Group #20244321 — subgroups

Subgroups — #20244321

Order 6016.

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

normal subgroups only

This group has 344 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	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093948	C₂	1	2	yes	maximal, complemented
3	C₂×#13093945	C₂	1	2	yes	maximal
4	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₄₇	—	2	1	no	—
9	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
10	#13093948	—	2	4	no	—
11	#13093945	—	2	4	no	—
12	C₄₇⋊#95	—	2	1	no	—
13	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	#124×C₄₇	—	2	2	no	—
16	#124×C₄₇	C₄	1	4	yes	complemented
17	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
18	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	—	2	1	no	—
22	C₂×Q₈×C₄₇	—	2	2	no	—
23	C₂×Q₈×C₄₇	—	2	2	no	—
24	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₂×D₄×C₄₇	—	2	2	no	—
26	C₂×D₄×C₄₇	—	2	1	no	—
27	C₂×D₄×C₄₇	—	2	2	no	—
28	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₄₇	—	2	1	no	—
30	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₇	—	2	2	no	—
34	C₂²⋊C₄×C₄₇	—	2	4	no	—
35	C₂²⋊C₄×C₄₇	—	2	4	no	—
36	C₄²×C₄₇	C₂×C₄	1	2	yes	—
37	C₄²×C₄₇	D₄	1	2	yes	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₄⋊Dic₄₇	D₄	1	4	yes	—
40	C₄×Dic₄₇	—	4	2	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	—	2	2	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂³×C₄₇	—	2	2	no	—
45	C₂³×C₄₇	—	4	2	no	—
46	C₂³×C₄₇	—	4	2	no	—
47	C₂³×C₄₇	—	4	1	no	—
48	C₂³×C₄₇	—	4	1	no	—
49	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₄₇	—	2	4	no	—
51	Q₈×C₄₇	—	4	1	no	—
52	Q₈×C₄₇	—	4	1	no	—
53	D₄×C₄₇	—	2	4	no	—
54	D₄×C₄₇	—	4	1	no	—
55	D₄×C₄₇	—	4	1	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	4	2	no	—
61	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#41	1	2	yes	—
67	C₂×Dic₄₇	—	2	4	no	—
68	C₂×Dic₄₇	—	4	2	no	—
69	C₂×Dic₄₇	—	4	2	no	—
70	C₄₇⋊C₈	—	4	2	no	—
71	C₂²×C₄₇	—	4	2	no	—
72	C₂²×C₄₇	—	4	2	no	—
73	C₂²×C₄₇	—	4	2	no	—
74	C₂²×C₄₇	—	8	2	no	—
75	C₂²×C₄₇	—	8	2	no	—
76	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄₇	#121	1	2	yes	—
78	C₂²×C₄₇	#135	1	2	yes	—
79	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂²×C₄₇	#137	1	1	yes	characteristic
81	C₄×C₄₇	—	2	2	no	—
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	—	4	2	no	—
84	C₄×C₄₇	C₄×D₄	1	2	yes	—
85	C₄×C₄₇	#124	1	1	yes	characteristic
86	C₄×C₄₇	#127	1	1	yes	characteristic
87	Dic₄₇	—	4	4	no	—
88	#1983	—	47	1	no	maximal, Sylow
89	C₂×C₄₇	—	8	2	no	—
90	C₂×C₄₇	#390	1	1	yes	characteristic
91	C₂×C₄₇	#416	1	1	yes	characteristic
92	C₂×C₄₇	#417	1	1	yes	characteristic
93	C₂×C₄₇	#463	1	2	yes	—
94	C₂×C₄₇	#464	1	2	yes	—
95	C₂×#124	D₄₇	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	47	1	no	—
97	C₂×#103	—	47	2	no	—
98	C₂×#102	—	47	2	no	—
99	C₄×C₄⋊C₄	—	47	1	no	—
100	C₄₇	#1983	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
102	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
103	C₂²×C₈	—	47	2	no	—
104	#124	—	2	2	no	—
105	#124	Dic₄₇	1	4	yes	complemented
106	C₂×C₄⋊C₄	—	47	2	no	—
107	C₂×C₂²⋊C₄	—	2	1	no	—
108	C₂×C₂²⋊C₄	—	2	1	no	—
109	C₂×C₄²	—	94	1	no	—
110	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
111	C₄⋊C₈	—	94	2	no	—
112	#103	—	94	4	no	—
113	#102	—	94	4	no	—
114	#95	—	94	1	no	—
115	C₂⁴	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	94	2	no	—
126	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
129	C₂×C₈	—	94	2	no	—
130	C₂×C₈	—	94	2	no	—
131	C₂×C₈	—	94	2	no	—
132	C₄⋊C₄	—	47	4	no	—
133	C₄⋊C₄	—	94	2	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	—	188	2	no	—
137	C₄²	C₂×Dic₄₇	1	2	yes	—
138	C₄²	C₂²⋊D₄₇	1	2	yes	—
139	C₂³	—	2	2	no	—
140	C₂³	—	4	2	no	—
141	C₂³	—	4	2	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
145	Q₈	—	2	4	no	—
146	Q₈	—	4	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	2	4	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	94	4	no	—
157	C₂×C₄	—	188	2	no	—
158	C₂×C₄	—	188	2	no	—
159	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
160	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
161	C₂×C₄	D₄×D₄₇	1	2	yes	—
162	C₂×C₄	#10625352	1	2	yes	—
163	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
165	C₈	—	188	2	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	8	2	no	—
171	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
172	C₂²	#13093995	1	2	yes	—
173	C₂²	#13094044	1	2	yes	—
174	C₂²	#13094049	1	1	yes	characteristic
175	C₂²	#13094054	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	188	4	no	—
180	C₄	D₄×Dic₄₇	1	2	yes	—
181	C₄	#13094011	1	1	yes	characteristic
182	C₄	#13094020	1	1	yes	characteristic
183	C₂	—	8	2	no	—
184	C₂	#15689026	1	1	yes	characteristic
185	C₂	#15689119	1	1	yes	characteristic
186	C₂	#15689124	1	1	yes	characteristic
187	C₂	#15689335	1	2	yes	—
188	C₂	#15689342	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×#13093948

