Group #20249139 — subgroups

Subgroups — #20249139

Order 6016.

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

normal subgroups only

This group has 728 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×#13093995	C₂	1	2	yes	maximal, complemented
2	C₂×#13093981	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689037	C₂	1	4	yes	maximal, complemented
4	#15689031	C₂	1	4	yes	maximal, complemented
5	#15689026	C₂	1	2	yes	maximal, complemented
6	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#127×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₄₇	C₂²	1	4	yes	—
9	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
10	#13093995	—	2	8	no	—
11	#13093981	—	2	4	no	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
16	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₄₇⋊#95	C₂²	1	4	yes	—
19	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×D₄×C₄₇	C₂²	1	4	yes	—
21	#127×C₄₇	—	2	4	no	—
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂⁴×C₄₇	C₂³	1	4	yes	—
24	C₂³×D₄₇	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₄₇	—	2	8	no	—
26	C₂×D₄×C₄₇	—	2	4	no	—
27	C₂×D₄×C₄₇	—	2	4	no	—
28	C₂×D₄×C₄₇	—	2	4	no	—
29	C₂×D₄×C₄₇	—	4	8	no	—
30	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₂²⋊D₄₇	—	2	8	no	—
33	C₂×C₂²⋊D₄₇	—	4	8	no	—
34	C₂²×Dic₄₇	C₂³	1	4	yes	—
35	C₂²×Dic₄₇	C₂³	1	2	yes	—
36	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×D₄₇	—	2	2	no	—
38	C₂×C₄×D₄₇	—	2	2	no	—
39	C₂×C₄×D₄₇	D₄	1	4	yes	complemented
40	C₄²×C₄₇	D₄	1	4	yes	complemented
41	C₂²⋊Dic₄₇	—	2	8	no	—
42	C₂²⋊Dic₄₇	—	4	8	no	—
43	(C₂×D₄₇)⋊C₄	—	2	4	no	—
44	C₄⋊Dic₄₇	—	2	4	no	—
45	Dic₄₇⋊C₄	—	2	4	no	—
46	C₄×Dic₄₇	—	2	2	no	—
47	C₂³×C₄₇	—	2	4	no	—
48	C₂³×C₄₇	—	2	4	no	—
49	C₂³×C₄₇	—	2	4	no	—
50	C₂³×C₄₇	—	2	8	no	—
51	C₂³×C₄₇	—	2	8	no	—
52	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂²×D₄₇	—	2	1	no	—
54	C₂²×D₄₇	—	2	1	no	—
55	C₂²×D₄₇	—	2	4	no	—
56	C₂²×D₄₇	C₂×D₄	1	2	yes	—
57	D₄×C₄₇	—	4	16	no	—
58	D₄×C₄₇	—	4	8	no	—
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₄₇	#41	1	2	yes	—
64	C₂²⋊D₄₇	—	4	16	no	—
65	C₂×Dic₄₇	—	2	1	no	—
66	C₂×Dic₄₇	—	2	1	no	—
67	C₂×Dic₄₇	—	2	2	no	—
68	C₂×Dic₄₇	—	2	4	no	—
69	C₂×Dic₄₇	—	2	4	no	—
70	C₂×Dic₄₇	—	2	8	no	—
71	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
72	C₄×D₄₇	—	2	8	no	—
73	C₂²×C₄₇	—	4	4	no	—
74	C₂²×C₄₇	—	4	4	no	—
75	C₂²×C₄₇	—	4	8	no	—
76	C₂²×C₄₇	—	4	4	no	—
77	C₂²×C₄₇	—	4	8	no	—
78	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
81	C₂²×C₄₇	H(5,𝔽₂)	1	4	yes	—
82	C₂×D₄₇	—	2	2	no	—
83	C₂×D₄₇	—	2	4	no	—
84	C₂×D₄₇	—	4	2	no	—
85	C₂×D₄₇	—	4	2	no	—
86	C₄×C₄₇	—	2	4	no	—
87	C₄×C₄₇	#121	1	4	yes	—
88	Dic₄₇	—	2	2	no	—
89	Dic₄₇	—	4	2	no	—
90	Dic₄₇	—	4	4	no	—
91	#2619	—	47	1	no	maximal, Sylow
92	C₂×C₄₇	—	8	4	no	—
93	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
