Group #20251416 — subgroups

Subgroups — #20251416

Order 6016.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13094003	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689059	C₂	1	2	yes	maximal, complemented
3	#15689054	C₂	1	2	yes	maximal, complemented
4	#15689037	C₂	1	2	yes	maximal, complemented
5	#15689037	C₂	1	4	yes	maximal, complemented
6	#15689035	C₂	1	2	yes	maximal, complemented
7	#15688986	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#121×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
10	#13094003	—	2	4	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
17	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
18	C₄₇⋊#95	C₂²	1	2	yes	—
19	C₄₇⋊#95	C₂²	1	2	yes	—
20	C₄₇⋊#95	C₂²	1	4	yes	—
21	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
22	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
24	#121×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
27	C₂⁴×C₄₇	C₂³	1	2	yes	—
28	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₄₇	—	2	4	no	—
30	C₂×D₄×C₄₇	—	2	2	no	—
31	C₂×D₄×C₄₇	—	4	2	no	—
32	C₂×D₄×C₄₇	—	4	4	no	—
33	C₂²×C₄×C₄₇	—	4	2	no	—
34	C₂²×C₄×C₄₇	D₄	1	4	yes	—
35	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₄₇	C₂³	1	4	yes	—
39	C₂²×Dic₄₇	C₂³	1	2	yes	—
40	C₂²×Dic₄₇	C₂³	1	2	yes	—
41	C₂×Dic₉₄	—	2	2	no	—
42	C₂×Dic₉₄	—	4	2	no	—
43	C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂²⋊C₄×C₄₇	—	2	4	no	—
45	C₂²⋊Dic₄₇	—	2	4	no	—
46	C₂²⋊Dic₄₇	—	4	4	no	—
47	C₂²⋊Dic₄₇	—	4	2	no	—
48	C₂²⋊Dic₄₇	—	4	4	no	—
49	C₄⋊Dic₄₇	—	2	2	no	—
50	Dic₄₇⋊C₄	—	2	4	no	—
51	Dic₄₇⋊C₄	—	4	4	no	—
52	C₂³×C₄₇	—	2	2	no	—
53	C₂³×C₄₇	—	2	2	no	—
54	C₂³×C₄₇	—	2	2	no	—
55	C₂³×C₄₇	—	2	2	no	—
56	C₂³×C₄₇	—	2	4	no	—
57	C₂³×C₄₇	—	2	4	no	—
58	C₂³×C₄₇	—	2	4	no	—
59	C₂³×C₄₇	C₂×D₄	1	2	yes	—
60	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
61	D₄×C₄₇	—	4	4	no	—
62	D₄×C₄₇	—	4	8	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	4	no	—
67	C₂×C₄×C₄₇	—	4	4	no	—
68	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×Dic₄₇	—	2	2	no	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	4	no	—
74	C₂×Dic₄₇	—	2	2	no	—
75	C₂×Dic₄₇	—	2	4	no	—
76	C₂×Dic₄₇	—	2	4	no	—
77	C₂×Dic₄₇	—	2	4	no	—
78	Dic₉₄	—	4	4	no	—
79	C₂²×C₄₇	—	2	2	no	—
80	C₂²×C₄₇	—	2	2	no	—
81	C₂²×C₄₇	—	2	2	no	—
82	C₂²×C₄₇	—	4	4	no	—
83	C₂²×C₄₇	—	4	2	no	—
84	C₂²×C₄₇	—	4	2	no	—
85	C₂²×C₄₇	—	4	2	no	—
86	C₂²×C₄₇	—	4	8	no	—
87	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
89	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
90	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	4	2	no	—
94	Dic₄₇	—	4	2	no	—
95	Dic₄₇	—	4	2	no	—
96	Dic₄₇	—	4	4	no	—
97	#2816	—	47	1	no	maximal, Sylow
98	C₂×C₄₇	—	4	2	no	—
99	C₂×C₄₇	—	8	2	no	—
100	C₂×C₄₇	#534	1	1	yes	characteristic
101	C₂×C₄₇	#535	1	1	yes	characteristic
102	C₂×C₄₇	#536	1	1	yes	characteristic
103	C₂×C₄₇	#562	1	4	yes	—
104	C₂×C₂²⋊Q₈	—	47	1	no	—
105	C₂×#121	D₄₇	1	1	yes	characteristic
106	#399	—	47	2	no	—
107	#398	—	47	2	no	—
108	#394	—	47	2	no	—
109	#394	—	47	4	no	—
110	#393	—	47	2	no	—
111	#381	—	47	1	no	—
112	C₄₇	#2816	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	47	1	no	—
114	C₂²×D₄	C₂×D₄₇	1	2	yes	—
115	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
116	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
117	C₂²⋊Q₈	—	94	4	no	—
118	#121	—	2	4	no	—
119	C₂×C₄⋊C₄	—	47	2	no	—
120	C₂×C₄⋊C₄	—	47	1	no	—
121	C₂×C₄⋊C₄	—	47	4	no	—
122	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
123	C₂×C₂²⋊C₄	—	47	2	no	—
124	C₂×C₂²⋊C₄	—	47	2	no	—
125	C₂×C₂²⋊C₄	—	47	4	no	—
126	C₂×C₂²⋊C₄	—	47	4	no	—
127	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
128	#95	—	47	2	no	—
129	#95	—	47	2	no	—
130	#95	—	47	4	no	—
131	C₂⁴	C₂²×D₄₇	1	2	yes	—
132	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
133	C₂×Q₈	—	94	2	no	—
134	C₂×Q₈	—	188	2	no	—
135	C₂×D₄	—	2	4	no	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	4	2	no	—
138	C₂×D₄	—	4	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	47	4	no	—
141	C₂²×C₄	—	47	2	no	—
142	C₂²×C₄	—	47	2	no	—
143	C₂²×C₄	C₂²⋊D₄₇	1	4	yes	—
144	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
145	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	94	4	no	—
149	C₄⋊C₄	—	94	2	no	—
150	C₄⋊C₄	—	188	4	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	94	4	no	—
153	C₂²⋊C₄	—	188	2	no	—
154	C₂²⋊C₄	—	188	4	no	—
155	C₂²⋊C₄	—	188	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	4	no	—
161	C₂³	—	2	4	no	—
162	C₂³	—	2	4	no	—
163	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
164	C₂³	C₂³×D₄₇	1	1	yes	characteristic
165	Q₈	—	188	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	4	8	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	94	2	no	—
174	C₂×C₄	—	94	2	no	—
175	C₂×C₄	—	94	4	no	—
176	C₂×C₄	—	94	2	no	—
177	C₂×C₄	—	94	4	no	—
178	C₂×C₄	—	94	4	no	—
179	C₂×C₄	—	94	2	no	—
180	C₂×C₄	—	94	4	no	—
181	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	4	no	—
187	C₂²	—	4	8	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂²	#13094069	1	2	yes	—
193	C₂²	#13094069	1	2	yes	—
194	C₂²	#13094071	1	2	yes	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	188	4	no	—
199	C₄	—	188	2	no	—
200	C₄	—	188	2	no	—
201	C₂	—	4	2	no	—
202	C₂	—	8	2	no	—
203	C₂	#15689682	1	1	yes	characteristic
204	C₂	#15689690	1	1	yes	characteristic
205	C₂	#15689694	1	1	yes	characteristic
206	C₂	#15689868	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094003

