29	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	—	2	2	no	—
32	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₆₁	—	2	8	no	—
38	C₂×C₈×C₆₁	—	2	2	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	2	8	no	—
42	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
44	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₆₁	#41	1	4	yes	—
46	C₄⋊C₄×C₆₁	—	2	4	no	—
47	C₂²⋊C₄×C₆₁	—	2	4	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₁₆×C₆₁	—	4	4	no	—
51	C₂³×C₆₁	—	2	2	no	—
52	C₂³×C₆₁	—	4	2	no	—
53	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	2	2	no	—
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	2	2	no	—
58	C₂×C₄×C₆₁	—	2	4	no	—
59	C₂×C₄×C₆₁	—	2	4	no	—
60	C₂×C₄×C₆₁	—	4	4	no	—
61	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#117	1	2	yes	—
64	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₆₁	—	2	2	no	—
67	C₈×C₆₁	—	2	4	no	—
68	C₈×C₆₁	#123	1	4	yes	—
69	#12847	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	—	2	2	no	—
72	C₂²×C₆₁	—	2	4	no	—
73	C₂²×C₆₁	—	4	4	no	—
74	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₆₁	#431	1	2	yes	—
76	C₄×C₆₁	—	2	2	no	—
77	C₄×C₆₁	—	4	4	no	—
78	C₄×C₆₁	#381	1	1	yes	characteristic
79	C₄×C₆₁	#407	1	1	yes	characteristic
80	C₄×C₆₁	#432	1	2	yes	—
81	C₂×#405	C₂×C₆₁	1	1	yes	characteristic
82	#2112	C₂×C₆₁	1	2	yes	—
83	#1377	C₂×C₆₁	1	4	yes	complemented
84	C₂×C₆₁	—	2	2	no	—
85	C₂×C₆₁	—	4	2	no	—
86	C₂×C₆₁	#1817	1	1	yes	characteristic
87	C₂×C₆₁	#2115	1	2	yes	—
88	C₂²×M₁₆	C₄×C₆₁	1	1	yes	characteristic
89	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
90	C₂×#117	C₄×C₆₁	1	1	yes	characteristic
91	C₂×M₃₂	C₂²×C₆₁	1	4	yes	—
92	#405	—	2	4	no	—
93	C₂×#97	C₂²×C₆₁	1	2	yes	—
94	C₂×C₄×C₈	C₄×C₆₁	1	2	yes	—
95	C₂²⋊C₁₆	—	2	4	no	—
96	C₆₁	#12847	1	1	yes	characteristic, direct product factor, Sylow
97	C₂³×C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
103	C₂²×C₈	D₄×C₆₁	1	4	yes	—
104	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	C₈×C₆₁	1	2	yes	—
108	C₂×C₂²⋊C₄	C₈×C₆₁	1	2	yes	—
109	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
110	M₃₂	—	2	8	no	—
111	C₂×C₁₆	—	2	4	no	—
112	#97	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₂⁴	C₂×C₈×C₆₁	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
118	C₂²×C₄	C₂×C₈×C₆₁	1	1	yes	characteristic
119	C₂²×C₄	C₂×C₈×C₆₁	1	4	yes	—
120	C₂²×C₄	M₁₆×C₆₁	1	2	yes	—
121	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
128	C₂×C₈	M₁₆×C₆₁	1	4	yes	—
129	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
130	C₂×C₈	#41×C₆₁	1	4	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	4	4	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂²⋊C₈×C₆₁	1	2	yes	—
139	C₂³	C₂²×C₈×C₆₁	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
148	C₂×C₄	#117×C₆₁	1	2	yes	—
149	C₂×C₄	C₂×M₁₆×C₆₁	1	2	yes	—
150	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	#123×C₆₁	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
159	C₂²	C₆₁×#431	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	C₆₁×#381	1	1	yes	characteristic
163	C₄	C₆₁×#407	1	1	yes	characteristic
164	C₄	C₆₁×#432	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	4	2	no	—
167	C₂	C₆₁×#1817	1	1	yes	characteristic
168	C₂	C₆₁×#2115	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₆₁×#405