94	C₂×C₄₇	#535	1	2	yes	—
95	C₂×C₄₇	#535	1	2	yes	—
96	C₂×C₄₇	#540	1	2	yes	—
97	D₄₇	—	4	2	no	—
98	C₂×#127	D₄₇	1	1	yes	characteristic, complemented
99	C₂×#121	—	47	2	no	—
100	C₂×#117	—	47	1	no	—
101	#394	—	47	4	no	—
102	#392	—	47	4	no	—
103	#390	—	47	2	no	—
104	#382	—	47	1	no	—
105	C₄₇	#2619	1	1	yes	characteristic, complemented, Sylow
106	C₂²×D₄	—	47	4	no	—
107	C₂²×D₄	C₂×D₄₇	1	2	yes	—
108	C₂²×D₄	C₂×D₄₇	1	4	yes	—
109	C₂³×C₄	—	47	1	no	—
110	#127	—	2	4	no	—
111	#121	—	94	8	no	—
112	#117	—	94	4	no	—
113	C₂×C₄⋊C₄	—	47	2	no	—
114	C₂×C₄⋊C₄	—	47	2	no	—
115	C₂×C₂²⋊C₄	—	47	4	no	—
116	C₂×C₂²⋊C₄	—	47	2	no	—
117	C₂×C₂²⋊C₄	—	47	8	no	—
118	C₂×C₄²	—	47	1	no	—
119	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
120	#95	—	47	4	no	—
121	C₂⁴	—	47	1	no	—
122	C₂⁴	C₂²×D₄₇	1	4	yes	—
123	C₂×D₄	—	2	4	no	—
124	C₂×D₄	—	2	4	no	—
125	C₂×D₄	—	2	8	no	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	—	4	8	no	—
128	C₂×D₄	—	94	8	no	—
129	C₂×D₄	—	188	8	no	—
130	C₂²×C₄	—	47	1	no	—
131	C₂²×C₄	—	47	4	no	—
132	C₂²×C₄	—	47	4	no	—
133	C₂²×C₄	—	47	2	no	—
134	C₂²×C₄	—	94	2	no	—
135	C₂²×C₄	—	94	2	no	—
136	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
137	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₄⋊C₄	—	94	4	no	—
139	C₄⋊C₄	—	94	4	no	—
140	C₂²⋊C₄	—	94	8	no	—
141	C₂²⋊C₄	—	94	4	no	—
142	C₂²⋊C₄	—	188	8	no	—
143	C₄²	—	94	2	no	—
144	C₄²	C₂²⋊D₄₇	1	4	yes	complemented
145	C₂³	—	2	4	no	—
146	C₂³	—	2	4	no	—
147	C₂³	—	2	4	no	—
148	C₂³	—	2	8	no	—
149	C₂³	—	2	8	no	—
150	C₂³	—	47	2	no	—
151	C₂³	—	94	1	no	—
152	C₂³	—	94	1	no	—
153	C₂³	—	94	4	no	—
154	C₂³	C₂³×D₄₇	1	1	yes	characteristic
155	D₄	—	4	16	no	—
156	D₄	—	4	8	no	—
157	D₄	—	188	16	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	47	2	no	—
160	C₂×C₄	—	94	1	no	—
161	C₂×C₄	—	94	1	no	—
162	C₂×C₄	—	94	8	no	—
163	C₂×C₄	—	94	4	no	—
164	C₂×C₄	—	94	2	no	—
165	C₂×C₄	—	94	4	no	—
166	C₂×C₄	—	94	8	no	—
167	C₂×C₄	D₄×D₄₇	1	2	yes	—
168	C₂×C₄	#10625352	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
170	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
171	C₂²	—	4	4	no	—
172	C₂²	—	4	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	—	4	8	no	—
175	C₂²	—	4	8	no	—
176	C₂²	—	94	2	no	—
177	C₂²	—	94	4	no	—
178	C₂²	—	188	2	no	—
179	C₂²	—	188	2	no	—
180	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
181	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂²	C₂×#10625352	1	1	yes	characteristic
183	C₂²	#13094069	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	94	2	no	—
186	C₄	—	188	4	no	—
187	C₄	—	188	2	no	—
188	C₄	#13093995	1	4	yes	—
189	C₂	—	8	4	no	—
190	C₂	—	188	2	no	—
191	C₂	C₂×#13093995	1	1	yes	characteristic
192	C₂	#15689686	1	2	yes	—
193	C₂	#15689690	1	2	yes	—
194	C₂	#15689727	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093995

