Group #20246005 — subgroups

Subgroups — #20246005

Order 6016.

600 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 229 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×#13093980	C₂	1	2	yes	maximal
2	C₂²×Dic₄₇⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#15689020	C₂	1	4	yes	maximal, complemented
4	#15689018	C₂	1	4	yes	maximal, complemented
5	#15688983	C₂	1	1	yes	maximal, characteristic
6	#15688981	C₂	1	2	yes	maximal
7	C₄₇×#380	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
10	#13093980	—	2	8	no	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	C₄	1	8	yes	complemented
13	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
15	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
19	C₄₇⋊#95	C₂²	1	4	yes	—
20	C₄₇⋊#95	C₂²	1	2	yes	—
21	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
24	#95×C₄₇	C₂²	1	2	yes	—
25	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	—	4	2	no	—
27	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₇	—	4	2	no	—
32	C₂²×Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂²×Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂²×Dic₄₇	D₄	1	4	yes	—
35	C₂²×Dic₄₇	Q₈	1	4	yes	—
36	C₂²×Dic₄₇	C₂³	1	4	yes	—
37	C₂²×Dic₄₇	C₂³	1	2	yes	—
38	C₂²×Dic₄₇	C₂³	1	2	yes	—
39	C₄⋊C₄×C₄₇	—	4	2	no	—
40	C₂²⋊C₄×C₄₇	—	2	4	no	—
41	C₂²⋊Dic₄₇	—	2	4	no	—
42	C₄⋊Dic₄₇	—	4	2	no	—
43	Dic₄₇⋊C₄	—	2	8	no	—
44	Dic₄₇⋊C₄	—	4	4	no	—
45	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
46	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
47	C₄×Dic₄₇	—	2	8	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂³×C₄₇	C₂×D₄	1	2	yes	—
52	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
53	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	—	2	4	no	—
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	—	4	4	no	—
60	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂×Dic₄₇	—	2	4	no	—
63	C₂×Dic₄₇	—	2	4	no	—
64	C₂×Dic₄₇	—	2	4	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	4	4	no	—
67	C₂×Dic₄₇	C₄⋊C₄	1	16	yes	complemented
68	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
69	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
70	C₂×Dic₄₇	C₂×D₄	1	4	yes	—
71	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	—	2	4	no	—
75	C₂²×C₄₇	—	2	4	no	—
76	C₂²×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
77	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
78	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
81	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	4	4	no	—
85	Dic₄₇	—	2	4	no	—
86	Dic₄₇	—	4	4	no	—
87	Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
88	#2316	—	47	1	no	maximal, Sylow
89	C₂×C₄₇	—	2	4	no	—
90	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄₇	#518	1	1	yes	characteristic
92	C₂×C₄₇	#519	1	1	yes	characteristic
93	C₂×C₄₇	#537	1	2	yes	—
94	C₂×C₄₇	#544	1	2	yes	—
95	C₂×#117	—	47	2	no	—
96	C₂²×C₄⋊C₄	—	47	1	no	—
97	#389	—	47	4	no	—
98	#389	—	47	4	no	—
99	#380	—	47	1	no	—
100	#380	—	47	2	no	—
101	#380	D₄₇	1	1	yes	characteristic
102	C₄₇	#2316	1	1	yes	characteristic, complemented, Sylow
103	C₂³×C₄	—	47	1	no	—
104	C₂³×C₄	—	47	1	no	—
105	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
106	#117	—	94	8	no	—
107	C₂×C₄⋊C₄	—	47	8	no	—
108	C₂×C₄⋊C₄	—	47	2	no	—
109	C₂×C₄⋊C₄	—	47	2	no	—
110	C₂×C₄⋊C₄	—	47	4	no	—
111	C₂×C₄⋊C₄	—	47	4	no	—
112	C₂×C₄⋊C₄	—	47	2	no	—
113	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
114	C₂×C₂²⋊C₄	—	47	2	no	—
115	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
116	C₂×C₄²	—	47	4	no	—
117	#95	—	47	4	no	—
118	#95	—	47	2	no	—
119	#95	C₂×D₄₇	1	2	yes	—
120	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	47	4	no	—
123	C₂²×C₄	—	47	4	no	—
124	C₂²×C₄	—	47	4	no	—
125	C₂²×C₄	—	47	2	no	—
126	C₂²×C₄	—	47	2	no	—
127	C₂²×C₄	—	47	4	no	—
128	C₂²×C₄	—	47	4	no	—
129	C₂²×C₄	—	188	2	no	—
130	C₂²×C₄	C₄×D₄₇	1	4	yes	—
131	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
132	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	47	8	no	—
136	C₄⋊C₄	—	47	8	no	—
137	C₄⋊C₄	—	94	8	no	—
138	C₄⋊C₄	—	188	4	no	—
139	C₄⋊C₄	—	188	2	no	—
140	C₂²⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	94	4	no	—
142	C₄²	—	94	8	no	—
143	C₂³	—	2	2	no	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂×C₄×D₄₇	1	2	yes	—
146	C₂³	D₄×D₄₇	1	2	yes	—
147	C₂³	Q₈×D₄₇	1	2	yes	—
148	C₂³	C₂³×D₄₇	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	47	16	no	—
156	C₂×C₄	—	47	4	no	—
157	C₂×C₄	—	47	4	no	—
158	C₂×C₄	—	47	4	no	—
159	C₂×C₄	—	47	4	no	—
160	C₂×C₄	—	94	4	no	—
161	C₂×C₄	—	94	4	no	—
162	C₂×C₄	—	94	4	no	—
163	C₂×C₄	—	94	4	no	—
164	C₂×C₄	—	188	4	no	—
165	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
166	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	C₄⋊C₄×D₄₇	1	4	yes	—
172	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
173	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
174	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
175	C₂²	#13094069	1	2	yes	—
176	C₂²	#13094071	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	4	no	—
180	C₄	—	47	8	no	—
181	C₄	—	94	4	no	—
182	C₄	—	188	4	no	—
183	C₂	—	2	4	no	—
184	C₂	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
185	C₂	#15689596	1	1	yes	characteristic
186	C₂	#15689600	1	1	yes	characteristic
187	C₂	#15689700	1	2	yes	—
188	C₂	#15689752	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093980

