Group #20244267 — subgroups

Subgroups — #20244267

Order 6016.

340 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 105 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×#10625325	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093947	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13093944	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689028	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689024	C₂	1	1	yes	maximal, characteristic
7	C₂×#107×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
9	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×#10625325	—	2	2	no	—
11	C₂×#10625325	—	2	2	no	—
12	C₂×#10625325	C₄	1	4	yes	complemented
13	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
15	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
16	#13093947	—	2	2	no	—
17	#13093944	—	2	2	no	—
18	#13093934	—	2	2	no	—
19	C₄₇⋊#95	—	2	2	no	—
20	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
23	#107×C₄₇	—	2	2	no	—
24	C₂³×D₄₇	—	2	1	no	—
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₄₇	—	2	1	no	—
29	C₂²×Dic₄₇	D₄	1	2	yes	—
30	C₂×D₁₈₈	—	2	1	no	—
31	C₂×D₁₈₈	—	2	1	no	—
32	C₂×D₁₈₈	—	2	1	no	—
33	C₂×D₁₈₈	—	2	1	no	—
34	C₂×D₁₈₈	—	2	1	no	—
35	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
36	C₂×Dic₉₄	—	2	1	no	—
37	C₂×Dic₉₄	—	2	1	no	—
38	C₂×Dic₉₄	—	2	1	no	—
39	C₂×Dic₉₄	—	2	1	no	—
40	C₂×Dic₉₄	—	2	1	no	—
41	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₈×C₄₇	D₄	1	2	yes	—
44	C₂×C₈×C₄₇	D₄	1	2	yes	—
45	C₄⋊C₄×C₄₇	—	2	1	no	—
46	C₄⋊C₄×C₄₇	—	2	1	no	—
47	C₄⋊C₄×C₄₇	—	2	1	no	—
48	C₄⋊C₄×C₄₇	—	2	1	no	—
49	(C₂×D₄₇)⋊C₄	—	4	4	no	—
50	C₄×Dic₄₇	—	2	1	no	—
51	C₄×Dic₄₇	—	2	1	no	—
52	C₂×C₄₇⋊C₈	—	2	1	no	—
53	C₂×C₄₇⋊C₈	—	2	1	no	—
54	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
55	#10625325	—	2	8	no	—
56	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×D₄₇	—	2	2	no	—
58	C₂²×D₄₇	—	4	1	no	—
59	C₂²×D₄₇	—	4	1	no	—
60	C₂²×D₄₇	—	4	2	no	—
61	C₂²×D₄₇	—	4	2	no	—
62	C₂×C₄×C₄₇	—	4	1	no	—
63	C₂×C₄×C₄₇	—	4	1	no	—
64	C₂×C₄×C₄₇	—	4	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	—	2	2	no	—
75	C₂×Dic₄₇	—	2	2	no	—
76	C₂×Dic₄₇	—	4	1	no	—
77	C₂×Dic₄₇	—	4	1	no	—
78	D₁₈₈	—	2	2	no	—
79	D₁₈₈	—	2	2	no	—
80	D₁₈₈	—	4	1	no	—
81	D₁₈₈	—	4	1	no	—
82	Dic₉₄	—	2	2	no	—
83	Dic₉₄	—	2	2	no	—
84	Dic₉₄	—	4	1	no	—
85	Dic₉₄	—	4	1	no	—
86	C₈×C₄₇	C₂²⋊C₄	1	4	yes	complemented
87	C₄₇⋊C₈	—	2	2	no	—
88	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₄₇	#121	1	1	yes	characteristic
90	C₂²×C₄₇	#127	1	1	yes	characteristic
91	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
92	C₂²×C₄₇	#137	1	2	yes	—
93	C₂×D₄₇	—	4	2	no	—
94	C₂×D₄₇	—	4	4	no	—
95	C₂×D₄₇	—	8	2	no	—
96	C₂×D₄₇	—	8	1	no	—
97	C₂×D₄₇	—	8	1	no	—
98	C₄×C₄₇	—	8	1	no	—
99	C₄×C₄₇	—	8	1	no	—
100	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
102	C₄×C₄₇	#121	1	1	yes	characteristic
103	C₄×C₄₇	#124	1	1	yes	characteristic
104	Dic₄₇	—	4	2	no	—
105	Dic₄₇	—	4	2	no	—
106	#1973	—	47	1	no	maximal, Sylow
107	C₂×C₄₇	#390	1	1	yes	characteristic
108	C₂×C₄₇	#440	1	2	yes	—
