Group #20242107 — subgroups

Subgroups — #20242107

Order 6016.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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	#15689671	C₂	1	1	yes	maximal, characteristic
2	#15689179	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689145	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688870	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688868	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688867	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625352	—	2	1	no	—
9	C₂×#10625324	—	2	1	no	—
10	#13094024	C₂²	1	1	yes	characteristic
11	#13094011	—	2	1	no	—
12	#13093995	—	2	1	no	—
13	#13093995	C₄	1	2	yes	—
14	D₄×Dic₄₇	—	2	1	no	—
15	D₄×Dic₄₇	C₄	1	2	yes	—
16	C₄²×D₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
19	Dic₄₇⋊C₈	—	2	1	no	—
20	(C₂×D₄₇)⋊C₈	—	2	1	no	—
21	#13093934	—	2	1	no	—
22	#127×C₄₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₄×C₄₇	—	2	1	no	—
27	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊D₄₇	—	2	2	no	—
29	C₂²×Dic₄₇	—	2	2	no	—
30	#10625352	—	4	2	no	—
31	#10625352	—	4	2	no	—
32	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
35	C₂×Dic₉₄	—	2	1	no	—
36	C₂×C₈×C₄₇	—	2	1	no	—
37	C₂×C₈×C₄₇	—	2	1	no	—
38	C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₄₇	—	2	2	no	—
40	C₂²⋊Dic₄₇	—	2	2	no	—
41	C₄⋊Dic₄₇	—	2	1	no	—
42	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
43	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₄₇	D₄	1	1	yes	characteristic
45	C₄×Dic₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	#10625324	—	4	2	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂²×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	D₄×C₄₇	—	4	2	no	—
52	D₄×C₄₇	—	4	1	no	—
53	D₄×C₄₇	—	4	1	no	—
54	D₄×C₄₇	—	4	1	no	—
55	D₄×C₄₇	—	4	1	no	—
56	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₄₇	—	4	4	no	—
60	C₂×Dic₄₇	—	2	2	no	—
61	C₂×Dic₄₇	—	4	4	no	—
62	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄×D₄₇	—	2	1	no	—
66	C₄×D₄₇	—	2	1	no	—
67	C₄×D₄₇	—	2	1	no	—
68	C₄×D₄₇	—	2	1	no	—
69	C₄×D₄₇	—	4	1	no	—
70	Dic₉₄	—	4	1	no	—
71	Dic₉₄	—	4	1	no	—
72	C₈×C₄₇	—	4	1	no	—
73	C₈×C₄₇	—	4	1	no	—
74	C₄₇⋊C₈	—	4	1	no	—
75	C₄₇⋊C₈	—	4	1	no	—
76	C₂²×C₄₇	—	4	2	no	—
77	C₂²×C₄₇	—	4	2	no	—
78	C₂²×C₄₇	—	4	2	no	—
79	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×D₄₇	—	2	1	no	—
81	C₂×D₄₇	—	2	1	no	—
82	C₂×D₄₇	—	2	1	no	—
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₄×C₄₇	#137	1	1	yes	characteristic
88	Dic₄₇	—	2	1	no	—
89	Dic₄₇	—	2	1	no	—
90	Dic₄₇	—	2	1	no	—
91	Dic₄₇	—	4	2	no	—
92	#1557	—	47	1	no	maximal, Sylow
93	C₂×C₄₇	—	8	2	no	—
94	C₂×C₄₇	#413	1	1	yes	characteristic
95	C₂×C₄₇	#417	1	1	yes	characteristic
96	C₂×C₄₇	#418	1	1	yes	characteristic
97	D₄₇	—	4	1	no	—
98	#532	—	47	1	no	—
99	#432	—	47	1	no	—
100	#423	—	47	1	no	—
101	#332	—	47	1	no	—
102	#332	—	47	1	no	—
103	#331	—	47	1	no	—
104	#331	D₄₇	1	1	yes	characteristic, complemented
105	C₄₇	#1557	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	—	94	1	no	—
107	C₂×M₁₆	—	94	1	no	—
108	C₄⋊Q₈	—	47	1	no	—
109	#127	C₂×D₄₇	1	1	yes	characteristic
110	#124	—	94	1	no	—
111	#121	—	47	2	no	—
112	#121	—	94	1	no	—
113	C₄×D₄	—	47	2	no	—
114	C₄×D₄	—	94	1	no	—
115	C₂×C₄²	—	47	1	no	—
116	C₄⋊C₈	—	47	1	no	—
117	C₄⋊C₈	—	47	1	no	—
118	C₄⋊C₈	—	94	1	no	—
119	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
120	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
121	C₂²⋊C₈	—	94	1	no	—
122	#97	—	94	1	no	—
123	#41	—	188	2	no	—
124	#41	—	188	2	no	—
125	C₂×Q₈	—	94	1	no	—
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	—	2	1	no	—
128	C₂×D₄	—	94	2	no	—
129	C₂×D₄	C₄×D₄₇	1	2	yes	—
130	C₂²×C₄	—	47	1	no	—
131	C₂²×C₄	—	47	1	no	—
132	C₂²×C₄	—	47	1	no	—
133	C₂²×C₄	—	94	2	no	—
134	M₁₆	—	188	2	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	94	1	no	—
138	C₂×C₈	—	94	1	no	—
139	C₄⋊C₄	—	47	2	no	—
140	C₄⋊C₄	—	94	1	no	—
141	C₂²⋊C₄	—	94	2	no	—
142	C₂²⋊C₄	—	94	2	no	—
143	C₄²	—	47	1	no	—
144	C₄²	—	47	1	no	—
145	C₄²	—	47	1	no	—
146	C₄²	C₂²×D₄₇	1	1	yes	characteristic
147	C₂³	—	2	2	no	—
148	C₂³	—	47	1	no	—
149	Q₈	—	188	1	no	—
150	Q₈	—	188	1	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	4	1	no	—
155	D₄	—	4	1	no	—
156	D₄	—	188	4	no	—
157	C₂×C₄	—	47	1	no	—
158	C₂×C₄	—	47	1	no	—
159	C₂×C₄	—	47	1	no	—
160	C₂×C₄	—	94	1	no	—
161	C₂×C₄	—	94	1	no	—
162	C₂×C₄	—	94	1	no	—
163	C₂×C₄	—	94	1	no	—
164	C₂×C₄	—	94	2	no	—
165	C₂×C₄	—	188	1	no	—
166	C₂×C₄	—	188	4	no	—
167	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
168	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
169	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
170	C₈	—	4	1	no	—
171	C₈	—	4	1	no	—
172	C₈	—	188	1	no	—
173	C₈	—	188	1	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	94	1	no	—
178	C₂²	—	94	1	no	—
179	C₂²	—	94	1	no	—
180	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	94	1	no	—
183	C₄	—	94	1	no	—
184	C₄	—	94	1	no	—
185	C₄	—	188	2	no	—
186	C₄	#13094048	1	1	yes	characteristic
187	C₄	#13094048	1	1	yes	characteristic
188	C₄	#13094051	1	1	yes	characteristic
189	C₄	#13094053	1	1	yes	characteristic
190	C₂	—	8	2	no	—
191	C₂	—	188	1	no	—
192	C₂	#15689104	1	1	yes	characteristic
193	C₂	#15689121	1	1	yes	characteristic
194	C₂	#15689126	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689671