C₂

yes

maximal

C₂²×Dic₄₇⋊C₄

C₂

yes

maximal, characteristic

#15689020

C₂

yes

maximal, complemented

#15689018

C₂

yes

maximal, complemented

#15688983

C₂

yes

maximal, characteristic

#15688981

C₂

yes

maximal

C₄₇×#380

C₂

yes

maximal, characteristic

C₂³×Dic₄₇

C₂²

yes

characteristic

C₂³×Dic₄₇

C₂²

yes

characteristic

#13093980

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₄

yes

complemented

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

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

C₂×C₄

yes

—

C₂²×Dic₄₇

D₄

yes

—

C₂²×Dic₄₇

Q₈

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

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

C₄⋊C₄

yes

complemented

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×C₄⋊C₄

yes

—

#2316

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#544

yes

—

C₂×#117

—

C₂²×C₄⋊C₄

—

#389

—

#389

—

#380

—

100

#380

—

101

#380

D₄₇

yes

characteristic

102

C₄₇

#2316

yes

characteristic, complemented, Sylow

103

C₂³×C₄

—

104

C₂³×C₄

—

105

C₂³×C₄

C₂×D₄₇

yes

characteristic

106

#117

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

114

C₂×C₂²⋊C₄

—

115

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

116

C₂×C₄²

—

117

#95

—

118

#95

—

119

#95

C₂×D₄₇

yes

—

120

C₂⁴

C₂²×D₄₇

yes

characteristic

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

188

—

130

C₂²×C₄

C₄×D₄₇

yes

—

131

C₂²×C₄

C₂²×D₄₇

yes

—

132

C₂²×C₄

C₂²×D₄₇

yes

characteristic

133

C₂²×C₄

C₂²×D₄₇

yes

characteristic

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

188

—

139

C₄⋊C₄

—

188

—

140

C₂²⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂×C₄×D₄₇

yes

—

146

C₂³

D₄×D₄₇

yes

—

147

C₂³

Q₈×D₄₇

yes

—

148

C₂³

C₂³×D₄₇

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

188

—

165

C₂×C₄

C₂×C₄×D₄₇

yes

—

166

C₂×C₄

C₂×C₄×D₄₇

yes

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

C₄⋊C₄×D₄₇

yes

—

172

C₂²

C₂²×C₄×D₄₇

yes

characteristic

173

C₂²

C₂×D₄×D₄₇

yes

characteristic

174

C₂²

C₂×Q₈×D₄₇

yes

characteristic

175

C₂²

#13094069

yes

—

176

C₂²

#13094071

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

188

—

183

C₂

—

184

C₂

C₂×C₄⋊C₄×D₄₇

yes

characteristic

185

C₂

#15689596

yes

characteristic

186

C₂

#15689600

yes

characteristic

187

C₂

#15689700

yes

—

188

C₂

#15689752

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#380	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#380	D₄₇
hypocenter	C₄₇	#2316
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #380

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#380

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #380

N	H ≅ G/N	type
3. #15689020	C₂	semidirect
4. #15689018	C₂	semidirect
12. C₂×Dic₄₇⋊C₄	C₄	semidirect
67. C₂×Dic₄₇	C₄⋊C₄	semidirect
102. C₄₇	#2316	semidirect

H ≅ G/N

type

3. #15689020

C₂

semidirect

4. #15689018

C₂

semidirect

12. C₂×Dic₄₇⋊C₄

C₄

semidirect

67. C₂×Dic₄₇

C₄⋊C₄

semidirect

102. C₄₇

#2316

semidirect

Subgroups — #20246005

Conjugacy classes of subgroups

Complements