109	C₂×C₄₇	#468	1	2	yes	—
110	C₂×C₄₇	#496	1	1	yes	characteristic
111	C₂×C₄₇	#497	1	1	yes	characteristic
112	D₄₇	—	8	2	no	—
113	C₂²×SD₁₆	—	47	1	no	—
114	C₂×#107	D₄₇	1	1	yes	characteristic, complemented
115	C₂×#103	—	47	1	no	—
116	C₂×#102	—	47	1	no	—
117	C₂×#97	—	47	1	no	—
118	#391	—	47	1	no	—
119	#390	—	47	1	no	—
120	C₄₇	#1973	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	47	1	no	—
122	C₂²×D₄	—	47	1	no	—
123	C₂×SD₁₆	—	47	4	no	—
124	C₂×SD₁₆	—	94	2	no	—
125	C₂×SD₁₆	—	94	2	no	—
126	C₂²×C₈	—	47	1	no	—
127	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	—	94	2	no	—
131	C₂×C₄²	—	47	1	no	—
132	#107	—	2	2	no	—
133	#103	—	94	2	no	—
134	#102	—	94	2	no	—
135	#97	—	94	2	no	—
136	#95	—	94	2	no	—
137	C₂⁴	—	94	1	no	—
138	C₂×Q₈	—	47	2	no	—
139	C₂×Q₈	—	94	1	no	—
140	C₂×Q₈	—	94	1	no	—
141	C₂×Q₈	—	94	1	no	—
142	C₂×Q₈	—	94	1	no	—
143	C₂×Q₈	—	94	1	no	—
144	C₂×D₄	—	47	2	no	—
145	C₂×D₄	—	94	1	no	—
146	C₂×D₄	—	94	1	no	—
147	C₂×D₄	—	94	1	no	—
148	C₂×D₄	—	94	1	no	—
149	C₂×D₄	—	94	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	47	2	no	—
153	C₂²×C₄	—	94	1	no	—
154	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
155	SD₁₆	—	94	8	no	—
156	C₂×C₈	—	47	2	no	—
157	C₂×C₈	—	94	1	no	—
158	C₂×C₈	—	94	1	no	—
159	C₂×C₈	C₄×D₄₇	1	2	yes	—
160	C₂×C₈	D₁₈₈	1	2	yes	—
161	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₂²⋊C₄	—	188	4	no	—
167	C₄²	—	94	1	no	—
168	C₄²	—	94	1	no	—
169	C₂³	—	94	2	no	—
170	C₂³	—	188	1	no	—
171	C₂³	—	188	1	no	—
172	C₂³	—	188	2	no	—
173	C₂³	—	188	2	no	—
174	C₂³	D₄×D₄₇	1	1	yes	characteristic
175	Q₈	—	94	2	no	—
176	Q₈	—	94	2	no	—
177	Q₈	—	188	1	no	—
178	Q₈	—	188	1	no	—
179	D₄	—	94	2	no	—
180	D₄	—	94	2	no	—
181	D₄	—	188	1	no	—
182	D₄	—	188	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	94	2	no	—
188	C₂×C₄	—	94	2	no	—
189	C₂×C₄	—	94	2	no	—
190	C₂×C₄	—	94	2	no	—
191	C₂×C₄	—	188	1	no	—
192	C₂×C₄	—	188	1	no	—
193	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
194	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
195	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄	#10625354	1	1	yes	characteristic
197	C₂×C₄	#10625354	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
199	C₈	—	94	2	no	—
200	C₈	(C₂×D₄₇)⋊C₄	1	4	yes	complemented
201	C₂²	—	376	1	no	—
202	C₂²	—	376	1	no	—
203	C₂²	—	376	2	no	—
204	C₂²	—	188	4	no	—
205	C₂²	—	188	2	no	—
206	C₂²	#13093986	1	1	yes	characteristic
207	C₂²	#13093999	1	1	yes	characteristic
208	C₂²	#13094020	1	1	yes	characteristic
209	C₂²	#13094048	1	2	yes	—
210	C₂²	#13094056	1	2	yes	—
211	C₄	—	8	1	no	—
212	C₄	—	8	1	no	—
213	C₄	—	188	2	no	—
214	C₄	—	188	2	no	—
215	C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
216	C₄	#13093986	1	1	yes	characteristic
217	C₄	#13093999	1	1	yes	characteristic
218	C₄	#13094013	1	1	yes	characteristic
219	C₂	—	376	2	no	—
220	C₂	#15689024	1	1	yes	characteristic
221	C₂	#15689222	1	2	yes	—
222	C₂	#15689370	1	2	yes	—
223	C₂	#15689511	1	1	yes	characteristic
224	C₂	#15689516	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625325

