Group #20254834 — subgroups

Subgroups — #20254834

Order 6016.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#15689735	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689427	C₂	1	2	yes	maximal, complemented
3	#15689423	C₂	1	2	yes	maximal, complemented
4	#15689367	C₂	1	2	yes	maximal, complemented
5	#15689350	C₂	1	2	yes	maximal, complemented
6	#15689328	C₂	1	2	yes	maximal, complemented
7	#15689320	C₂	1	2	yes	maximal, complemented
8	#15689150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#540	C₂	1	1	yes	maximal, characteristic
10	C₂×#10625335	C₂²	1	2	yes	—
11	C₂×#10625335	C₂²	1	2	yes	—
12	C₂×#10625329	C₂²	1	1	yes	characteristic
13	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#13094024	C₂²	1	1	yes	characteristic
15	#13094017	—	2	2	no	—
16	#13094016	C₂²	1	1	yes	characteristic
17	#13094014	—	2	2	no	—
18	#13094003	C₂²	1	2	yes	—
19	C₄×Dic₉₄	C₂²	1	2	yes	—
20	#13093957	C₂²	1	2	yes	—
21	#13093955	C₂²	1	2	yes	—
22	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
23	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
24	#13093947	C₂²	1	2	yes	—
25	#13093947	C₂²	1	2	yes	—
26	#13093945	C₂²	1	2	yes	—
27	#13093945	C₂²	1	2	yes	—
28	#127×C₄₇	C₂²	1	1	yes	characteristic
29	#125×C₄₇	C₂²	1	1	yes	characteristic
30	#123×C₄₇	—	2	2	no	—
31	#121×C₄₇	—	2	2	no	—
32	#121×C₄₇	C₂²	1	2	yes	—
33	C₄×D₄×C₄₇	C₂²	1	2	yes	—
34	#117×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×D₄×C₄₇	—	2	2	no	—
36	C₂×D₄×C₄₇	—	2	2	no	—
37	C₂×D₄×C₄₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×Dic₉₄	C₂³	1	2	yes	—
41	C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
43	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
44	C₂²⋊C₄×C₄₇	—	2	2	no	—
45	C₂²⋊C₄×C₄₇	—	2	2	no	—
46	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
47	C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₄₇	—	2	2	no	—
50	#10625335	—	4	2	no	—
51	#10625335	—	4	2	no	—
52	C₄⋊Dic₄₇	—	2	2	no	—
53	C₄⋊Dic₄₇	C₂³	1	2	yes	—
54	Dic₄₇⋊C₄	—	2	2	no	—
55	Dic₄₇⋊C₄	—	2	2	no	—
56	Dic₄₇⋊C₄	—	2	2	no	—
57	C₄×Dic₄₇	—	2	2	no	—
58	#10625329	—	4	1	no	—
59	C₂×C₄₇⋊C₈	—	4	1	no	—
60	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
61	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
62	C₂³×C₄₇	—	2	2	no	—
63	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
64	D₄×C₄₇	—	2	2	no	—
65	D₄×C₄₇	—	2	2	no	—
66	D₄×C₄₇	—	8	2	no	—
67	D₄×C₄₇	—	8	2	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	8	2	no	—
72	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂×Dic₄₇	—	2	2	no	—
77	C₂×Dic₄₇	—	2	2	no	—
78	Dic₉₄	—	2	2	no	—
79	Dic₉₄	—	2	2	no	—
80	C₄₇⋊C₈	—	2	2	no	—
81	C₄₇⋊C₈	—	2	2	no	—
82	C₂²×C₄₇	—	2	1	no	—
83	C₂²×C₄₇	—	2	2	no	—
84	C₂²×C₄₇	—	4	2	no	—
