Group #20252020 — subgroups

Subgroups — #20252020

Order 6016.

484 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 197 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×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689179	C₂	1	4	yes	maximal, complemented
3	#15689175	C₂	1	4	yes	maximal, complemented
4	#15689147	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689086	C₂	1	2	yes	maximal
6	M₁₆×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₄₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×D₄₇	C₄	1	2	yes	—
9	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
10	C₂×#10625329	C₂²	1	2	yes	—
11	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₄²×D₄₇	C₄	1	2	yes	—
14	C₄²×D₄₇	C₄	1	2	yes	—
15	C₄²×D₄₇	C₂²	1	2	yes	—
16	C₄²×D₄₇	C₂²	1	2	yes	—
17	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	complemented
18	(C₂×D₄₇)⋊C₈	C₂²	1	8	yes	—
19	#13093934	C₂²	1	4	yes	complemented
20	C₈×Dic₄₇	C₂²	1	4	yes	complemented
21	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₄₇	C₂²	1	4	yes	complemented
24	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₄₇	—	2	2	no	—
30	C₂×C₄×D₄₇	—	2	2	no	—
31	C₂×C₄×D₄₇	—	2	8	no	—
32	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₄₇	C₂³	1	2	yes	—
38	M₁₆×C₄₇	—	2	4	no	—
39	C₂×C₈×C₄₇	C₂³	1	4	yes	—
40	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₄₇	C₂³	1	2	yes	—
43	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
44	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
45	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
46	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
48	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	#10625329	—	2	4	no	—
51	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
52	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×D₄₇	—	2	2	no	—
54	C₂²×D₄₇	—	2	4	no	—
55	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₄₇	#41	1	2	yes	—
61	C₂×C₄×C₄₇	#41	1	2	yes	—
62	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₄₇	—	2	2	no	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
66	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
67	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
68	C₄×D₄₇	—	2	4	no	—
69	C₄×D₄₇	—	2	4	no	—
70	C₄×D₄₇	—	2	4	no	—
71	C₄×D₄₇	—	2	2	no	—
72	C₄×D₄₇	—	2	2	no	—
73	C₄×D₄₇	—	2	4	no	—
74	C₄×D₄₇	M₁₆	1	8	yes	complemented
75	C₈×C₄₇	—	2	4	no	—
76	C₄₇⋊C₈	—	2	4	no	—
77	C₂²×C₄₇	—	2	2	no	—
78	C₂²×C₄₇	#117	1	2	yes	—
79	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×D₄₇	—	2	8	no	—
81	C₂×D₄₇	—	2	2	no	—
82	C₂×D₄₇	—	2	2	no	—
83	C₂×D₄₇	C₂×M₁₆	1	4	yes	—
84	C₄×C₄₇	—	2	2	no	—
85	C₄×C₄₇	#117	1	2	yes	—
86	C₄×C₄₇	C₂×M₁₆	1	4	yes	—
87	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
88	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
89	Dic₄₇	—	2	2	no	—
90	Dic₄₇	—	2	2	no	—
