Group #20245983 — subgroups

Subgroups — #20245983

Order 6016.

672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 189 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×(C₂×D₄₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#15689014	C₂	1	4	yes	maximal, complemented
3	#15689012	C₂	1	4	yes	maximal, complemented
4	#15688981	C₂	1	2	yes	maximal, complemented
5	#15688978	C₂	1	2	yes	maximal
6	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×D₄₇	C₂²	1	1	yes	characteristic
9	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	complemented
11	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×(C₂×D₄₇)⋊C₄	C₄	1	8	yes	complemented
13	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
14	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
15	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
16	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄×Dic₄₇	C₂²	1	2	yes	complemented
18	C₄₇⋊#95	C₂²	1	2	yes	complemented
19	C₄₇⋊#95	C₂²	1	4	yes	complemented
20	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	2	yes	complemented
23	#95×C₄₇	C₂²	1	2	yes	complemented
24	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
25	C₂³×D₄₇	—	2	4	no	—
26	C₂³×D₄₇	—	4	2	no	—
27	C₂³×D₄₇	—	4	2	no	—
28	C₂³×D₄₇	C₂×C₄	1	4	yes	—
29	C₂³×D₄₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₇	—	4	2	no	—
31	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₄₇	—	4	2	no	—
36	C₂²×Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂²×Dic₄₇	C₂³	1	4	yes	—
38	C₂²×Dic₄₇	C₂³	1	2	yes	—
39	C₂²⋊C₄×C₄₇	—	2	4	no	—
40	C₄²×C₄₇	—	4	2	no	—
41	C₂²⋊Dic₄₇	—	2	4	no	—
42	(C₂×D₄₇)⋊C₄	—	4	4	no	—
43	(C₂×D₄₇)⋊C₄	—	4	4	no	—
44	(C₂×D₄₇)⋊C₄	C₂×C₄	1	8	yes	—
45	(C₂×D₄₇)⋊C₄	C₂×C₄	1	8	yes	—
46	Dic₄₇⋊C₄	—	4	4	no	—
47	C₄×Dic₄₇	—	4	2	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂³×C₄₇	#41	1	4	yes	—
52	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂²×D₄₇	—	2	8	no	—
54	C₂²×D₄₇	—	2	4	no	—
55	C₂²×D₄₇	—	2	4	no	—
56	C₂²×D₄₇	—	2	4	no	—
57	C₂²×D₄₇	—	4	4	no	—
58	C₂²×D₄₇	—	4	4	no	—
59	C₂²×D₄₇	—	4	16	no	—
60	C₂²×D₄₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	4	no	—
64	C₂×C₄×C₄₇	—	2	4	no	—
65	C₂×C₄×C₄₇	—	2	4	no	—
66	C₂×C₄×C₄₇	—	4	4	no	—
67	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂×Dic₄₇	—	2	4	no	—
70	C₂×Dic₄₇	—	2	4	no	—
71	C₂×Dic₄₇	—	2	4	no	—
72	C₂×Dic₄₇	—	2	4	no	—
73	C₂×Dic₄₇	—	4	4	no	—
74	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	—	2	8	no	—
78	C₂²×C₄₇	#117	1	4	yes	—
79	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
80	C₂²×C₄₇	C₂×#41	1	2	yes	—
81	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
82	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
83	C₂×D₄₇	—	2	4	no	—
84	C₂×D₄₇	—	2	4	no	—
85	C₂×D₄₇	—	2	4	no	—
86	C₂×D₄₇	—	4	16	no	—
87	C₂×D₄₇	—	4	8	no	—
88	C₄×C₄₇	—	2	2	no	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	4	4	no	—
