Group #20243970 — subgroups

Subgroups — #20243970

Order 6016.

230 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13093955	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688885	C₂	1	1	yes	maximal, characteristic
4	#15688885	C₂	1	1	yes	maximal, characteristic
5	#15688885	C₂	1	1	yes	maximal, characteristic
6	#15688878	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#389	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	C₂×C₄×Dic₄₇	—	2	1	no	—
13	#13093955	C₄	1	4	yes	complemented
14	C₄₇⋊#95	—	2	1	no	—
15	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	#95×C₄₇	—	2	1	no	—
20	C₂²×C₄×C₄₇	—	2	1	no	—
21	C₂²×C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	1	no	—
26	C₂²×Dic₄₇	—	2	1	no	—
27	C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	—	4	2	no	—
31	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₄²×C₄₇	—	2	1	no	—
33	C₄²×C₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₄⋊Dic₄₇	—	2	1	no	—
36	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₄₇	Q₈	1	2	yes	—
38	C₄×Dic₄₇	—	4	2	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	—	2	1	no	—
41	C₂×C₄₇⋊C₈	—	2	1	no	—
42	C₂×C₄₇⋊C₈	—	2	1	no	—
43	C₂×C₄₇⋊C₈	—	2	1	no	—
44	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
45	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	4	1	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	—	4	1	no	—
55	C₂×C₄×C₄₇	—	4	1	no	—
56	C₂×C₄×C₄₇	—	4	1	no	—
57	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
58	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂×Dic₄₇	—	2	2	no	—
65	C₂×Dic₄₇	—	2	2	no	—
66	C₂×Dic₄₇	—	4	1	no	—
67	C₂×Dic₄₇	—	4	1	no	—
68	C₂×Dic₄₇	—	4	1	no	—
69	C₂×Dic₄₇	—	4	1	no	—
70	C₄₇⋊C₈	—	2	2	no	—
71	C₄₇⋊C₈	—	4	1	no	—
72	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₄₇	#123	1	1	yes	characteristic
75	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂²×C₄₇	#135	1	1	yes	characteristic
77	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₄₇	#137	1	1	yes	characteristic
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	—	4	2	no	—
81	C₄×C₄₇	—	4	1	no	—
82	C₄×C₄₇	—	8	1	no	—
83	C₄×C₄₇	#117	1	1	yes	characteristic
84	C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
85	C₄×C₄₇	#125	1	1	yes	characteristic
86	C₄×C₄₇	#126	1	1	yes	characteristic
87	Dic₄₇	—	4	2	no	—
88	Dic₄₇	—	4	2	no	—
89	#1920	—	47	1	no	maximal, Sylow
90	C₂×C₄₇	#387	1	1	yes	characteristic
91	C₂×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
92	C₂×C₄₇	#440	1	1	yes	characteristic
93	C₂×C₄₇	#476	1	1	yes	characteristic
94	C₂×C₄₇	#479	1	1	yes	characteristic
95	C₂×C₄₇	#482	1	1	yes	characteristic
96	C₂×C₄₇	#483	1	1	yes	characteristic
97	C₂×#106	—	47	1	no	—
98	#389	D₄₇	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	47	1	no	—
100	#340	—	47	1	no	—
101	#340	—	47	1	no	—
102	#340	—	47	1	no	—
103	#336	—	47	1	no	—
104	C₄₇	#1920	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	47	1	no	—
106	C₂²×C₈	—	47	1	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	47	1	no	—
109	C₂×C₄⋊C₄	—	47	1	no	—
110	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
112	C₂×C₄²	—	94	1	no	—
113	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
114	#106	—	47	4	no	—
115	#95	—	2	1	no	—
116	#95	—	94	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	94	1	no	—
120	C₂²×C₄	—	94	1	no	—
121	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
124	C₂×C₈	—	47	2	no	—
125	C₂×C₈	—	94	1	no	—
126	C₂×C₈	—	94	1	no	—
127	C₂×C₈	—	94	1	no	—
128	C₂×C₈	—	94	1	no	—
129	C₂×C₈	—	94	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	47	2	no	—
135	C₄⋊C₄	—	47	2	no	—
136	C₄⋊C₄	—	94	1	no	—
137	C₄⋊C₄	—	94	1	no	—
138	C₄⋊C₄	C₄×D₄₇	1	2	yes	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	188	2	no	—
142	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	94	2	no	—
155	C₂×C₄	—	94	2	no	—
156	C₂×C₄	—	188	1	no	—
157	C₂×C₄	—	188	1	no	—
158	C₂×C₄	—	188	1	no	—
159	C₂×C₄	—	188	1	no	—
160	C₂×C₄	#10625336	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
162	C₂×C₄	#10625350	1	1	yes	characteristic
163	C₂×C₄	#10625352	1	1	yes	characteristic
164	C₂×C₄	#10625352	1	1	yes	characteristic
165	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
166	C₂×C₄	#10625354	1	1	yes	characteristic
167	C₈	—	94	2	no	—
168	C₈	—	188	1	no	—
169	C₂²	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂²	#13094000	1	1	yes	characteristic
171	C₂²	#13094005	1	1	yes	characteristic
172	C₂²	C₂×#10625336	1	1	yes	characteristic
173	C₂²	#13094044	1	1	yes	characteristic
174	C₂²	#13094050	1	1	yes	characteristic
175	C₂²	#13094054	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	1	no	—
179	C₄	—	8	1	no	—
180	C₄	—	188	2	no	—
181	C₄	—	188	2	no	—
182	C₄	#13093982	1	1	yes	characteristic
183	C₄	#13093989	1	1	yes	characteristic
184	C₄	#13094014	1	1	yes	characteristic
185	C₄	#13094019	1	1	yes	characteristic
186	C₂	#15689004	1	1	yes	characteristic
187	C₂	C₄×#10625336	1	1	yes	characteristic
188	C₂	#15689223	1	1	yes	characteristic
189	C₂	#15689410	1	1	yes	characteristic
190	C₂	#15689427	1	1	yes	characteristic
191	C₂	#15689444	1	1	yes	characteristic
192	C₂	#15689451	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093955

