Group #20244909 — subgroups

Subgroups — #20244909

Order 6016.

274 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 85 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×#13093957	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093955	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13093945	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093945	C₂	1	1	yes	maximal, characteristic
5	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688878	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#390	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	#13093957	—	2	2	no	—
13	#13093955	—	2	2	no	—
14	#13093945	—	2	2	no	—
15	#13093945	—	2	2	no	—
16	C₄₇⋊#95	—	2	2	no	—
17	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄₇	—	2	1	no	—
22	C₂×D₄×C₄₇	—	2	1	no	—
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₄×C₄₇	—	2	1	no	—
27	C₂×D₄×C₄₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₄₇	—	2	1	no	—
29	C₂²×C₄×C₄₇	D₄	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₇	—	2	1	no	—
32	C₂²×Dic₄₇	—	2	1	no	—
33	C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂²⋊C₄×C₄₇	—	4	4	no	—
36	C₄²×C₄₇	—	2	1	no	—
37	C₄²×C₄₇	—	2	1	no	—
38	C₄⋊Dic₄₇	—	2	1	no	—
39	C₄⋊Dic₄₇	—	2	1	no	—
40	C₄⋊Dic₄₇	—	2	1	no	—
41	C₄⋊Dic₄₇	—	2	1	no	—
42	C₂×C₄₇⋊C₈	—	2	1	no	—
43	C₂×C₄₇⋊C₈	—	2	1	no	—
44	C₂×C₄₇⋊C₈	—	2	1	no	—
45	C₂×C₄₇⋊C₈	—	2	1	no	—
46	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
47	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	4	1	no	—
50	C₂³×C₄₇	—	4	1	no	—
51	C₂³×C₄₇	—	4	2	no	—
52	C₂³×C₄₇	—	4	2	no	—
53	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
54	D₄×C₄₇	—	2	2	no	—
55	D₄×C₄₇	—	2	2	no	—
56	D₄×C₄₇	—	4	1	no	—
57	D₄×C₄₇	—	4	1	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	4	1	no	—
62	C₂×C₄×C₄₇	—	4	2	no	—
63	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂×Dic₄₇	—	4	1	no	—
70	C₂×Dic₄₇	—	4	1	no	—
71	C₂×Dic₄₇	—	4	2	no	—
72	C₂×Dic₄₇	—	4	2	no	—
73	C₄₇⋊C₈	—	2	2	no	—
74	C₄₇⋊C₈	—	2	2	no	—
75	C₂²×C₄₇	—	4	2	no	—
76	C₂²×C₄₇	—	4	4	no	—
77	C₂²×C₄₇	—	8	2	no	—
78	C₂²×C₄₇	—	8	1	no	—
79	C₂²×C₄₇	—	8	1	no	—
80	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂²×C₄₇	#123	1	1	yes	characteristic
82	C₂²×C₄₇	#127	1	1	yes	characteristic
83	C₂²×C₄₇	#135	1	2	yes	—
84	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
85	C₄×C₄₇	—	4	2	no	—
86	C₄×C₄₇	—	8	1	no	—
87	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
89	C₄×C₄₇	#123	1	1	yes	characteristic
90	C₄×C₄₇	#123	1	1	yes	characteristic
91	Dic₄₇	—	8	1	no	—
92	Dic₄₇	—	8	1	no	—
93	#2092	—	47	1	no	maximal, Sylow
94	C₂×C₄₇	—	8	2	no	—
95	C₂×C₄₇	#399	1	1	yes	characteristic
96	C₂×C₄₇	#478	1	2	yes	—
97	C₂×C₄₇	#482	1	2	yes	—
98	C₂×C₄₇	#495	1	1	yes	characteristic
99	C₂×C₄₇	#496	1	1	yes	characteristic
100	C₂×#107	—	47	1	no	—
101	C₂×#106	—	47	1	no	—
102	C₂×#102	—	47	1	no	—
103	C₂×#102	—	47	1	no	—
104	#390	D₄₇	1	1	yes	characteristic
105	#382	—	47	1	no	—
106	#336	—	47	1	no	—
107	C₄₇	#2092	1	1	yes	characteristic, complemented, Sylow
108	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
109	C₂²×C₈	—	47	1	no	—
110	C₂²×C₈	—	47	1	no	—
111	C₂×C₄⋊C₄	—	47	1	no	—
112	C₂×C₄⋊C₄	—	47	1	no	—
113	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
114	C₂×C₂²⋊C₄	—	2	2	no	—
115	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
116	#107	—	94	2	no	—
117	#106	—	94	2	no	—
118	#102	—	94	2	no	—
119	#102	—	94	2	no	—
120	#95	—	94	2	no	—
121	C₂⁴	—	2	1	no	—
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	—	2	1	no	—
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	Dic₉₄	1	2	yes	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	94	1	no	—
130	C₂²×C₄	—	94	1	no	—
131	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
132	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
133	C₂×C₈	—	47	2	no	—
134	C₂×C₈	—	47	2	no	—
135	C₂×C₈	—	94	1	no	—
136	C₂×C₈	—	94	1	no	—
137	C₂×C₈	—	94	1	no	—
138	C₂×C₈	—	94	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	94	1	no	—
142	C₄⋊C₄	—	94	1	no	—
143	C₄⋊C₄	—	94	1	no	—
144	C₄⋊C₄	—	94	1	no	—
145	C₂²⋊C₄	—	4	4	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₂³	—	2	2	no	—
149	C₂³	—	4	1	no	—
150	C₂³	—	4	1	no	—
151	C₂³	—	4	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
154	D₄	—	2	2	no	—
155	D₄	—	2	2	no	—
156	D₄	—	4	1	no	—
157	D₄	—	4	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	188	1	no	—
164	C₂×C₄	—	188	1	no	—
165	C₂×C₄	—	188	2	no	—
166	C₂×C₄	—	188	2	no	—
167	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
168	C₂×C₄	#10625350	1	1	yes	characteristic
169	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
170	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
171	C₂×C₄	#10625352	1	1	yes	characteristic
172	C₂×C₄	#10625352	1	1	yes	characteristic
173	C₈	—	94	2	no	—
174	C₈	—	94	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	4	no	—
177	C₂²	—	8	1	no	—
178	C₂²	—	8	1	no	—
179	C₂²	—	8	2	no	—
180	C₂²	#13094003	1	1	yes	characteristic
181	C₂²	#13094005	1	1	yes	characteristic
182	C₂²	#13094020	1	1	yes	characteristic
183	C₂²	#13094044	1	2	yes	—
184	C₂²	#13094050	1	2	yes	—
185	C₄	—	4	2	no	—
186	C₄	—	8	1	no	—
187	C₄	—	376	1	no	—
188	C₄	—	376	1	no	—
189	C₄	#13094001	1	1	yes	characteristic
190	C₄	#13094001	1	1	yes	characteristic
191	C₄	#13094006	1	1	yes	characteristic
192	C₄	#13094006	1	1	yes	characteristic
193	C₂	—	8	2	no	—
194	C₂	#15689059	1	1	yes	characteristic
195	C₂	#15689423	1	2	yes	—
196	C₂	#15689444	1	2	yes	—
197	C₂	#15689505	1	1	yes	characteristic
198	C₂	#15689510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093957

