Group #20256465 — subgroups

Subgroups — #20256465

Order 6016.

356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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	#15689845	C₂	1	2	yes	maximal, complemented
2	#15689516	C₂	1	2	yes	maximal, complemented
3	#15689507	C₂	1	1	yes	maximal, characteristic
4	#15689345	C₂	1	2	yes	maximal, complemented
5	#15689341	C₂	1	2	yes	maximal
6	#15689332	C₂	1	2	yes	maximal, complemented
7	#15689253	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689222	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	C₂×#10625337	C₂²	1	2	yes	—
12	C₂×#10625325	C₂²	1	2	yes	—
13	C₂×#10625336	C₂²	1	2	yes	—
14	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
15	#13094022	C₂²	1	2	yes	—
16	#13094019	—	2	2	no	—
17	#13094018	—	2	2	no	—
18	#13094013	C₂²	1	2	yes	—
19	#13094012	C₂²	1	2	yes	—
20	#13094002	—	2	2	no	—
21	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
22	#13093989	C₂²	1	2	yes	—
23	#13093986	C₂²	1	2	yes	—
24	#13093981	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
26	Dic₄₇⋊C₈	C₂²	1	2	yes	—
27	#13093947	C₂²	1	2	yes	—
28	#13093944	C₂²	1	2	yes	—
29	#13093934	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
32	#107×C₄₇	C₂²	1	1	yes	characteristic
33	#103×C₄₇	C₂²	1	2	yes	—
34	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
36	C₂×D₁₈₈	C₂³	1	2	yes	—
37	C₂×C₄×D₄₇	—	2	2	no	—
38	C₂×Dic₉₄	D₄	1	1	yes	characteristic
39	C₂×Dic₉₄	C₂³	1	2	yes	—
40	Q₁₆×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
45	C₄²×C₄₇	—	2	2	no	—
46	C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#10625337	—	4	2	no	—
48	#10625336	—	4	2	no	—
49	(C₂×D₄₇)⋊C₄	—	2	2	no	—
50	(C₂×D₄₇)⋊C₄	—	2	2	no	—
51	(C₂×D₄₇)⋊C₄	—	2	2	no	—
52	(C₂×D₄₇)⋊C₄	—	2	2	no	—
53	(C₂×D₄₇)⋊C₄	—	2	2	no	—
54	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
55	Dic₄₇⋊C₄	—	2	2	no	—
56	Dic₄₇⋊C₄	—	2	2	no	—
57	Dic₄₇⋊C₄	—	2	2	no	—
58	Dic₄₇⋊C₄	D₄	1	2	yes	—
59	C₄×Dic₄₇	—	2	2	no	—
60	C₄×Dic₄₇	C₂³	1	2	yes	—
61	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
62	Dic₁₈₈	—	4	1	no	—
63	D₃₇₆	—	4	1	no	—
64	#10625325	—	2	4	no	—
65	C₂²×D₄₇	—	2	2	no	—
66	Q₈×C₄₇	—	2	2	no	—
67	Q₈×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
75	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
76	D₁₈₈	—	2	2	no	—
77	D₁₈₈	—	2	2	no	—
78	C₄×D₄₇	—	4	4	no	—
79	Dic₉₄	—	2	2	no	—
80	Dic₉₄	—	2	2	no	—
81	Dic₉₄	—	4	1	no	—
82	C₈×C₄₇	—	2	1	no	—
83	C₈×C₄₇	#41	1	2	yes	—
84	C₄₇⋊C₈	—	2	2	no	—
85	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×D₄₇	—	4	2	no	—
87	C₂×D₄₇	—	4	2	no	—
88	C₂×D₄₇	—	4	2	no	—
89	C₄×C₄₇	—	2	1	no	—
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	—	4	2	no	—
93	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
94	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
95	Dic₄₇	—	2	2	no	—
