Group #20244459 — subgroups

Subgroups — #20244459

Order 6016.

510 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 99 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×#10625334	C₂	1	2	yes	maximal, complemented
2	C₂×#13093945	C₂	1	2	yes	maximal, complemented
3	#15689025	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×#10625334	—	2	4	no	—
8	C₂×#10625334	—	2	4	no	—
9	C₂×#10625334	—	2	4	no	—
10	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
13	#13093945	—	2	4	no	—
14	C₂²×D₄×C₄₇	—	2	2	no	—
15	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
16	#127×C₄₇	—	2	4	no	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄₇	—	2	2	no	—
19	C₂³×D₄₇	—	2	1	no	—
20	C₂×D₄×C₄₇	—	2	8	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	—	2	2	no	—
23	C₂×D₄×C₄₇	—	2	2	no	—
24	C₂×D₄×C₄₇	—	2	2	no	—
25	C₂×D₄×C₄₇	—	2	2	no	—
26	C₂×D₄×C₄₇	—	4	4	no	—
27	C₂×D₄×C₄₇	—	4	4	no	—
28	C₂×D₄×C₄₇	D₄	1	4	yes	—
29	C₂²×C₄×C₄₇	D₄	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₇	—	2	1	no	—
32	C₂×D₁₈₈	—	2	2	no	—
33	C₂×D₁₈₈	—	2	2	no	—
34	C₂×D₁₈₈	—	2	1	no	—
35	C₂×D₁₈₈	D₄	1	2	yes	complemented
36	C₄²×C₄₇	—	2	2	no	—
37	#10625334	—	2	16	no	—
38	(C₂×D₄₇)⋊C₄	—	4	4	no	—
39	C₄⋊Dic₄₇	—	2	2	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	D₄	1	4	yes	complemented
43	C₂³×C₄₇	—	2	4	no	—
44	C₂³×C₄₇	—	4	2	no	—
45	C₂³×C₄₇	—	4	2	no	—
46	C₂³×C₄₇	—	4	4	no	—
47	C₂³×C₄₇	—	4	4	no	—
48	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₄₇	—	2	2	no	—
50	C₂²×D₄₇	—	4	1	no	—
51	C₂²×D₄₇	—	4	1	no	—
52	C₂²×D₄₇	—	4	2	no	—
53	C₂²×D₄₇	—	4	2	no	—
54	D₄×C₄₇	—	2	4	no	—
55	D₄×C₄₇	—	2	4	no	—
56	D₄×C₄₇	—	4	16	no	—
57	D₄×C₄₇	—	4	2	no	—
58	D₄×C₄₇	—	4	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	D₈	1	4	yes	complemented
62	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂×Dic₄₇	—	4	1	no	—
67	C₂×Dic₄₇	—	4	2	no	—
68	D₁₈₈	—	2	4	no	—
69	D₁₈₈	—	4	2	no	—
70	C₄₇⋊C₈	—	2	4	no	—
71	C₂²×C₄₇	—	4	4	no	—
72	C₂²×C₄₇	—	4	8	no	—
73	C₂²×C₄₇	—	8	4	no	—
74	C₂²×C₄₇	—	8	2	no	—
75	C₂²×C₄₇	—	8	2	no	—
76	C₂²×C₄₇	#120	1	1	yes	characteristic
77	C₂²×C₄₇	#121	1	1	yes	characteristic
78	C₂²×C₄₇	#127	1	1	yes	characteristic
79	C₂²×C₄₇	C₂×D₈	1	2	yes	—
80	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×D₄₇	—	4	2	no	—
82	C₂×D₄₇	—	4	4	no	—
83	C₂×D₄₇	—	8	2	no	—
84	C₂×D₄₇	—	8	1	no	—
85	C₂×D₄₇	—	8	1	no	—
86	C₄×C₄₇	—	2	4	no	—
87	C₄×C₄₇	#120	1	2	yes	—
88	C₄×C₄₇	#121	1	2	yes	—
89	Dic₄₇	—	8	1	no	—
90	#2011	—	47	1	no	maximal, Sylow
91	C₂×C₄₇	—	8	4	no	—
92	C₂×C₄₇	#392	1	1	yes	characteristic
93	C₂×C₄₇	#447	1	2	yes	—
94	C₂×C₄₇	#459	1	2	yes	—
95	C₂×C₄₇	#493	1	1	yes	characteristic
96	C₂×C₄₇	#496	1	1	yes	characteristic