91	Dic₄₇	—	2	4	no	—
92	Dic₄₇	—	4	4	no	—
93	#2312	—	47	1	no	maximal, Sylow
94	C₂×C₄₇	—	2	4	no	—
95	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
96	C₂×C₄₇	#518	1	1	yes	characteristic
97	C₂×C₄₇	#518	1	1	yes	characteristic
98	C₂×C₄₇	#538	1	4	yes	—
99	D₄₇	—	4	4	no	—
100	C₂²×C₂²⋊C₄	—	47	1	no	—
101	#388	—	47	4	no	—
102	#388	—	47	4	no	—
103	#380	—	47	2	no	—
104	#379	—	47	2	no	—
105	C₄×C₂²⋊C₄	—	47	1	no	—
106	C₄×C₂²⋊C₄	D₄₇	1	1	yes	characteristic, complemented
107	C₄₇	#2312	1	1	yes	characteristic, complemented, Sylow
108	C₂⁵	—	47	1	no	—
109	C₂³×C₄	—	47	1	no	—
110	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	47	4	no	—
112	C₂×C₂²⋊C₄	—	47	8	no	—
113	C₂×C₂²⋊C₄	—	47	2	no	—
114	C₂×C₂²⋊C₄	—	47	2	no	—
115	C₂×C₂²⋊C₄	—	47	2	no	—
116	C₂×C₂²⋊C₄	—	47	4	no	—
117	C₂×C₂²⋊C₄	—	47	4	no	—
118	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
119	C₂×C₄²	—	47	2	no	—
120	C₂×C₄²	C₂×D₄₇	1	2	yes	complemented
121	#95	—	47	2	no	—
122	#95	—	47	4	no	—
123	#95	C₂×D₄₇	1	2	yes	complemented
124	C₂⁴	—	47	2	no	—
125	C₂⁴	—	47	4	no	—
126	C₂⁴	—	94	4	no	—
127	C₂⁴	—	188	2	no	—
128	C₂⁴	—	188	2	no	—
129	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	47	2	no	—
132	C₂²×C₄	—	47	4	no	—
133	C₂²×C₄	—	47	4	no	—
134	C₂²×C₄	—	188	2	no	—
135	C₂²×C₄	C₄×D₄₇	1	4	yes	—
136	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
137	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	C₄⋊C₄	—	188	4	no	—
140	C₂²⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	47	8	no	—
142	C₂²⋊C₄	—	47	8	no	—
143	C₂²⋊C₄	—	94	4	no	—
144	C₂²⋊C₄	—	188	4	no	—
145	C₂²⋊C₄	—	188	4	no	—
146	C₄²	—	4	2	no	—
147	C₄²	—	188	2	no	—
148	C₂³	—	2	2	no	—
149	C₂³	—	2	2	no	—
150	C₂³	—	47	4	no	—
151	C₂³	—	94	8	no	—
152	C₂³	—	94	4	no	—
153	C₂³	—	94	4	no	—
154	C₂³	—	94	4	no	—
155	C₂³	—	188	4	no	—
156	C₂³	—	188	4	no	—
157	C₂³	—	188	16	no	—
158	C₂³	C₂×C₄×D₄₇	1	2	yes	—
159	C₂³	#10625350	1	4	yes	—
160	C₂³	C₂³×D₄₇	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	47	4	no	—
168	C₂×C₄	—	94	4	no	—
169	C₂×C₄	—	94	4	no	—
170	C₂×C₄	—	94	4	no	—
171	C₂×C₄	—	94	4	no	—
172	C₂×C₄	—	188	4	no	—
173	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
174	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	94	4	no	—
179	C₂²	—	94	4	no	—
180	C₂²	—	94	4	no	—
181	C₂²	—	188	16	no	—
182	C₂²	—	188	8	no	—
183	C₂²	#13093981	1	4	yes	—
184	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
185	C₂²	C₂×#10625350	1	2	yes	—
186	C₂²	#13094069	1	2	yes	—
187	C₂²	#13094070	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	4	no	—
191	C₄	—	94	4	no	—
192	C₄	—	188	4	no	—
193	C₂	—	2	4	no	—
194	C₂	—	188	4	no	—
195	C₂	C₂×#13093981	1	1	yes	characteristic
196	C₂	#15689596	1	1	yes	characteristic
197	C₂	#15689597	1	1	yes	characteristic
198	C₂	#15689706	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×(C₂×D₄₇)⋊C₄