85	C₂²×C₄₇	—	4	2	no	—
86	C₂²×C₄₇	—	4	2	no	—
87	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₄×C₄₇	—	2	1	no	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
93	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
94	Dic₄₇	—	4	2	no	—
95	Dic₄₇	—	4	2	no	—
96	#3260	—	47	1	no	maximal, Sylow
97	C₂×C₄₇	—	4	1	no	—
98	C₂×C₄₇	—	8	2	no	—
99	C₂×C₄₇	#537	1	1	yes	characteristic
100	C₂×C₄₇	#577	1	2	yes	—
101	#541	—	47	1	no	—
102	#540	D₄₇	1	1	yes	characteristic
103	#479	—	47	2	no	—
104	#478	—	47	2	no	—
105	#468	—	47	2	no	—
106	#465	—	47	2	no	—
107	#461	—	47	2	no	—
108	#460	—	47	2	no	—
109	#424	—	47	1	no	—
110	C₄₇	#3260	1	1	yes	characteristic, complemented, Sylow
111	C₂×SD₁₆	—	47	2	no	—
112	C₂×SD₁₆	—	47	2	no	—
113	C₂×M₁₆	—	47	1	no	—
114	C₂²×C₈	—	47	1	no	—
115	C₄⋊Q₈	—	47	1	no	—
116	#127	C₂×D₄₇	1	1	yes	characteristic
117	#126	—	94	2	no	—
118	#125	—	47	1	no	—
119	#125	—	94	2	no	—
120	#125	C₂×D₄₇	1	1	yes	characteristic
121	#123	—	2	2	no	—
122	C₂²⋊Q₈	—	47	2	no	—
123	#121	—	2	2	no	—
124	#121	C₂×D₄₇	1	2	yes	—
125	C₄×Q₈	—	47	2	no	—
126	C₄×D₄	C₂×D₄₇	1	2	yes	—
127	#117	C₂×D₄₇	1	1	yes	characteristic
128	#107	—	47	2	no	—
129	#106	—	47	2	no	—
130	C₄⋊C₈	—	47	2	no	—
131	C₄⋊C₈	—	47	2	no	—
132	#103	—	47	2	no	—
133	#103	—	47	2	no	—
134	#102	—	47	2	no	—
135	#102	—	47	2	no	—
136	C₂×Q₈	—	47	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	C₂²×D₄₇	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
142	SD₁₆	—	188	2	no	—
143	SD₁₆	—	188	2	no	—
144	M₁₆	—	188	1	no	—
145	C₂×C₈	—	47	2	no	—
146	C₂×C₈	—	47	2	no	—
147	C₂×C₈	—	188	1	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	47	2	no	—
150	C₄⋊C₄	—	94	2	no	—
151	C₄⋊C₄	—	94	2	no	—
152	C₄⋊C₄	—	94	2	no	—
153	C₄⋊C₄	—	94	2	no	—
154	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
155	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	94	2	no	—
159	C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
160	C₄²	—	94	2	no	—
161	C₄²	C₂²×D₄₇	1	1	yes	characteristic
162	C₄²	C₂²×D₄₇	1	1	yes	characteristic
163	C₂³	—	2	2	no	—
164	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
165	Q₈	—	94	2	no	—
166	Q₈	—	94	2	no	—
167	D₄	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	8	2	no	—
170	D₄	—	8	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	94	2	no	—
176	C₂×C₄	—	94	2	no	—
177	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
181	C₈	—	94	2	no	—
182	C₈	—	94	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	—	188	2	no	—
194	C₄	—	188	2	no	—
195	C₄	#13094070	1	1	yes	characteristic
196	C₄	#13094071	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	8	2	no	—
199	C₂	#15689701	1	1	yes	characteristic
200	C₂	#15689934	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689735