97	D₄₇	—	8	2	no	—
98	C₂²×D₈	—	47	2	no	—
99	C₂×#127	D₄₇	1	1	yes	characteristic, complemented
100	C₂×#102	—	47	2	no	—
101	#390	—	47	1	no	—
102	#336	—	47	1	no	—
103	C₄₇	#2011	1	1	yes	characteristic, complemented, Sylow
104	C₂²×D₄	—	2	2	no	—
105	C₂²×D₄	—	47	1	no	—
106	C₂²×D₄	C₂×D₄₇	1	2	yes	—
107	C₂×D₈	—	94	4	no	—
108	C₂×D₈	—	94	4	no	—
109	C₂×D₈	—	94	4	no	—
110	C₂²×C₈	—	47	2	no	—
111	#127	—	2	4	no	—
112	C₂×C₄⋊C₄	—	47	1	no	—
113	C₂×C₂²⋊C₄	—	94	2	no	—
114	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
115	#102	—	94	4	no	—
116	C₂⁴	—	2	2	no	—
117	C₂⁴	—	94	1	no	—
118	C₂×D₄	—	2	8	no	—
119	C₂×D₄	—	2	2	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	—	4	4	no	—
125	C₂×D₄	—	4	4	no	—
126	C₂×D₄	—	47	2	no	—
127	C₂×D₄	—	94	2	no	—
128	C₂×D₄	—	94	1	no	—
129	C₂×D₄	—	94	2	no	—
130	C₂×D₄	C₂²⋊D₄₇	1	4	yes	—
131	C₂²×C₄	—	94	1	no	—
132	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
133	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
134	D₈	—	94	16	no	—
135	C₂×C₈	—	47	4	no	—
136	C₂×C₈	—	94	2	no	—
137	C₂×C₈	—	94	2	no	—
138	C₄⋊C₄	—	94	2	no	—
139	C₂²⋊C₄	—	188	4	no	—
140	C₄²	—	2	2	no	—
141	C₂³	—	2	4	no	—
142	C₂³	—	4	2	no	—
143	C₂³	—	4	2	no	—
144	C₂³	—	4	4	no	—
145	C₂³	—	4	4	no	—
146	C₂³	—	94	2	no	—
147	C₂³	—	188	1	no	—
148	C₂³	—	188	1	no	—
149	C₂³	—	188	2	no	—
150	C₂³	—	188	2	no	—
151	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
152	D₄	—	2	4	no	—
153	D₄	—	2	4	no	—
154	D₄	—	4	16	no	—
155	D₄	—	4	2	no	—
156	D₄	—	4	2	no	—
157	D₄	—	94	4	no	—
158	D₄	—	188	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	188	1	no	—
162	C₂×C₄	—	188	2	no	—
163	C₂×C₄	#10625334	1	4	yes	complemented
164	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
165	C₂×C₄	D₄×D₄₇	1	2	yes	—
166	C₂×C₄	#10625352	1	1	yes	characteristic
167	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
168	C₈	—	94	4	no	—
169	C₂²	—	4	8	no	—
170	C₂²	—	4	4	no	—
171	C₂²	—	8	2	no	—
172	C₂²	—	8	2	no	—
173	C₂²	—	8	4	no	—
174	C₂²	—	376	1	no	—
175	C₂²	—	376	1	no	—
176	C₂²	—	376	2	no	—
177	C₂²	—	188	2	no	—
178	C₂²	—	188	4	no	—
179	C₂²	#13093993	1	1	yes	characteristic
180	C₂²	#13093995	1	1	yes	characteristic
181	C₂²	#13094020	1	1	yes	characteristic
182	C₂²	C₂×#10625334	1	2	yes	—
183	C₂²	#13094050	1	2	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	376	1	no	—
186	C₄	#13093992	1	2	yes	—
187	C₄	#13093996	1	2	yes	—
188	C₂	—	8	4	no	—
189	C₂	—	376	2	no	—
190	C₂	#15689031	1	1	yes	characteristic
191	C₂	#15689257	1	2	yes	—
192	C₂	#15689315	1	2	yes	—
193	C₂	#15689499	1	1	yes	characteristic
194	C₂	#15689510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625334