C₂

yes

maximal, characteristic, complemented

#15689059

C₂

yes

maximal, complemented

#15689054

C₂

yes

maximal, complemented

#15689037

C₂

yes

maximal, complemented

#15689037

C₂

yes

maximal, complemented

#15689035

C₂

yes

maximal, complemented

#15688986

C₂

yes

maximal, characteristic, complemented

C₂×#121×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic

#13094003

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×D₄×C₄₇

C₂²

yes

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#121×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2816

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

100

C₂×C₄₇

#534

yes

characteristic

101

C₂×C₄₇

#535

yes

characteristic

102

C₂×C₄₇

#536

yes

characteristic

103

C₂×C₄₇

#562

yes

—

104

C₂×C₂²⋊Q₈

—

105

C₂×#121

D₄₇

yes

characteristic

106

#399

—

107

#398

—

108

#394

—

109

#394

—

110

#393

—

111

#381

—

112

C₄₇

#2816

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂²×D₄

C₂×D₄₇

yes

—

115

C₂²×D₄

C₂×D₄₇

yes

characteristic

116

C₂³×C₄

C₂×D₄₇

yes

characteristic

117

C₂²⋊Q₈

—

118

#121

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

123

C₂×C₂²⋊C₄

—

124

C₂×C₂²⋊C₄

—

125

C₂×C₂²⋊C₄

—

126

C₂×C₂²⋊C₄

—

127

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

128

#95

—

129

#95

—

130

#95

—

131

C₂⁴

C₂²×D₄₇

yes

—

132

C₂⁴

C₂²×D₄₇

yes

characteristic

133

C₂×Q₈

—

134

C₂×Q₈

—

188

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂²⋊D₄₇

yes

—

144

C₂²×C₄

C₂²×D₄₇

yes

—

145

C₂²×C₄

C₂²×D₄₇

yes

characteristic

146

C₂²×C₄

C₂²×D₄₇

yes

characteristic

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

188

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

153

C₂²⋊C₄

—

188

—

154

C₂²⋊C₄

—

188

—

155

C₂²⋊C₄

—

188

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

C₂×C₂²⋊D₄₇

yes

—

164

C₂³

C₂³×D₄₇

yes

characteristic

165

Q₈

—

188

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

182

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

192

C₂²

#13094069

yes

—

193

C₂²

#13094069

yes

—

194

C₂²

#13094071

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

188

—

199

C₄

—

188

—

200

C₄

—

188

—

201

C₂

—

202

C₂

—

203

C₂

#15689682

yes

characteristic

204

C₂

#15689690

yes

characteristic

205

C₂

#15689694

yes

characteristic

206

C₂

#15689868

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#121×C₄₇	C₂
hypercenter	C₂×#121	D₄₇
hypocenter	C₄₇	#2816
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#121

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×#121×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#121

N	H ≅ G/N	type
1. C₂×#13094003	C₂	semidirect
2. #15689059	C₂	semidirect
3. #15689054	C₂	semidirect
4. #15689037	C₂	semidirect
5. #15689037	C₂	semidirect
6. #15689035	C₂	semidirect
7. #15688986	C₂	semidirect
112. C₄₇	#2816	semidirect

H ≅ G/N

type

1. C₂×#13094003

C₂

semidirect

2. #15689059

C₂

semidirect

3. #15689054

C₂

semidirect

4. #15689037

C₂

semidirect

5. #15689037

C₂

semidirect

6. #15689035

C₂

semidirect

7. #15688986

C₂

semidirect

112. C₄₇

#2816

semidirect

Subgroups — #20251416

Conjugacy classes of subgroups

Complements