C₂

yes

maximal, characteristic

#15689179

C₂

yes

maximal, characteristic, complemented

#15689145

C₂

yes

maximal, characteristic, complemented

#15688870

C₂

yes

maximal, characteristic, complemented

#15688868

C₂

yes

maximal, characteristic, complemented

#15688867

C₂

yes

maximal, characteristic, complemented

C₄₇×#331

C₂

yes

maximal, characteristic, complemented

C₂×#10625352

—

C₂×#10625324

—

#13094024

C₂²

yes

—

—

C₄

yes

—

D₄×Dic₄₇

—

D₄×Dic₄₇

C₄

yes

—

C₄²×D₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

(C₂×D₄₇)⋊C₈

—

#13093934

—

#127×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

#10625352

—

#10625352

—

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂³×C₄₇

—

C₂²×D₄₇

C₂²⋊C₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#1557

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#532

—

#432

—

100

#423

—

101

#332

—

102

#332

—

103

#331

—

104

#331

D₄₇

yes

characteristic, complemented

105

C₄₇

#1557

yes

characteristic, complemented, Sylow

106

C₂×#41

—

107

C₂×M₁₆

—

108

C₄⋊Q₈

—

109

#127

C₂×D₄₇

yes

characteristic

110

#124

—

111

#121

—

112

#121

—

113

C₄×D₄

—

114

C₄×D₄

—

115

C₂×C₄²

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

C₂×D₄₇

yes

characteristic

120

C₄⋊C₈

C₂×D₄₇

yes

characteristic

121

C₂²⋊C₈

—

122

#97

—

123

#41

—

188

—

124

#41

—

188

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

C₄×D₄₇

yes

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

M₁₆

—

188

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂²×D₄₇

yes

characteristic

147

C₂³

—

148

C₂³

—

149

Q₈

—

188

—

150

Q₈

—

188

—

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

155

D₄

—

156

D₄

—

188

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

188

—

166

C₂×C₄

—

188

—

167

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

168

C₂×C₄

D₄×D₄₇

yes

characteristic

169

C₂×C₄

D₄×D₄₇

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

188

—

173

C₈

—

188

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

188

—

186

C₄

#13094048

yes

characteristic

187

C₄

#13094048

yes

characteristic

188

C₄

#13094051

yes

characteristic

189

C₄

#13094053

yes

characteristic

190

C₂

—

191

C₂

—

188

—

192

C₂

#15689104

yes

characteristic

193

C₂

#15689121

yes

characteristic

194

C₂

#15689126

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#331	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#331	D₄₇
hypocenter	C₄₇	#1557
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#331

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #15689179	C₂	semidirect
3. #15689145	C₂	semidirect
4. #15688870	C₂	semidirect
5. #15688868	C₂	semidirect
6. #15688867	C₂	semidirect
7. C₄₇×#331	C₂	semidirect
104. #331	D₄₇	semidirect
105. C₄₇	#1557	semidirect

H ≅ G/N

type

2. #15689179

C₂

semidirect

3. #15689145

C₂

semidirect

4. #15688870

C₂

semidirect

5. #15688868

C₂

semidirect

6. #15688867

C₂

semidirect

7. C₄₇×#331

C₂

semidirect

104. #331

D₄₇

semidirect

105. C₄₇

#1557

semidirect

Subgroups — #20242107

Conjugacy classes of subgroups

Complements