Group #20248876 — subgroups

Subgroups — #20248876

Order 6016.

520 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13094012	C₂	1	1	yes	maximal, characteristic
2	#15689049	C₂	1	2	yes	maximal, complemented
3	#15689034	C₂	1	2	yes	maximal, complemented
4	#15689029	C₂	1	2	yes	maximal, complemented
5	#15689025	C₂	1	2	yes	maximal
6	#15689013	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
10	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
11	#13094012	—	2	4	no	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	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	4	yes	—
17	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
18	C₄₇⋊#95	C₂²	1	2	yes	—
19	C₄₇⋊#95	C₂²	1	4	yes	—
20	C₄₇⋊#95	C₂²	1	4	yes	—
21	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₄₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂³×D₄₇	C₂³	1	2	yes	—
28	C₂×Q₈×C₄₇	—	2	2	no	—
29	C₂×Q₈×C₄₇	—	2	4	no	—
30	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₄₇	C₂³	1	2	yes	—
36	C₂²×Dic₄₇	C₂³	1	4	yes	—
37	C₂×D₁₈₈	—	2	2	no	—
38	C₂×D₁₈₈	—	2	4	no	—
39	C₂×Dic₉₄	—	2	2	no	—
40	C₂×Dic₉₄	—	2	4	no	—
41	C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	—	2	4	no	—
44	C₄²×C₄₇	D₄	1	4	yes	—
45	(C₂×D₄₇)⋊C₄	—	2	4	no	—
46	(C₂×D₄₇)⋊C₄	—	2	4	no	—
47	(C₂×D₄₇)⋊C₄	—	2	8	no	—
48	(C₂×D₄₇)⋊C₄	—	4	4	no	—
49	C₄⋊Dic₄₇	—	2	4	no	—
50	C₄×Dic₄₇	—	4	2	no	—
51	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂²×D₄₇	—	2	2	no	—
53	C₂²×D₄₇	—	2	4	no	—
54	C₂²×D₄₇	—	2	4	no	—
55	C₂²×D₄₇	—	2	2	no	—
56	C₂²×D₄₇	—	2	2	no	—
57	Q₈×C₄₇	—	2	8	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	1	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	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₄₇	#41	1	4	yes	—
70	C₂×C₄×C₄₇	#41	1	4	yes	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂×Dic₄₇	—	2	4	no	—
73	C₂×Dic₄₇	—	2	4	no	—
74	C₂×Dic₄₇	—	2	4	no	—
75	C₂×Dic₄₇	—	2	4	no	—
76	D₁₈₈	—	4	4	no	—
77	Dic₉₄	—	4	4	no	—
78	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₄₇	C₂×#41	1	2	yes	—
80	C₂²×C₄₇	C₂×#41	1	2	yes	—
81	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
83	C₂×D₄₇	—	4	2	no	—
84	C₂×D₄₇	—	4	2	no	—
85	C₂×D₄₇	—	4	2	no	—
86	C₂×D₄₇	—	4	4	no	—
87	C₂×D₄₇	—	4	4	no	—
88	C₄×C₄₇	—	2	2	no	—
89	C₄×C₄₇	—	2	4	no	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	#124	1	4	yes	—
93	Dic₄₇	—	4	2	no	—
94	Dic₄₇	—	4	4	no	—
95	#2592	—	47	1	no	maximal, Sylow
96	C₂×C₄₇	#525	1	1	yes	characteristic
97	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
98	C₂×C₄₇	#537	1	1	yes	characteristic
99	C₂×C₄₇	#556	1	2	yes	—
100	C₂×C₄₇	#559	1	2	yes	—
101	D₄₇	—	8	2	no	—
102	C₂×#124	—	47	1	no	—
103	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic, complemented
104	#397	—	47	2	no	—
105	#393	—	47	2	no	—
106	#391	—	47	2	no	—
107	#390	—	47	2	no	—
108	#388	—	47	4	no	—
109	C₄×C₄⋊C₄	—	47	1	no	—
110	C₄₇	#2592	1	1	yes	characteristic, complemented, Sylow
111	C₂²×Q₈	—	47	1	no	—
112	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
113	C₂²×D₄	—	47	1	no	—
114	#124	—	94	4	no	—
115	C₄×Q₈	—	2	4	no	—
116	C₂×C₄⋊C₄	—	47	2	no	—
117	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
118	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	—	47	2	no	—
120	C₂×C₂²⋊C₄	—	47	2	no	—
121	C₂×C₂²⋊C₄	—	47	4	no	—
122	C₂×C₂²⋊C₄	—	47	4	no	—
123	C₂×C₄²	—	47	2	no	—
124	C₂×C₄²	C₂×D₄₇	1	2	yes	—
125	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
126	#95	—	47	2	no	—
127	#95	—	47	4	no	—
128	#95	—	47	4	no	—
129	C₂⁴	—	47	2	no	—
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	94	4	no	—
133	C₂×Q₈	—	94	2	no	—
134	C₂×D₄	—	94	4	no	—
135	C₂×D₄	—	94	2	no	—
136	C₂²×C₄	—	47	2	no	—
137	C₂²×C₄	—	47	4	no	—
138	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
139	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
140	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	94	4	no	—
146	C₂²⋊C₄	—	94	8	no	—
147	C₂²⋊C₄	—	94	4	no	—
148	C₂²⋊C₄	—	94	4	no	—
149	C₂²⋊C₄	—	188	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	188	2	no	—
152	C₄²	C₂²⋊D₄₇	1	4	yes	—
153	C₂³	—	94	2	no	—
154	C₂³	—	94	4	no	—
155	C₂³	—	94	4	no	—
156	C₂³	—	94	2	no	—
157	C₂³	—	94	2	no	—
158	C₂³	C₂³×D₄₇	1	1	yes	characteristic
159	Q₈	—	2	8	no	—
160	Q₈	—	188	4	no	—
161	D₄	—	188	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	94	2	no	—
171	C₂×C₄	—	94	4	no	—
172	C₂×C₄	—	94	4	no	—
173	C₂×C₄	—	94	4	no	—
174	C₂×C₄	—	94	4	no	—
175	C₂×C₄	#10625350	1	4	yes	—
176	C₂×C₄	#10625354	1	4	yes	—
177	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
180	C₂²	—	188	4	no	—
181	C₂²	—	188	4	no	—
182	C₂²	—	188	2	no	—
183	C₂²	—	188	2	no	—
184	C₂²	—	188	2	no	—
185	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂²	C₂×#10625350	1	2	yes	—
187	C₂²	C₂×#10625354	1	2	yes	—
188	C₂²	#13094070	1	1	yes	characteristic
189	C₂²	#13094071	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	188	4	no	—
195	C₄	—	188	2	no	—
196	C₄	#13094013	1	4	yes	—
197	C₂	—	376	2	no	—
198	C₂	#15689634	1	1	yes	characteristic
199	C₂	C₂×#13094013	1	1	yes	characteristic
200	C₂	#15689701	1	1	yes	characteristic
201	C₂	#15689846	1	2	yes	—
202	C₂	#15689858	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094012

