Group #20242080 — subgroups

Subgroups — #20242080

Order 6016.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊C₈×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688871	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688871	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	C₄	1	2	yes	—
9	C₂×C₈×D₄₇	—	2	1	no	—
10	#13094024	C₂²	1	1	yes	characteristic
11	#13094011	C₄	1	2	yes	—
12	#13093995	—	2	1	no	—
13	#13093995	—	2	1	no	—
14	D₄×Dic₄₇	—	2	1	no	—
15	D₄×Dic₄₇	—	2	1	no	—
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	4	no	—
31	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
33	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
34	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₄₇	—	2	1	no	—
36	C₂×C₈×C₄₇	—	2	1	no	—
37	C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²⋊Dic₄₇	—	2	2	no	—
39	C₂²⋊Dic₄₇	—	2	2	no	—
40	C₄⋊Dic₄₇	—	2	1	no	—
41	C₄⋊Dic₄₇	—	2	1	no	—
42	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₄₇	D₄	1	1	yes	characteristic
44	C₄×Dic₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄₇⋊C₈	—	2	1	no	—
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₈×D₄₇	—	4	2	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂²×D₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	D₄×C₄₇	—	4	2	no	—
51	D₄×C₄₇	—	4	1	no	—
52	D₄×C₄₇	—	4	1	no	—
53	D₄×C₄₇	—	4	1	no	—
54	D₄×C₄₇	—	4	1	no	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²⋊D₄₇	—	4	4	no	—
59	C₂×Dic₄₇	—	2	2	no	—
60	C₂×Dic₄₇	—	4	4	no	—
61	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄×D₄₇	—	2	2	no	—
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	Dic₉₄	—	4	2	no	—
70	C₈×C₄₇	—	4	1	no	—
71	C₈×C₄₇	—	4	1	no	—
72	C₄₇⋊C₈	—	4	1	no	—
73	C₄₇⋊C₈	—	4	1	no	—
74	C₂²×C₄₇	—	4	2	no	—
75	C₂²×C₄₇	—	4	2	no	—
76	C₂²×C₄₇	—	4	2	no	—
77	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×D₄₇	—	2	1	no	—
79	C₂×D₄₇	—	2	1	no	—
80	C₂×D₄₇	#100	1	2	yes	—
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	#135	1	1	yes	characteristic
83	C₄×C₄₇	#135	1	1	yes	characteristic
84	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₄×C₄₇	#137	1	1	yes	characteristic
86	Dic₄₇	—	2	1	no	—
87	Dic₄₇	—	2	1	no	—
88	Dic₄₇	—	2	1	no	—
89	Dic₄₇	—	4	2	no	—
90	#1551	—	47	1	no	maximal, Sylow
91	C₂×C₄₇	—	8	2	no	—
92	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
93	C₂×C₄₇	#416	1	1	yes	characteristic
94	C₂×C₄₇	#417	1	1	yes	characteristic
95	D₄₇	—	2	2	no	—
96	#532	—	47	1	no	—
97	#432	—	47	1	no	—
98	C₂×C₄⋊C₈	—	47	1	no	—
99	#333	—	47	1	no	—
100	#333	—	47	1	no	—
101	#331	—	47	1	no	—
102	#331	D₄₇	1	1	yes	characteristic, complemented
103	C₄₇	#1551	1	1	yes	characteristic, complemented, Sylow
104	C₂×#41	—	47	2	no	—
105	C₂²×C₈	—	94	1	no	—
106	C₄⋊Q₈	—	47	1	no	—
107	#127	C₂×D₄₇	1	1	yes	characteristic
108	#124	—	47	2	no	—
109	#121	—	94	1	no	—
110	#121	—	94	1	no	—
111	C₄×D₄	—	94	1	no	—
112	C₄×D₄	—	94	1	no	—
113	C₂×C₄²	—	47	1	no	—
114	C₄⋊C₈	—	47	1	no	—
115	C₄⋊C₈	—	47	1	no	—
116	C₄⋊C₈	—	94	1	no	—
117	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
118	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
119	C₂²⋊C₈	—	94	1	no	—
120	#97	—	94	1	no	—
121	#41	—	188	4	no	—
122	C₂×Q₈	—	47	2	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	—	94	2	no	—
126	C₂×D₄	C₄×D₄₇	1	2	yes	—
127	C₂²×C₄	—	47	1	no	—
128	C₂²×C₄	—	47	1	no	—
129	C₂²×C₄	—	47	1	no	—
130	C₂²×C₄	—	94	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	94	1	no	—
134	C₂×C₈	—	94	1	no	—
135	C₂×C₈	—	188	2	no	—
136	C₄⋊C₄	—	94	1	no	—
137	C₄⋊C₄	—	94	1	no	—
138	C₂²⋊C₄	—	94	2	no	—
139	C₂²⋊C₄	—	94	2	no	—
140	C₄²	—	47	1	no	—
141	C₄²	—	47	1	no	—
142	C₄²	—	47	1	no	—
143	C₄²	C₂²×D₄₇	1	1	yes	characteristic
144	C₂³	—	2	2	no	—
145	C₂³	—	47	1	no	—
146	Q₈	—	188	2	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	188	4	no	—
153	C₂×C₄	—	47	1	no	—
154	C₂×C₄	—	47	1	no	—
155	C₂×C₄	—	47	1	no	—
156	C₂×C₄	—	94	2	no	—
157	C₂×C₄	—	94	1	no	—
158	C₂×C₄	—	94	1	no	—
159	C₂×C₄	—	94	1	no	—
160	C₂×C₄	—	94	1	no	—
161	C₂×C₄	—	94	2	no	—
162	C₂×C₄	—	188	4	no	—
163	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
164	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
165	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
166	C₈	—	4	1	no	—
167	C₈	—	4	1	no	—
168	C₈	—	188	1	no	—
169	C₈	—	188	1	no	—
170	C₂²	—	4	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	47	2	no	—
174	C₂²	—	94	1	no	—
175	C₂²	—	94	1	no	—
176	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	94	1	no	—
179	C₄	—	94	1	no	—
180	C₄	—	94	1	no	—
181	C₄	—	188	2	no	—
182	C₄	#13094043	1	1	yes	characteristic
183	C₄	#13094045	1	1	yes	characteristic
184	C₄	#13094048	1	1	yes	characteristic
185	C₄	#13094053	1	1	yes	characteristic
186	C₂	—	8	2	no	—
187	C₂	—	94	2	no	—
188	C₂	#100×D₄₇	1	1	yes	characteristic
189	C₂	#15689116	1	1	yes	characteristic
190	C₂	#15689121	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689671