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal, characteristic

#15688878

C₂

yes

maximal, characteristic

C₄₇×#389

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

characteristic

C₂×C₄×Dic₄₇

—

#13093955

C₄

yes

complemented

C₄₇⋊#95

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

Q₈

yes

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

#123

yes

characteristic

C₂²×C₄₇

C₂×SD₁₆

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#117

yes

characteristic

C₄×C₄₇

C₄×Q₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₄×SD₁₆

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

C₄×C₄⋊C₄

—

100

#340

—

101

#340

—

102

#340

—

103

#336

—

104

C₄₇

#1920

yes

characteristic, complemented, Sylow

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

C₂×D₄₇

yes

characteristic

114

#106

—

115

#95

—

116

#95

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

122

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

123

C₂²×C₄

C₂²×D₄₇

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

C₄×D₄₇

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

188

—

142

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

188

—

157

C₂×C₄

—

188

—

158

C₂×C₄

—

188

—

159

C₂×C₄

—

188

—

160

C₂×C₄

#10625336

yes

—

161

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

162

C₂×C₄

#10625350

yes

characteristic

163

C₂×C₄

#10625352

yes

characteristic

164

C₂×C₄

#10625352

yes

characteristic

165

C₂×C₄

Q₈×D₄₇

yes

characteristic

166

C₂×C₄

#10625354

yes

characteristic

167

C₈

—

168

C₈

—

188

—

169

C₂²

C₄×C₂²⋊D₄₇

yes

characteristic

170

C₂²

#13094000

yes

characteristic

171

C₂²

#13094005

yes

characteristic

172

C₂²

C₂×#10625336

yes

characteristic

173

C₂²

#13094044

yes

characteristic

174

C₂²

#13094050

yes

characteristic

175

C₂²

#13094054

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

188

—

181

C₄

—

188

—

182

C₄

#13093982

yes

characteristic

183

C₄

#13093989

yes

characteristic

184

C₄

#13094014

yes

characteristic

185

C₄

#13094019

yes

characteristic

186

C₂

#15689004

yes

characteristic

187

C₂

C₄×#10625336

yes

characteristic

188

C₂

#15689223

yes

characteristic

189

C₂

#15689410

yes

characteristic

190

C₂

#15689427

yes

characteristic

191

C₂

#15689444

yes

characteristic

192

C₂

#15689451

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#389	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#389	D₄₇
hypocenter	C₄₇	#1920
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#389

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
13. #13093955	C₄	semidirect
104. C₄₇	#1920	semidirect

H ≅ G/N

type

13. #13093955

C₄

semidirect

104. C₄₇

#1920

semidirect

Subgroups — #20243970

Conjugacy classes of subgroups

Complements