96	Dic₄₇	—	2	2	no	—
97	Dic₄₇	—	4	2	no	—
98	#3402	—	47	1	no	maximal, Sylow
99	C₂×C₄₇	#550	1	1	yes	characteristic
100	C₂×C₄₇	#575	1	1	yes	characteristic
101	C₂×C₄₇	#578	1	1	yes	characteristic
102	D₄₇	—	8	2	no	—
103	#556	—	47	2	no	—
104	#497	—	47	2	no	—
105	#495	—	47	1	no	—
106	#464	—	47	2	no	—
107	#463	—	47	2	no	—
108	#462	—	47	2	no	—
109	#446	—	47	1	no	—
110	#440	—	47	2	no	—
111	C₄×Q₁₆	D₄₇	1	1	yes	characteristic, complemented
112	C₄₇	#3402	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₁₆	—	47	2	no	—
114	C₂×Q₁₆	—	47	1	no	—
115	C₂×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
116	C₂×SD₁₆	—	47	2	no	—
117	C₂×SD₁₆	—	47	2	no	—
118	C₂×D₈	—	47	1	no	—
119	C₄⋊Q₈	—	47	2	no	—
120	#126	—	94	2	no	—
121	#126	—	94	2	no	—
122	#124	—	47	2	no	—
123	#124	—	47	2	no	—
124	C₂²⋊Q₈	—	94	2	no	—
125	C₄×Q₈	—	47	1	no	—
126	C₄×Q₈	—	47	2	no	—
127	C₄×Q₈	C₂×D₄₇	1	2	yes	—
128	C₄×D₄	—	47	2	no	—
129	#117	—	94	2	no	—
130	#107	C₂×D₄₇	1	1	yes	characteristic
131	C₄⋊C₈	—	47	2	no	—
132	C₄⋊C₈	—	47	1	no	—
133	#103	—	47	2	no	—
134	#103	C₂×D₄₇	1	2	yes	—
135	#102	—	47	2	no	—
136	#97	—	47	2	no	—
137	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
138	C₂×Q₈	—	47	1	no	—
139	C₂×Q₈	—	47	2	no	—
140	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
141	C₂×D₄	—	47	2	no	—
142	C₂²×C₄	—	94	2	no	—
143	Q₁₆	—	2	2	no	—
144	Q₁₆	—	188	1	no	—
145	Q₁₆	—	188	2	no	—
146	SD₁₆	—	94	4	no	—
147	SD₁₆	—	188	2	no	—
148	D₈	—	188	1	no	—
149	C₂×C₈	—	47	2	no	—
150	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	47	2	no	—
154	C₄⋊C₄	—	47	1	no	—
155	C₄⋊C₄	—	94	2	no	—
156	C₄⋊C₄	—	94	2	no	—
157	C₄⋊C₄	—	94	2	no	—
158	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
159	C₂²⋊C₄	—	94	2	no	—
160	C₂²⋊C₄	—	94	2	no	—
161	C₂²⋊C₄	—	94	2	no	—
162	C₂²⋊C₄	—	94	2	no	—
163	C₂²⋊C₄	—	94	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	47	2	no	—
166	C₄²	—	94	2	no	—
167	C₄²	C₂²×D₄₇	1	1	yes	characteristic
168	C₂³	—	94	2	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	94	2	no	—
172	Q₈	—	94	2	no	—
173	Q₈	—	188	1	no	—
174	D₄	—	94	2	no	—
175	D₄	—	94	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	47	2	no	—
179	C₂×C₄	—	47	2	no	—
180	C₂×C₄	—	94	2	no	—
181	C₂×C₄	—	188	4	no	—
182	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
183	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	94	2	no	—
187	C₈	#10625350	1	2	yes	—
188	C₂²	—	188	2	no	—
189	C₂²	—	188	2	no	—
190	C₂²	—	188	2	no	—
191	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	94	2	no	—
197	C₄	—	94	2	no	—
198	C₄	—	188	2	no	—
199	C₄	C₂×#10625350	1	1	yes	characteristic
200	C₄	#13094070	1	1	yes	characteristic
201	C₂	—	376	2	no	—
202	C₂	#15689794	1	1	yes	characteristic
203	C₂	#15689920	1	1	yes	characteristic
204	C₂	#15689936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689845