C₂

yes

maximal, characteristic

C₆₁×#2112

C₂

yes

maximal

C₆₁×#1377

C₂

yes

maximal, complemented

C₂²×M₁₆×C₆₁

C₄

yes

characteristic

C₂³×C₈×C₆₁

C₂²

yes

characteristic

C₂×#117×C₆₁

C₄

yes

characteristic

C₂×M₃₂×C₆₁

C₂²

yes

—

C₆₁×#405

—

C₂×#97×C₆₁

C₂²

yes

—

C₂×C₄×C₈×C₆₁

C₄

yes

—

C₆₁×C₂²⋊C₁₆

—

C₂³×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₆₁

—

C₂×M₁₆×C₆₁

—

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂²×C₈×C₆₁

C₂³

yes

characteristic

#117×C₆₁

—

#117×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₈

yes

—

C₂×C₂²⋊C₄×C₆₁

C₈

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

—

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

#97×C₆₁

—

C₄×C₈×C₆₁

—

C₂⁴×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₂×C₈×C₆₁

#41

yes

—

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₁₆×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²⋊C₈

yes

—

C₂³×C₆₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#123

yes

—

#12847

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₆₁

#431

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×#405

C₂×C₆₁

yes

characteristic

#2112

C₂×C₆₁

yes

—

#1377

C₂×C₆₁

yes

complemented

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1817

yes

characteristic

C₂×C₆₁

#2115

yes

—

C₂²×M₁₆

C₄×C₆₁

yes

characteristic

C₂³×C₈

C₂²×C₆₁

yes

characteristic

C₂×#117

C₄×C₆₁

yes

characteristic

C₂×M₃₂

C₂²×C₆₁

yes

—

#405

—

C₂×#97

C₂²×C₆₁

yes

—

C₂×C₄×C₈

C₄×C₆₁

yes

—

C₂²⋊C₁₆

—

C₆₁

#12847

yes

characteristic, direct product factor, Sylow

C₂³×C₄

C₂×C₄×C₆₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×C₄×C₆₁

yes

characteristic

102

C₂²×C₈

C₂×C₄×C₆₁

yes

—

103

C₂²×C₈

D₄×C₆₁

yes

—

104

C₂²×C₈

C₂³×C₆₁

yes

characteristic

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

C₈×C₆₁

yes

—

108

C₂×C₂²⋊C₄

C₈×C₆₁

yes

—

109

C₂×C₄²

C₂×C₄×C₆₁

yes

—

110

M₃₂

—

111

C₂×C₁₆

—

112

#97

—

113

C₄×C₈

—

114

C₂⁴

C₂×C₈×C₆₁

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²⋊C₄×C₆₁

yes

—

118

C₂²×C₄

C₂×C₈×C₆₁

yes

characteristic

119

C₂²×C₄

C₂×C₈×C₆₁

yes

—

120

C₂²×C₄

M₁₆×C₆₁

yes

—

121

C₂²×C₄

C₂²×C₄×C₆₁

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊C₄×C₆₁

yes

—

128

C₂×C₈

M₁₆×C₆₁

yes

—

129

C₂×C₈

C₂×D₄×C₆₁

yes

—

130

C₂×C₈

#41×C₆₁

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊C₈×C₆₁

yes

—

139

C₂³

C₂²×C₈×C₆₁

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊C₈×C₆₁

yes

—

147

C₂×C₄

C₂×C₂²⋊C₄×C₆₁

yes

characteristic

148

C₂×C₄

#117×C₆₁

yes

—

149

C₂×C₄

C₂×M₁₆×C₆₁

yes

—

150

C₂×C₄

C₂×M₁₆×C₆₁

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

#123×C₆₁

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₂²⋊C₈×C₆₁

yes

characteristic

159

C₂²

C₆₁×#431

yes

—

160

C₄

—

161

C₄

—

162

C₄

C₆₁×#381

yes

characteristic

163

C₄

C₆₁×#407

yes

characteristic

164

C₄

C₆₁×#432

yes

—

165

C₂

—

166

C₂

—

167

C₂

C₆₁×#1817

yes

characteristic

168

C₂

C₆₁×#2115

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₈×C₆₁

Frattini, cosocle

C₂²×C₈

C₂³×C₆₁

socle

C₂²×C₆₁

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₆₁

C₂×C₈

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

N	H ≅ G/N	type
3. C₆₁×#1377	C₂	semidirect
69. #12847	C₆₁	direct (N × H)
83. #1377	C₂×C₆₁	semidirect
96. C₆₁	#12847	direct (N × H)

H ≅ G/N

type

3. C₆₁×#1377

C₂

semidirect

69. #12847

C₆₁

direct (N × H)

83. #1377

C₂×C₆₁

semidirect

96. C₆₁

#12847

direct (N × H)

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

subgroup	index
C₂×C₆₁×#405	2
C₆₁×#2112	2
C₆₁×#1377	2
#12847	61

subgroup

index

C₂×C₆₁×#405

C₆₁×#2112

C₆₁×#1377

#12847

Subgroups — C₆₁×#12847 #29516280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#12847 #29516280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12847 #29516280