C₂

yes

maximal, complemented

C₂×#13093945

C₂

yes

maximal

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

#13093948

—

#13093945

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×D₄×C₄₇

C₂²

yes

characteristic

#124×C₄₇

—

#124×C₄₇

C₄

yes

complemented

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×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

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

#135

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1983

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#464

yes

—

C₂×#124

D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

C₂×#102

—

C₄×C₄⋊C₄

—

100

C₄₇

#1983

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

C₂×D₄₇

yes

characteristic

102

C₂²×D₄

C₂×D₄₇

yes

characteristic

103

C₂²×C₈

—

104

#124

—

105

#124

Dic₄₇

yes

complemented

106

C₂×C₄⋊C₄

—

107

C₂×C₂²⋊C₄

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂×D₄₇

yes

characteristic

111

C₄⋊C₈

—

112

#103

—

113

#102

—

114

#95

—

115

C₂⁴

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

C₂×Dic₄₇

yes

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

C₂×Dic₄₇

yes

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

127

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

128

C₂²×C₄

C₂²×D₄₇

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

188

—

137

C₄²

C₂×Dic₄₇

yes

—

138

C₄²

C₂²⋊D₄₇

yes

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

145

Q₈

—

146

Q₈

—

147

Q₈

—

148

D₄

—

149

D₄

—

150

D₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

188

—

158

C₂×C₄

—

188

—

159

C₂×C₄

C₂²⋊Dic₄₇

yes

—

160

C₂×C₄

C₂²⋊Dic₄₇

yes

—

161

C₂×C₄

D₄×D₄₇

yes

—

162

C₂×C₄

#10625352

yes

—

163

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

164

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

165

C₈

—

188

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

172

C₂²

#13093995

yes

—

173

C₂²

#13094044

yes

—

174

C₂²

#13094049

yes

characteristic

175

C₂²

#13094054

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

188

—

180

C₄

D₄×Dic₄₇

yes

—

181

C₄

#13094011

yes

characteristic

182

C₄

#13094020

yes

characteristic

183

C₂

—

184

C₂

#15689026

yes

characteristic

185

C₂

#15689119

yes

characteristic

186

C₂

#15689124

yes

characteristic

187

C₂

#15689335

yes

—

188

C₂

#15689342

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#124×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#124	D₄₇
hypocenter	C₄₇	#1983
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#124×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
2. C₂×#13093948	C₂	semidirect
4. C₄×C₄⋊Dic₄₇	C₂	semidirect
16. #124×C₄₇	C₄	semidirect
100. C₄₇	#1983	semidirect
105. #124	Dic₄₇	semidirect

H ≅ G/N

type

1. C₂×C₄⋊(C₄₇⋊C₈)

C₂

semidirect

2. C₂×#13093948

C₂

semidirect

4. C₄×C₄⋊Dic₄₇

C₂

semidirect

16. #124×C₄₇

C₄

semidirect

100. C₄₇

#1983

semidirect

105. #124

Dic₄₇

semidirect

Subgroups — #20244321

Conjugacy classes of subgroups

Complements