C₂

yes

maximal, complemented

C₂×#13093945

C₂

yes

maximal, complemented

#15689025

C₂

yes

maximal, characteristic, complemented

#15688878

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

—

—

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

—

#13093945

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂²

yes

—

#127×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂³×D₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

complemented

C₄²×C₄₇

—

#10625334

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

D₈

yes

complemented

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×D₈

yes

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#120

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

#2011

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#392

yes

characteristic

C₂×C₄₇

#447

yes

—

C₂×C₄₇

#459

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂²×D₈

—

C₂×#127

D₄₇

yes

characteristic, complemented

100

C₂×#102

—

101

#390

—

102

#336

—

103

C₄₇

#2011

yes

characteristic, complemented, Sylow

104

C₂²×D₄

—

105

C₂²×D₄

—

106

C₂²×D₄

C₂×D₄₇

yes

—

107

C₂×D₈

—

108

C₂×D₈

—

109

C₂×D₈

—

110

C₂²×C₈

—

111

#127

—

112

C₂×C₄⋊C₄

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

C₂×D₄₇

yes

characteristic

115

#102

—

116

C₂⁴

—

117

C₂⁴

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

C₂²⋊D₄₇

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²⋊D₄₇

yes

—

133

C₂²×C₄

C₂²×D₄₇

yes

characteristic

134

D₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

188

—

140

C₄²

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

188

—

148

C₂³

—

188

—

149

C₂³

—

188

—

150

C₂³

—

188

—

151

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

152

D₄

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

D₄

—

188

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

188

—

162

C₂×C₄

—

188

—

163

C₂×C₄

#10625334

yes

complemented

164

C₂×C₄

D₄×D₄₇

yes

characteristic

165

C₂×C₄

D₄×D₄₇

yes

—

166

C₂×C₄

#10625352

yes

characteristic

167

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

168

C₈

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

376

—

175

C₂²

—

376

—

176

C₂²

—

376

—

177

C₂²

—

188

—

178

C₂²

—

188

—

179

C₂²

#13093993

yes

characteristic

180

C₂²

#13093995

yes

characteristic

181

C₂²

#13094020

yes

characteristic

182

C₂²

C₂×#10625334

yes

—

183

C₂²

#13094050

yes

—

184

C₄

—

185

C₄

—

376

—

186

C₄

#13093992

yes

—

187

C₄

#13093996

yes

—

188

C₂

—

189

C₂

—

376

—

190

C₂

#15689031

yes

characteristic

191

C₂

#15689257

yes

—

192

C₂

#15689315

yes

—

193

C₂

#15689499

yes

characteristic

194

C₂

#15689510

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#127	D₄₇
hypocenter	C₄₇	#2011
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#127×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂²×#10625334	C₂	semidirect
2. C₂×#13093945	C₂	semidirect
3. #15689025	C₂	semidirect
4. #15688878	C₂	semidirect
5. C₂×#127×C₄₇	C₂	semidirect
35. C₂×D₁₈₈	D₄	semidirect
42. C₂×C₄₇⋊C₈	D₄	semidirect
61. C₂×C₄×C₄₇	D₈	semidirect
99. C₂×#127	D₄₇	semidirect
103. C₄₇	#2011	semidirect
163. C₂×C₄	#10625334	semidirect

H ≅ G/N

type

1. C₂²×#10625334

C₂

semidirect

2. C₂×#13093945

C₂

semidirect

3. #15689025

C₂

semidirect

4. #15688878

C₂

semidirect

5. C₂×#127×C₄₇

C₂

semidirect

35. C₂×D₁₈₈

D₄

semidirect

42. C₂×C₄₇⋊C₈

D₄

semidirect

61. C₂×C₄×C₄₇

D₈

semidirect

99. C₂×#127

D₄₇

semidirect

103. C₄₇

#2011

semidirect

163. C₂×C₄

#10625334

semidirect

Subgroups — #20244459

Conjugacy classes of subgroups

Complements