Group #20248598 — subgroups

Subgroups — #20248598

Order 6016.

500 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 185 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×#13094023	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689044	C₂	1	4	yes	maximal
4	#15689030	C₂	1	1	yes	maximal, characteristic
5	#15689020	C₂	1	4	yes	maximal
6	#15688993	C₂	1	1	yes	maximal, characteristic
7	#15688992	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₇⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₉₄	C₂²	1	2	yes	—
11	#13094023	—	2	4	no	—
12	Q₈×Dic₄₇	—	2	4	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
23	C₄₇⋊#95	C₂²	1	2	yes	—
24	C₄₇⋊#95	C₂²	1	4	yes	—
25	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	#95×C₄₇	C₂²	1	4	yes	—
29	C₂×Q₈×C₄₇	—	2	2	no	—
30	C₂×Q₈×C₄₇	—	2	2	no	—
31	C₂×Q₈×C₄₇	Q₈	1	4	yes	complemented
32	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×Dic₄₇	C₂³	1	2	yes	—
37	C₂²×Dic₄₇	C₂³	1	2	yes	—
38	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₄₇	C₂³	1	2	yes	—
40	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
41	C₂×Dic₉₄	—	2	4	no	—
42	C₂×Dic₉₄	—	4	4	no	—
43	C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄⋊Dic₄₇	—	2	2	no	—
46	C₄⋊Dic₄₇	—	2	2	no	—
47	C₄⋊Dic₄₇	—	2	4	no	—
48	C₄⋊Dic₄₇	—	4	4	no	—
49	Dic₄₇⋊C₄	—	2	4	no	—
50	Dic₄₇⋊C₄	—	4	4	no	—
51	Dic₄₇⋊C₄	Q₈	1	8	yes	complemented
52	C₄×Dic₄₇	—	2	2	no	—
53	C₄×Dic₄₇	—	2	4	no	—
54	C₄×Dic₄₇	D₄	1	4	yes	—
55	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
56	Q₈×C₄₇	—	2	8	no	—
57	C₂×C₄×C₄₇	—	2	2	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	2	no	—
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
67	C₂×C₄×C₄₇	#41	1	2	yes	—
68	C₂×Dic₄₇	—	2	2	no	—
69	C₂×Dic₄₇	—	2	2	no	—
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	4	no	—
75	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
76	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
77	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
78	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
79	Dic₉₄	—	4	8	no	—
80	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
82	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
84	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
85	C₄×C₄₇	—	2	4	no	—
86	C₄×C₄₇	—	2	4	no	—
87	C₄×C₄₇	—	4	2	no	—
88	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
89	Dic₄₇	—	2	2	no	—
90	Dic₄₇	—	2	4	no	—
91	Dic₄₇	—	4	2	no	—
92	Dic₄₇	—	4	2	no	—
93	Dic₄₇	C₄⋊Q₈	1	4	yes	—
94	#2567	—	47	1	no	maximal, Sylow
95	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄₇	#535	1	1	yes	characteristic
98	C₂×C₄₇	#554	1	2	yes	—
99	C₂×C₄₇	Q₈²	1	2	yes	—
100	C₂×C₄⋊Q₈	—	47	1	no	—
101	C₂×C₄×Q₈	—	47	1	no	—
102	#395	—	47	4	no	—
103	#391	—	47	1	no	—
104	#391	D₄₇	1	1	yes	characteristic
105	#389	—	47	4	no	—
106	#384	—	47	1	no	—
107	#384	—	47	1	no	—
108	C₄×C₄⋊C₄	—	47	1	no	—
109	C₄₇	#2567	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	47	2	no	—
111	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
112	C₄⋊Q₈	—	94	4	no	—
113	C₄×Q₈	—	94	4	no	—
114	C₂×C₄⋊C₄	—	47	2	no	—
115	C₂×C₄⋊C₄	—	47	1	no	—
116	C₂×C₄⋊C₄	—	47	2	no	—
117	C₂×C₄⋊C₄	—	47	2	no	—
118	C₂×C₄⋊C₄	—	47	1	no	—
119	C₂×C₄⋊C₄	—	47	4	no	—
120	C₂×C₄⋊C₄	—	47	4	no	—
121	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
122	C₂×C₄²	—	47	1	no	—
123	C₂×C₄²	—	47	2	no	—
124	C₂×C₄²	—	47	1	no	—
125	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
126	#95	—	47	2	no	—
127	#95	—	47	4	no	—
128	#95	C₂×D₄₇	1	4	yes	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	94	4	no	—
132	C₂×Q₈	—	188	4	no	—
133	C₂×Q₈	Dic₉₄	1	4	yes	complemented
134	C₂²×C₄	—	47	1	no	—
135	C₂²×C₄	—	47	2	no	—
136	C₂²×C₄	—	47	1	no	—
137	C₂²×C₄	—	47	2	no	—
138	C₂²×C₄	—	47	2	no	—
139	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
140	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
141	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
142	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	47	8	no	—
145	C₄⋊C₄	—	94	4	no	—
146	C₄⋊C₄	—	94	4	no	—
147	C₄⋊C₄	—	94	2	no	—
148	C₄⋊C₄	—	94	2	no	—
149	C₄⋊C₄	—	188	4	no	—
150	C₄⋊C₄	—	188	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	47	4	no	—
153	C₄²	—	94	4	no	—
154	C₄²	—	94	2	no	—
155	C₂³	C₂³×D₄₇	1	1	yes	characteristic
156	Q₈	—	2	8	no	—
157	Q₈	—	188	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	47	2	no	—
165	C₂×C₄	—	47	4	no	—
166	C₂×C₄	—	47	2	no	—
167	C₂×C₄	—	47	4	no	—
168	C₂×C₄	—	94	2	no	—
169	C₂×C₄	—	94	2	no	—
170	C₂×C₄	—	94	2	no	—
171	C₂×C₄	—	94	2	no	—
172	C₂×C₄	—	94	2	no	—
173	C₂×C₄	—	94	4	no	—
174	C₂×C₄	—	94	4	no	—
175	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
176	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
177	C₂×C₄	D₄×D₄₇	1	2	yes	—
178	C₂×C₄	#10625352	1	2	yes	—
179	C₂×C₄	Q₈×D₄₇	1	4	yes	—
180	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
181	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
182	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
183	C₂²	C₂×#10625352	1	1	yes	characteristic
184	C₂²	#13094070	1	2	yes	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	47	4	no	—
189	C₄	—	94	4	no	—
190	C₄	—	94	2	no	—
191	C₄	—	188	2	no	—
192	C₄	—	188	2	no	—
193	C₄	#13094001	1	4	yes	—
194	C₂	C₂×#13094001	1	1	yes	characteristic
195	C₂	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
196	C₂	#15689691	1	1	yes	characteristic
197	C₂	#15689831	1	2	yes	—
198	C₂	Q₈×Dic₉₄	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094023