C₂

yes

maximal, characteristic

#15689049

C₂

yes

maximal, complemented

#15689034

C₂

yes

maximal, complemented

#15689029

C₂

yes

maximal, complemented

#15689025

C₂

yes

maximal

#15689013

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

characteristic

#13094012

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

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

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×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×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#124

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×C₄₇

#559

yes

—

101

D₄₇

—

102

C₂×#124

—

103

C₂×C₄×Q₈

D₄₇

yes

characteristic, complemented

104

#397

—

105

#393

—

106

#391

—

107

#390

—

108

#388

—

109

C₄×C₄⋊C₄

—

110

C₄₇

#2592

yes

characteristic, complemented, Sylow

111

C₂²×Q₈

—

112

C₂²×Q₈

C₂×D₄₇

yes

characteristic

113

C₂²×D₄

—

114

#124

—

115

C₄×Q₈

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

118

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

119

C₂×C₂²⋊C₄

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₂²⋊C₄

—

122

C₂×C₂²⋊C₄

—

123

C₂×C₄²

—

124

C₂×C₄²

C₂×D₄₇

yes

—

125

C₂×C₄²

C₂×D₄₇

yes

characteristic

126

#95

—

127

#95

—

128

#95

—

129

C₂⁴

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²×D₄₇

yes

—

139

C₂²×C₄

C₂²×D₄₇

yes

—

140

C₂²×C₄

C₂²×D₄₇

yes

characteristic

141

C₂²×C₄

C₂²×D₄₇

yes

characteristic

142

C₂²×C₄

C₂²×D₄₇

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

188

—

150

C₄²

—

151

C₄²

—

188

—

152

C₄²

C₂²⋊D₄₇

yes

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

C₂³×D₄₇

yes

characteristic

159

Q₈

—

160

Q₈

—

188

—

161

D₄

—

188

—

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

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#10625350

yes

—

176

C₂×C₄

#10625354

yes

—

177

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

178

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

179

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

180

C₂²

—

188

—

181

C₂²

—

188

—

182

C₂²

—

188

—

183

C₂²

—

188

—

184

C₂²

—

188

—

185

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

186

C₂²

C₂×#10625350

yes

—

187

C₂²

C₂×#10625354

yes

—

188

C₂²

#13094070

yes

characteristic

189

C₂²

#13094071

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

188

—

195

C₄

—

188

—

196

C₄

#13094013

yes

—

197

C₂

—

376

—

198

C₂

#15689634

yes

characteristic

199

C₂

C₂×#13094013

yes

characteristic

200

C₂

#15689701

yes

characteristic

201

C₂

#15689846

yes

—

202

C₂

#15689858

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄×Q₈×C₄₇	C₂
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	#2592
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄×Q₈×C₄₇

C₂

hypercenter

C₂×C₄×Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
2. #15689049	C₂	semidirect
3. #15689034	C₂	semidirect
4. #15689029	C₂	semidirect
6. #15689013	C₂	semidirect
7. C₄×C₄⋊Dic₄₇	C₂	semidirect
8. C₂×C₄×Q₈×C₄₇	C₂	semidirect
103. C₂×C₄×Q₈	D₄₇	semidirect
110. C₄₇	#2592	semidirect

H ≅ G/N

type

2. #15689049

C₂

semidirect

3. #15689034

C₂

semidirect

4. #15689029

C₂

semidirect

6. #15689013

C₂

semidirect

7. C₄×C₄⋊Dic₄₇

C₂

semidirect

8. C₂×C₄×Q₈×C₄₇

C₂

semidirect

103. C₂×C₄×Q₈

D₄₇

semidirect

110. C₄₇

#2592

semidirect

Subgroups — #20248876

Conjugacy classes of subgroups

Complements