C₂

yes

maximal, characteristic, complemented

#15689427

C₂

yes

maximal, complemented

#15689423

C₂

yes

maximal, complemented

#15689367

C₂

yes

maximal, complemented

#15689350

C₂

yes

maximal, complemented

#15689328

C₂

yes

maximal, complemented

#15689320

C₂

yes

maximal, complemented

#15689150

C₂

yes

maximal, characteristic, complemented

C₄₇×#540

C₂

yes

maximal, characteristic

C₂×#10625335

C₂²

yes

—

C₂×#10625335

C₂²

yes

—

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

#13094024

C₂²

yes

characteristic

#13094017

—

#13094016

C₂²

yes

characteristic

#13094014

—

#13094003

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

#13093957

C₂²

yes

—

#13093955

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093947

C₂²

yes

—

#13093947

C₂²

yes

—

#13093945

C₂²

yes

—

#13093945

C₂²

yes

—

#127×C₄₇

C₂²

yes

characteristic

#125×C₄₇

C₂²

yes

characteristic

#123×C₄₇

—

#121×C₄₇

—

#121×C₄₇

C₂²

yes

—

C₄×D₄×C₄₇

C₂²

yes

—

#117×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

#10625335

—

#10625335

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#3260

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#537

yes

characteristic

100

C₂×C₄₇

#577

yes

—

101

#541

—

102

#540

D₄₇

yes

characteristic

103

#479

—

104

#478

—

105

#468

—

106

#465

—

107

#461

—

108

#460

—

109

#424

—

110

C₄₇

#3260

yes

characteristic, complemented, Sylow

111

C₂×SD₁₆

—

112

C₂×SD₁₆

—

113

C₂×M₁₆

—

114

C₂²×C₈

—

115

C₄⋊Q₈

—

116

#127

C₂×D₄₇

yes

characteristic

117

#126

—

118

#125

—

119

#125

—

120

#125

C₂×D₄₇

yes

characteristic

121

#123

—

122

C₂²⋊Q₈

—

123

#121

—

124

#121

C₂×D₄₇

yes

—

125

C₄×Q₈

—

126

C₄×D₄

C₂×D₄₇

yes

—

127

#117

C₂×D₄₇

yes

characteristic

128

#107

—

129

#106

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

#103

—

134

#102

—

135

#102

—

136

C₂×Q₈

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

C₂²×D₄₇

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²×D₄₇

yes

characteristic

142

SD₁₆

—

188

—

143

SD₁₆

—

188

—

144

M₁₆

—

188

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

188

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

C₂²⋊D₄₇

yes

—

155

C₄⋊C₄

C₂²×D₄₇

yes

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

C₂²⋊D₄₇

yes

—

160

C₄²

—

161

C₄²

C₂²×D₄₇

yes

characteristic

162

C₄²

C₂²×D₄₇

yes

characteristic

163

C₂³

—

164

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

165

Q₈

—

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

178

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

179

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

180

C₂×C₄

C₂³×D₄₇

yes

characteristic

181

C₈

—

182

C₈

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

188

—

194

C₄

—

188

—

195

C₄

#13094070

yes

characteristic

196

C₄

#13094071

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#15689701

yes

characteristic

200

C₂

#15689934

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#540	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#540	D₄₇
hypocenter	C₄₇	#3260
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #540

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#540

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #540

N	H ≅ G/N	type
1. #15689735	C₂	semidirect
2. #15689427	C₂	semidirect
3. #15689423	C₂	semidirect
4. #15689367	C₂	semidirect
5. #15689350	C₂	semidirect
6. #15689328	C₂	semidirect
7. #15689320	C₂	semidirect
8. #15689150	C₂	semidirect
110. C₄₇	#3260	semidirect

H ≅ G/N

type

1. #15689735

C₂

semidirect

2. #15689427

C₂

semidirect

3. #15689423

C₂

semidirect

4. #15689367

C₂

semidirect

5. #15689350

C₂

semidirect

6. #15689328

C₂

semidirect

7. #15689320

C₂

semidirect

8. #15689150

C₂

semidirect

110. C₄₇

#3260

semidirect

Subgroups — #20254834

Conjugacy classes of subgroups

Complements