C₂

yes

maximal, complemented

C₂×#13093981

C₂

yes

maximal, characteristic, complemented

#15689037

C₂

yes

maximal, complemented

#15689031

C₂

yes

maximal, complemented

#15689026

C₂

yes

maximal, complemented

#15688989

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₄₇

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

characteristic

#13093995

—

#13093981

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄₇⋊#95

C₂²

yes

—

C₂²×D₄×C₄₇

C₂²

yes

—

C₂²×D₄×C₄₇

C₂²

yes

—

#127×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂³

yes

—

C₂³×D₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

complemented

C₄²×C₄₇

D₄

yes

complemented

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×D₄

yes

—

D₄×C₄₇

—

D₄×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

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₄×D₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2619

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#121

yes

characteristic

C₂×C₄₇

#535

yes

—

C₂×C₄₇

#535

yes

—

C₂×C₄₇

#540

yes

—

D₄₇

—

C₂×#127

D₄₇

yes

characteristic, complemented

C₂×#121

—

100

C₂×#117

—

101

#394

—

102

#392

—

103

#390

—

104

#382

—

105

C₄₇

#2619

yes

characteristic, complemented, Sylow

106

C₂²×D₄

—

107

C₂²×D₄

C₂×D₄₇

yes

—

108

C₂²×D₄

C₂×D₄₇

yes

—

109

C₂³×C₄

—

110

#127

—

111

#121

—

112

#117

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₂²⋊C₄

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₂²⋊C₄

—

118

C₂×C₄²

—

119

C₂×C₄²

C₂×D₄₇

yes

characteristic

120

#95

—

121

C₂⁴

—

122

C₂⁴

C₂²×D₄₇

yes

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

188

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂²×D₄₇

yes

—

137

C₂²×C₄

C₂²×D₄₇

yes

characteristic

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₂²⋊C₄

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

—

188

—

143

C₄²

—

144

C₄²

C₂²⋊D₄₇

yes

complemented

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

C₂³×D₄₇

yes

characteristic

155

D₄

—

156

D₄

—

157

D₄

—

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

D₄×D₄₇

yes

—

168

C₂×C₄

#10625352

yes

—

169

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

170

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

188

—

179

C₂²

—

188

—

180

C₂²

C₂×D₄×D₄₇

yes

characteristic

181

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

182

C₂²

C₂×#10625352

yes

characteristic

183

C₂²

#13094069

yes

—

184

C₄

—

185

C₄

—

186

C₄

—

188

—

187

C₄

—

188

—

188

C₄

#13093995

yes

—

189

C₂

—

190

C₂

—

188

—

191

C₂

C₂×#13093995

yes

characteristic

192

C₂

#15689686

yes

—

193

C₂

#15689690

yes

—

194

C₂

#15689727

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#127×C₄₇	C₂
hypercenter	C₂×#127	D₄₇
hypocenter	C₄₇	#2619
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×#127×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#13093995	C₂	semidirect
2. C₂×#13093981	C₂	semidirect
3. #15689037	C₂	semidirect
4. #15689031	C₂	semidirect
5. #15689026	C₂	semidirect
6. #15688989	C₂	semidirect
7. C₂×#127×C₄₇	C₂	semidirect
39. C₂×C₄×D₄₇	D₄	semidirect
40. C₄²×C₄₇	D₄	semidirect
98. C₂×#127	D₄₇	semidirect
105. C₄₇	#2619	semidirect
144. C₄²	C₂²⋊D₄₇	semidirect

H ≅ G/N

type

1. C₂×#13093995

C₂

semidirect

2. C₂×#13093981

C₂

semidirect

3. #15689037

C₂

semidirect

4. #15689031

C₂

semidirect

5. #15689026

C₂

semidirect

6. #15688989

C₂

semidirect

7. C₂×#127×C₄₇

C₂

semidirect

39. C₂×C₄×D₄₇

D₄

semidirect

40. C₄²×C₄₇

D₄

semidirect

98. C₂×#127

D₄₇

semidirect

105. C₄₇

#2619

semidirect

144. C₄²

C₂²⋊D₄₇

semidirect

Subgroups — #20249139

Conjugacy classes of subgroups

Complements