C₂

yes

maximal, characteristic, complemented

C₂×#13093955

C₂

yes

maximal, characteristic, complemented

C₂×#13093945

C₂

yes

maximal, characteristic

C₂×#13093945

C₂

yes

maximal, characteristic

#15688989

C₂

yes

maximal, characteristic, complemented

#15688878

C₂

yes

maximal, characteristic, complemented

C₄₇×#390

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

—

—

—

C₄₇⋊#95

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

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×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²⋊Q₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#2092

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#399

yes

characteristic

C₂×C₄₇

#478

yes

—

C₂×C₄₇

#482

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

100

—

101

C₂×#106

—

102

C₂×#102

—

103

C₂×#102

—

104

#390

D₄₇

yes

characteristic

105

#382

—

106

#336

—

107

C₄₇

#2092

yes

characteristic, complemented, Sylow

108

C₂²×D₄

C₂×D₄₇

yes

characteristic

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

114

C₂×C₂²⋊C₄

—

115

C₂×C₄²

C₂×D₄₇

yes

characteristic

116

#107

—

117

#106

—

118

#102

—

119

#102

—

120

#95

—

121

C₂⁴

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

Dic₉₄

yes

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂²⋊D₄₇

yes

—

132

C₂²×C₄

C₂²×D₄₇

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

188

—

164

C₂×C₄

—

188

—

165

C₂×C₄

—

188

—

166

C₂×C₄

—

188

—

167

C₂×C₄

C₂×Dic₉₄

yes

characteristic

168

C₂×C₄

#10625350

yes

characteristic

169

C₂×C₄

D₄×D₄₇

yes

characteristic

170

C₂×C₄

D₄×D₄₇

yes

characteristic

171

C₂×C₄

#10625352

yes

characteristic

172

C₂×C₄

#10625352

yes

characteristic

173

C₈

—

174

C₈

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

#13094003

yes

characteristic

181

C₂²

#13094005

yes

characteristic

182

C₂²

#13094020

yes

characteristic

183

C₂²

#13094044

yes

—

184

C₂²

#13094050

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

376

—

188

C₄

—

376

—

189

C₄

#13094001

yes

characteristic

190

C₄

#13094001

yes

characteristic

191

C₄

#13094006

yes

characteristic

192

C₄

#13094006

yes

characteristic

193

C₂

—

194

C₂

#15689059

yes

characteristic

195

C₂

#15689423

yes

—

196

C₂

#15689444

yes

—

197

C₂

#15689505

yes

characteristic

198

C₂

#15689510

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#390	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#390	D₄₇
hypocenter	C₄₇	#2092
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#390

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#13093957	C₂	semidirect
2. C₂×#13093955	C₂	semidirect
5. #15688989	C₂	semidirect
6. #15688878	C₂	semidirect
107. C₄₇	#2092	semidirect

H ≅ G/N

type

1. C₂×#13093957

C₂

semidirect

2. C₂×#13093955

C₂

semidirect

5. #15688989

C₂

semidirect

6. #15688878

C₂

semidirect

107. C₄₇

#2092

semidirect

Subgroups — #20244909

Conjugacy classes of subgroups

Complements