C₂

yes

maximal, characteristic

#15689179

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×D₄₇

C₂

yes

maximal, characteristic, complemented

#15688871

C₂

yes

maximal, characteristic, complemented

#15688871

C₂

yes

maximal, characteristic, complemented

#15688867

C₂

yes

maximal, characteristic, complemented

C₄₇×#331

C₂

yes

maximal, characteristic, complemented

C₂×#10625352

C₄

yes

—

C₂×C₈×D₄₇

—

#13094024

C₂²

yes

characteristic

#13094011

C₄

yes

—

#13093995

—

#13093995

—

D₄×Dic₄₇

—

D₄×Dic₄₇

—

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

—

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×Dic₉₄

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₈×D₄₇

—

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

—

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

#100

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#1551

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#532

—

#432

—

C₂×C₄⋊C₈

—

#333

—

100

#333

—

101

#331

—

102

#331

D₄₇

yes

characteristic, complemented

103

C₄₇

#1551

yes

characteristic, complemented, Sylow

104

C₂×#41

—

105

C₂²×C₈

—

106

C₄⋊Q₈

—

107

#127

C₂×D₄₇

yes

characteristic

108

#124

—

109

#121

—

110

#121

—

111

C₄×D₄

—

112

C₄×D₄

—

113

C₂×C₄²

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

C₂×D₄₇

yes

characteristic

118

C₄⋊C₈

C₂×D₄₇

yes

characteristic

119

C₂²⋊C₈

—

120

#97

—

121

#41

—

188

—

122

C₂×Q₈

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

C₄×D₄₇

yes

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

188

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

C₂²×D₄₇

yes

characteristic

144

C₂³

—

145

C₂³

—

146

Q₈

—

188

—

147

D₄

—

148

D₄

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

188

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

188

—

163

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

164

C₂×C₄

D₄×D₄₇

yes

characteristic

165

C₂×C₄

D₄×D₄₇

yes

characteristic

166

C₈

—

167

C₈

—

168

C₈

—

188

—

169

C₈

—

188

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

188

—

182

C₄

#13094043

yes

characteristic

183

C₄

#13094045

yes

characteristic

184

C₄

#13094048

yes

characteristic

185

C₄

#13094053

yes

characteristic

186

C₂

—

187

C₂

—

188

C₂

#100×D₄₇

yes

characteristic

189

C₂

#15689116

yes

characteristic

190

C₂

#15689121

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	#1551
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. C₄⋊C₈×D₄₇	C₂	semidirect
4. #15688871	C₂	semidirect
5. #15688871	C₂	semidirect
6. #15688867	C₂	semidirect
7. C₄₇×#331	C₂	semidirect
102. #331	D₄₇	semidirect
103. C₄₇	#1551	semidirect

H ≅ G/N

type

2. #15689179

C₂

semidirect

3. C₄⋊C₈×D₄₇

C₂

semidirect

4. #15688871

C₂

semidirect

5. #15688871

C₂

semidirect

6. #15688867

C₂

semidirect

7. C₄₇×#331

C₂

semidirect

102. #331

D₄₇

semidirect

103. C₄₇

#1551

semidirect

Subgroups — #20242080

Conjugacy classes of subgroups

Complements