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₄₇

C₂

yes

maximal, characteristic

#15689044

C₂

yes

maximal

#15689030

C₂

yes

maximal, characteristic

#15689020

C₂

yes

maximal

#15688993

C₂

yes

maximal, characteristic

#15688992

C₂

yes

maximal, characteristic

C₄×Dic₄₇⋊C₄

C₂

yes

maximal, characteristic

C₄₇×#391

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

#13094023

—

Q₈×Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

Q₈

yes

complemented

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

Q₈

yes

complemented

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×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×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₄⋊Q₈

yes

—

#2567

—

maximal, Sylow

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₇

C₂×C₄⋊Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

Q₈²

yes

—

100

C₂×C₄⋊Q₈

—

101

C₂×C₄×Q₈

—

102

#395

—

103

#391

—

104

#391

D₄₇

yes

characteristic

105

#389

—

106

#384

—

107

#384

—

108

C₄×C₄⋊C₄

—

109

C₄₇

#2567

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₄₇

yes

characteristic

112

C₄⋊Q₈

—

113

C₄×Q₈

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

122

C₂×C₄²

—

123

C₂×C₄²

—

124

C₂×C₄²

—

125

C₂×C₄²

C₂×D₄₇

yes

characteristic

126

#95

—

127

#95

—

128

#95

C₂×D₄₇

yes

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

188

—

133

C₂×Q₈

Dic₉₄

yes

complemented

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²×D₄₇

yes

—

140

C₂²×C₄

C₂²×D₄₇

yes

—

141

C₂²×C₄

C₂²×D₄₇

yes

—

142

C₂²×C₄

C₂²×D₄₇

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

188

—

150

C₄⋊C₄

—

188

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₂³

C₂³×D₄₇

yes

characteristic

156

Q₈

—

157

Q₈

—

188

—

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

—

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

C₂×Dic₉₄

yes

—

176

C₂×C₄

C₂×Dic₉₄

yes

—

177

C₂×C₄

D₄×D₄₇

yes

—

178

C₂×C₄

#10625352

yes

—

179

C₂×C₄

Q₈×D₄₇

yes

—

180

C₂²

C₂×D₄×D₄₇

yes

characteristic

181

C₂²

C₂×Q₈×D₄₇

yes

—

182

C₂²

C₂²×Dic₉₄

yes

characteristic

183

C₂²

C₂×#10625352

yes

characteristic

184

C₂²

#13094070

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

188

—

192

C₄

—

188

—

193

C₄

#13094001

yes

—

194

C₂

C₂×#13094001

yes

characteristic

195

C₂

C₄⋊Q₈×D₄₇

yes

characteristic

196

C₂

#15689691

yes

characteristic

197

C₂

#15689831

yes

—

198

C₂

Q₈×Dic₉₄

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#391	C₂
hypercenter	#391	D₄₇
hypocenter	C₄₇	#2567
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#391

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
31. C₂×Q₈×C₄₇	Q₈	semidirect
51. Dic₄₇⋊C₄	Q₈	semidirect
109. C₄₇	#2567	semidirect
133. C₂×Q₈	Dic₉₄	semidirect

H ≅ G/N

type

31. C₂×Q₈×C₄₇

Q₈

semidirect

51. Dic₄₇⋊C₄

Q₈

semidirect

109. C₄₇

#2567

semidirect

133. C₂×Q₈

Dic₉₄

semidirect

Subgroups — #20248598

Conjugacy classes of subgroups

Complements