91	Dic₄₇	C₂×M₁₆	1	4	yes	—
92	#2920	—	47	1	no	maximal, Sylow
93	C₂×C₄₇	—	2	2	no	—
94	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
95	C₂×C₄₇	C₂²×M₁₆	1	2	yes	—
96	D₄₇	—	2	4	no	—
97	C₂²×C₄²	—	47	1	no	—
98	#432	—	47	4	no	—
99	#431	—	47	4	no	—
100	#423	—	47	1	no	—
101	#423	D₄₇	1	1	yes	characteristic, complemented
102	#407	—	47	2	no	—
103	C₄×M₁₆	—	47	2	no	—
104	C₄₇	#2920	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₄	—	47	2	no	—
106	C₂³×C₄	—	47	1	no	—
107	C₂×M₁₆	—	47	2	no	—
108	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
109	C₂×C₄²	—	47	2	no	—
110	C₂×C₄²	—	47	1	no	—
111	C₂×C₄²	—	47	2	no	—
112	C₂×C₄²	—	47	2	no	—
113	C₂×C₄²	—	47	2	no	—
114	C₂×C₄²	—	47	2	no	—
115	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
116	C₄⋊C₈	—	47	4	no	—
117	C₄⋊C₈	C₂×D₄₇	1	4	yes	complemented
118	C₂²⋊C₈	—	47	8	no	—
119	#97	—	47	4	no	—
120	C₄×C₈	—	47	4	no	—
121	C₂⁴	—	47	1	no	—
122	C₂²×C₄	—	47	2	no	—
123	C₂²×C₄	—	47	2	no	—
124	C₂²×C₄	—	47	1	no	—
125	C₂²×C₄	—	47	2	no	—
126	C₂²×C₄	—	47	2	no	—
127	C₂²×C₄	—	47	2	no	—
128	C₂²×C₄	—	47	2	no	—
129	C₂²×C₄	—	47	2	no	—
130	C₂²×C₄	—	94	8	no	—
131	C₂²×C₄	—	94	2	no	—
132	C₂²×C₄	—	94	2	no	—
133	C₂²×C₄	C₄×D₄₇	1	2	yes	—
134	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
135	M₁₆	—	2	4	no	—
136	M₁₆	—	94	4	no	—
137	C₂×C₈	—	47	4	no	—
138	C₂×C₈	C₂²×D₄₇	1	4	yes	—
139	C₄²	—	47	2	no	—
140	C₄²	—	47	2	no	—
141	C₄²	—	47	2	no	—
142	C₄²	—	47	2	no	—
143	C₄²	—	47	2	no	—
144	C₄²	—	47	1	no	—
145	C₄²	—	47	1	no	—
146	C₄²	C₄×D₄₇	1	1	yes	characteristic
147	C₄²	C₄×D₄₇	1	1	yes	characteristic
148	C₄²	C₂²×D₄₇	1	2	yes	—
149	C₂³	—	47	2	no	—
150	C₂³	—	94	4	no	—
151	C₂³	—	94	2	no	—
152	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	47	2	no	—
155	C₂×C₄	—	47	8	no	—
156	C₂×C₄	—	47	2	no	—
157	C₂×C₄	—	47	2	no	—
158	C₂×C₄	—	94	4	no	—
159	C₂×C₄	—	94	4	no	—
160	C₂×C₄	—	94	4	no	—
161	C₂×C₄	—	94	2	no	—
162	C₂×C₄	—	94	4	no	—
163	C₂×C₄	—	94	4	no	—
164	C₂×C₄	—	94	2	no	—
165	C₂×C₄	—	94	2	no	—
166	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
167	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
169	C₂×C₄	#10625352	1	2	yes	—
170	C₂×C₄	#10625354	1	2	yes	—
171	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
172	C₈	—	2	4	no	—
173	C₈	—	94	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	47	4	no	—
176	C₂²	—	94	8	no	—
177	C₂²	—	94	2	no	—
178	C₂²	—	94	2	no	—
179	C₂²	#13093982	1	2	yes	—
180	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	47	4	no	—
183	C₄	—	94	2	no	—
184	C₄	—	94	2	no	—
185	C₄	#13093982	1	2	yes	—
186	C₄	M₁₆×D₄₇	1	4	yes	—
187	C₄	C₂×#10625352	1	1	yes	characteristic
188	C₄	C₂×#10625354	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	94	4	no	—
191	C₂	C₂×#13093982	1	1	yes	characteristic
192	C₂	C₂×M₁₆×D₄₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic

#15689179

C₂

yes

maximal, complemented

#15689175

C₂

yes

maximal, complemented

#15689147

C₂

yes

maximal, characteristic, complemented

#15689086

C₂

yes

maximal

M₁₆×Dic₄₇

C₂

yes

maximal, complemented

C₄₇×#423

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₄

yes

—

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

C₂×C₄×Dic₄₇

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄²×D₄₇

C₄

yes

—

C₄²×D₄₇

C₄

yes

—

C₄²×D₄₇

C₂²

yes

—

C₄²×D₄₇

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

(C₂×D₄₇)⋊C₈

C₂²

yes

—

#13093934

C₂²

yes

complemented

C₈×Dic₄₇

C₂²

yes

complemented

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

complemented

C₂³×D₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂³

yes

characteristic

C₄×Dic₄₇

C₂³

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

M₁₆

yes

complemented

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

#117

yes

—

C₄×C₄₇

C₂×M₁₆

yes

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×M₁₆

yes

—

#2920

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#117

yes

characteristic

C₂×C₄₇

C₂²×M₁₆

yes

—

D₄₇

—

C₂²×C₄²

—

#432

—

#431

—

100

#423

—

101

#423

D₄₇

yes

characteristic, complemented

102

#407

—

103

C₄×M₁₆

—

104

C₄₇

#2920

yes

characteristic, complemented, Sylow

105

C₂³×C₄

—

106

C₂³×C₄

—

107

C₂×M₁₆

—

108

C₂×M₁₆

C₂×D₄₇

yes

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×D₄₇

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

C₂×D₄₇

yes

complemented

118

C₂²⋊C₈

—

119

#97

—

120

C₄×C₈

—

121

C₂⁴

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₄×D₄₇

yes

—

134

C₂²×C₄

C₂²×D₄₇

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

C₂²×D₄₇

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₄×D₄₇

yes

characteristic

147

C₄²

C₄×D₄₇

yes

characteristic

148

C₄²

C₂²×D₄₇

yes

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

C₂×C₄×D₄₇

yes

characteristic

153

C₂×C₄

—

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

C₂×C₄×D₄₇

yes

—

167

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

168

C₂×C₄

C₂×C₄×D₄₇

yes

—

169

C₂×C₄

#10625352

yes

—

170

C₂×C₄

#10625354

yes

—

171

C₂×C₄

C₂³×D₄₇

yes

characteristic

172

C₈

—

173

C₈

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

#13093982

yes

—

180

C₂²

C₂²×C₄×D₄₇

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#13093982

yes

—

186

C₄

M₁₆×D₄₇

yes

—

187

C₄

C₂×#10625352

yes

characteristic

188

C₄

C₂×#10625354

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

C₂×#13093982

yes

characteristic

192

C₂

C₂×M₁₆×D₄₇

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#423	C₂
hypercenter	#423	D₄₇
hypocenter	C₄₇	#2920
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#423

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #15689179	C₂	semidirect
3. #15689175	C₂	semidirect
4. #15689147	C₂	semidirect
6. M₁₆×Dic₄₇	C₂	semidirect
7. C₄₇×#423	C₂	semidirect
17. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
19. #13093934	C₂²	semidirect
20. C₈×Dic₄₇	C₂²	semidirect
23. C₄⋊C₈×C₄₇	C₂²	semidirect
74. C₄×D₄₇	M₁₆	semidirect
101. #423	D₄₇	semidirect
104. C₄₇	#2920	semidirect
117. C₄⋊C₈	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689179

C₂

semidirect

3. #15689175

C₂

semidirect

4. #15689147

C₂

semidirect

6. M₁₆×Dic₄₇

C₂

semidirect

7. C₄₇×#423

C₂

semidirect

17. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

19. #13093934

C₂²

semidirect

20. C₈×Dic₄₇

C₂²

semidirect

23. C₄⋊C₈×C₄₇

C₂²

semidirect

74. C₄×D₄₇

M₁₆

semidirect

101. #423

D₄₇

semidirect

104. C₄₇

#2920

semidirect

117. C₄⋊C₈

C₂×D₄₇

semidirect

Subgroups — #20252020

Conjugacy classes of subgroups

Complements