C₂

yes

maximal, characteristic

C₂×#13093947

C₂

yes

maximal, characteristic, complemented

C₂×#13093944

C₂

yes

maximal, characteristic

C₂×#13093934

C₂

yes

maximal, characteristic, complemented

#15689028

C₂

yes

maximal, characteristic, complemented

#15689024

C₂

yes

maximal, characteristic

C₂×#107×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

—

C₂×C₄×Dic₄₇

C₂²

yes

—

—

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

#107×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

#10625325

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₂²⋊C₄

yes

complemented

C₄₇⋊C₈

—

C₂²×C₄₇

C₄×D₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#137

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

100

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

101

C₄×C₄₇

C₄×D₄

yes

characteristic

102

C₄×C₄₇

#121

yes

characteristic

103

C₄×C₄₇

#124

yes

characteristic

104

Dic₄₇

—

105

Dic₄₇

—

106

#1973

—

maximal, Sylow

107

C₂×C₄₇

#390

yes

characteristic

108

C₂×C₄₇

#440

yes

—

109

C₂×C₄₇

#468

yes

—

110

C₂×C₄₇

#496

yes

characteristic

111

C₂×C₄₇

#497

yes

characteristic

112

D₄₇

—

113

C₂²×SD₁₆

—

114

C₂×#107

D₄₇

yes

characteristic, complemented

115

C₂×#103

—

116

C₂×#102

—

117

C₂×#97

—

118

#391

—

119

#390

—

120

C₄₇

#1973

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×D₄

—

123

C₂×SD₁₆

—

124

C₂×SD₁₆

—

125

C₂×SD₁₆

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂×D₄₇

yes

characteristic

128

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

129

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

130

C₂×C₂²⋊C₄

—

131

C₂×C₄²

—

132

#107

—

133

#103

—

134

#102

—

135

#97

—

136

#95

—

137

C₂⁴

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

C₂²×D₄₇

yes

characteristic

155

SD₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₄×D₄₇

yes

—

160

C₂×C₈

D₁₈₈

yes

—

161

C₂×C₈

C₂²⋊D₄₇

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₂²⋊C₄

—

188

—

167

C₄²

—

168

C₄²

—

169

C₂³

—

170

C₂³

—

188

—

171

C₂³

—

188

—

172

C₂³

—

188

—

173

C₂³

—

188

—

174

C₂³

D₄×D₄₇

yes

characteristic

175

Q₈

—

176

Q₈

—

177

Q₈

—

188

—

178

Q₈

—

188

—

179

D₄

—

180

D₄

—

181

D₄

—

188

—

182

D₄

—

188

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

188

—

192

C₂×C₄

—

188

—

193

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

194

C₂×C₄

C₂×D₁₈₈

yes

characteristic

195

C₂×C₄

D₄×D₄₇

yes

characteristic

196

C₂×C₄

#10625354

yes

characteristic

197

C₂×C₄

#10625354

yes

characteristic

198

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

199

C₈

—

200

C₈

(C₂×D₄₇)⋊C₄

yes

complemented

201

C₂²

—

376

—

202

C₂²

—

376

—

203

C₂²

—

376

—

204

C₂²

—

188

—

205

C₂²

—

188

—

206

C₂²

#13093986

yes

characteristic

207

C₂²

#13093999

yes

characteristic

208

C₂²

#13094020

yes

characteristic

209

C₂²

#13094048

yes

—

210

C₂²

#13094056

yes

—

211

C₄

—

212

C₄

—

213

C₄

—

188

—

214

C₄

—

188

—

215

C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

216

C₄

#13093986

yes

characteristic

217

C₄

#13093999

yes

characteristic

218

C₄

#13094013

yes

characteristic

219

C₂

—

376

—

220

C₂

#15689024

yes

characteristic

221

C₂

#15689222

yes

—

222

C₂

#15689370

yes

—

223

C₂

#15689511

yes

characteristic

224

C₂

#15689516

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#107×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#107	D₄₇
hypocenter	C₄₇	#1973
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#107×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. C₂×#13093947	C₂	semidirect
4. C₂×#13093934	C₂	semidirect
5. #15689028	C₂	semidirect
7. C₂×#107×C₄₇	C₂	semidirect
12. C₂×#10625325	C₄	semidirect
86. C₈×C₄₇	C₂²⋊C₄	semidirect
114. C₂×#107	D₄₇	semidirect
120. C₄₇	#1973	semidirect
200. C₈	(C₂×D₄₇)⋊C₄	semidirect

H ≅ G/N

type

2. C₂×#13093947

C₂

semidirect

4. C₂×#13093934

C₂

semidirect

5. #15689028

C₂

semidirect

7. C₂×#107×C₄₇

C₂

semidirect

12. C₂×#10625325

C₄

semidirect

86. C₈×C₄₇

C₂²⋊C₄

semidirect

114. C₂×#107

D₄₇

semidirect

120. C₄₇

#1973

semidirect

200. C₈

(C₂×D₄₇)⋊C₄

semidirect

Subgroups — #20244267

Conjugacy classes of subgroups

Complements