C₂

yes

maximal, complemented

#15689516

C₂

yes

maximal, complemented

#15689507

C₂

yes

maximal, characteristic

#15689345

C₂

yes

maximal, complemented

#15689341

C₂

yes

maximal

#15689332

C₂

yes

maximal, complemented

#15689253

C₂

yes

maximal, characteristic, complemented

#15689222

C₂

yes

maximal, complemented

C₄×Q₁₆×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₈₈

C₂²

yes

characteristic

C₂×#10625337

C₂²

yes

—

C₂×#10625325

C₂²

yes

—

C₂×#10625336

C₂²

yes

—

C₂×D₃₇₆

C₂²

yes

characteristic

#13094022

C₂²

yes

—

#13094019

—

#13094018

—

#13094013

C₂²

yes

—

#13094012

C₂²

yes

—

#13094002

—

C₄×Dic₉₄

C₂²

yes

characteristic

#13093989

C₂²

yes

—

#13093986

C₂²

yes

—

#13093981

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₂²

yes

—

#13093947

C₂²

yes

—

#13093944

C₂²

yes

—

#13093934

C₂²

yes

—

C₂×Q₁₆×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

characteristic

#103×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

D₄

yes

characteristic

C₂×Dic₉₄

C₂³

yes

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

#10625336

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

characteristic

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

Dic₁₈₈

—

D₃₇₆

—

#10625325

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#41

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

100

C₂×C₄₇

#575

yes

characteristic

101

C₂×C₄₇

#578

yes

characteristic

102

D₄₇

—

103

#556

—

104

#497

—

105

#495

—

106

#464

—

107

#463

—

108

#462

—

109

#446

—

110

#440

—

111

C₄×Q₁₆

D₄₇

yes

characteristic, complemented

112

C₄₇

#3402

yes

characteristic, complemented, Sylow

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

C₂×D₄₇

yes

characteristic

116

C₂×SD₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₄⋊Q₈

—

120

#126

—

121

#126

—

122

#124

—

123

#124

—

124

C₂²⋊Q₈

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

C₂×D₄₇

yes

—

128

C₄×D₄

—

129

#117

—

130

#107

C₂×D₄₇

yes

characteristic

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

#103

—

134

#103

C₂×D₄₇

yes

—

135

#102

—

136

#97

—

137

C₄×C₈

C₂×D₄₇

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

C₂²×D₄₇

yes

—

141

C₂×D₄

—

142

C₂²×C₄

—

143

Q₁₆

—

144

Q₁₆

—

188

—

145

Q₁₆

—

188

—

146

SD₁₆

—

147

SD₁₆

—

188

—

148

D₈

—

188

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₄₇

yes

characteristic

151

C₂×C₈

C₂²×D₄₇

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

C₂²×D₄₇

yes

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

C₂²×D₄₇

yes

characteristic

168

C₂³

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

188

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

188

—

182

C₂×C₄

D₄×D₄₇

yes

characteristic

183

C₂×C₄

D₄×D₄₇

yes

characteristic

184

C₂×C₄

C₂³×D₄₇

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

#10625350

yes

—

188

C₂²

—

188

—

189

C₂²

—

188

—

190

C₂²

—

188

—

191

C₂²

C₂×D₄×D₄₇

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

188

—

199

C₄

C₂×#10625350

yes

characteristic

200

C₄

#13094070

yes

characteristic

201

C₂

—

376

—

202

C₂

#15689794

yes

characteristic

203

C₂

#15689920

yes

characteristic

204

C₂

#15689936

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄×Q₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#3402
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄×Q₁₆×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

hypercenter

C₄×Q₁₆

D₄₇

hypocenter

C₄₇

#3402

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
1. #15689845	C₂	semidirect
2. #15689516	C₂	semidirect
4. #15689345	C₂	semidirect
6. #15689332	C₂	semidirect
7. #15689253	C₂	semidirect
8. #15689222	C₂	semidirect
9. C₄×Q₁₆×C₄₇	C₂	semidirect
111. C₄×Q₁₆	D₄₇	semidirect
112. C₄₇	#3402	semidirect

H ≅ G/N

type

1. #15689845

C₂

semidirect

2. #15689516

C₂

semidirect

4. #15689345

C₂

semidirect

6. #15689332

C₂

semidirect

7. #15689253

C₂

semidirect

8. #15689222

C₂

semidirect

9. C₄×Q₁₆×C₄₇

C₂

semidirect

111. C₄×Q₁₆

D₄₇

semidirect

112. C₄₇

#3402

semidirect

Subgroups — #20256465

Conjugacy classes of subgroups

Complements