C₂

yes

maximal, characteristic

#15689014

C₂

yes

maximal, complemented

#15689012

C₂

yes

maximal, complemented

#15688981

C₂

yes

maximal, complemented

#15688978

C₂

yes

maximal

C₄×C₂²⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊C₄×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₄₇

C₂²

yes

characteristic

C₂³×Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

complemented

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₄

yes

complemented

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

complemented

C₄₇⋊#95

C₂²

yes

complemented

C₄₇⋊#95

C₂²

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

complemented

#95×C₄₇

C₂²

yes

complemented

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂³×D₄₇

—

C₂³×D₄₇

—

C₂³×D₄₇

—

C₂³×D₄₇

C₂×C₄

yes

—

C₂³×D₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#2312

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

100

C₂²×C₂²⋊C₄

—

101

#388

—

102

#388

—

103

#380

—

104

#379

—

105

C₄×C₂²⋊C₄

—

106

C₄×C₂²⋊C₄

D₄₇

yes

characteristic, complemented

107

C₄₇

#2312

yes

characteristic, complemented, Sylow

108

C₂⁵

—

109

C₂³×C₄

—

110

C₂³×C₄

C₂×D₄₇

yes

characteristic

111

C₂×C₄⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₂²⋊C₄

—

115

C₂×C₂²⋊C₄

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₂²⋊C₄

—

118

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂×D₄₇

yes

complemented

121

#95

—

122

#95

—

123

#95

C₂×D₄₇

yes

complemented

124

C₂⁴

—

125

C₂⁴

—

126

C₂⁴

—

127

C₂⁴

—

188

—

128

C₂⁴

—

188

—

129

C₂⁴

C₂²×D₄₇

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

188

—

135

C₂²×C₄

C₄×D₄₇

yes

—

136

C₂²×C₄

C₂²×D₄₇

yes

—

137

C₂²×C₄

C₂²×D₄₇

yes

characteristic

138

C₂²×C₄

C₂²×D₄₇

yes

characteristic

139

C₄⋊C₄

—

188

—

140

C₂²⋊C₄

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

—

143

C₂²⋊C₄

—

144

C₂²⋊C₄

—

188

—

145

C₂²⋊C₄

—

188

—

146

C₄²

—

147

C₄²

—

188

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

188

—

156

C₂³

—

188

—

157

C₂³

—

188

—

158

C₂³

C₂×C₄×D₄₇

yes

—

159

C₂³

#10625350

yes

—

160

C₂³

C₂³×D₄₇

yes

characteristic

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

—

171

C₂×C₄

—

172

C₂×C₄

—

188

—

173

C₂×C₄

C₂×C₄×D₄₇

yes

—

174

C₂×C₄

C₂×C₄×D₄₇

yes

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

188

—

182

C₂²

—

188

—

183

C₂²

#13093981

yes

—

184

C₂²

C₂²×C₄×D₄₇

yes

characteristic

185

C₂²

C₂×#10625350

yes

—

186

C₂²

#13094069

yes

—

187

C₂²

#13094070

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

188

—

193

C₂

—

194

C₂

—

188

—

195

C₂

C₂×#13093981

yes

characteristic

196

C₂

#15689596

yes

characteristic

197

C₂

#15689597

yes

characteristic

198

C₂

#15689706

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₄×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#2312
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₄×C₂²⋊C₄×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₄×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. #15689014	C₂	semidirect
3. #15689012	C₂	semidirect
4. #15688981	C₂	semidirect
6. C₄×C₂²⋊Dic₄₇	C₂	semidirect
7. C₄×C₂²⋊C₄×C₄₇	C₂	semidirect
10. C₂×Dic₄₇⋊C₄	C₂²	semidirect
12. C₂×(C₂×D₄₇)⋊C₄	C₄	semidirect
17. C₂×C₄×Dic₄₇	C₂²	semidirect
18. C₄₇⋊#95	C₂²	semidirect
19. C₄₇⋊#95	C₂²	semidirect
22. C₂×C₄²×C₄₇	C₂²	semidirect
23. #95×C₄₇	C₂²	semidirect
106. C₄×C₂²⋊C₄	D₄₇	semidirect
107. C₄₇	#2312	semidirect
120. C₂×C₄²	C₂×D₄₇	semidirect
123. #95	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689014

C₂

semidirect

3. #15689012

C₂

semidirect

4. #15688981

C₂

semidirect

6. C₄×C₂²⋊Dic₄₇

C₂

semidirect

7. C₄×C₂²⋊C₄×C₄₇

C₂

semidirect

10. C₂×Dic₄₇⋊C₄

C₂²

semidirect

12. C₂×(C₂×D₄₇)⋊C₄

C₄

semidirect

17. C₂×C₄×Dic₄₇

C₂²

semidirect

18. C₄₇⋊#95

C₂²

semidirect

19. C₄₇⋊#95

C₂²

semidirect

22. C₂×C₄²×C₄₇

C₂²

semidirect

23. #95×C₄₇

C₂²

semidirect

106. C₄×C₂²⋊C₄

D₄₇

semidirect

107. C₄₇

#2312

semidirect

120. C₂×C₄²

C₂×D₄₇

semidirect

123. #95

C₂×D₄₇

semidirect

Subgroups — #20245983

Conjugacy classes of subgroups

Complements