Group #20251741 — subgroups

Subgroups — #20251741

Order 6016.

644 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 209 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×#13094024	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094004	C₂	1	2	yes	maximal
3	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689028	C₂	1	2	yes	maximal, complemented
5	#15689022	C₂	1	4	yes	maximal, complemented
6	#15688980	C₂	1	4	yes	maximal
7	C₄₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₉₄	C₂²	1	2	yes	—
9	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
11	#13094024	—	2	4	no	—
12	#13094004	—	2	8	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
15	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₄²×D₄₇	—	2	4	no	—
19	C₄₇⋊#95	C₂²	1	8	yes	—
20	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×D₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂²×Dic₄₇	C₂³	1	4	yes	—
28	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₄₇	C₂³	1	2	yes	—
30	C₂×C₄×D₄₇	—	2	2	no	—
31	C₂×C₄×D₄₇	—	2	4	no	—
32	C₂×C₄×D₄₇	—	2	2	no	—
33	C₂×C₄×D₄₇	—	2	4	no	—
34	C₂×C₄×D₄₇	D₄	1	4	yes	—
35	C₂×C₄×D₄₇	Q₈	1	8	yes	complemented
36	C₂×Dic₉₄	—	2	4	no	—
37	C₂×Dic₉₄	—	2	4	no	—
38	C₂×Dic₉₄	—	2	4	no	—
39	C₄⋊C₄×C₄₇	—	2	4	no	—
40	C₄⋊C₄×C₄₇	—	2	8	no	—
41	C₄²×C₄₇	D₄	1	4	yes	—
42	(C₂×D₄₇)⋊C₄	—	2	8	no	—
43	C₄⋊Dic₄₇	—	2	8	no	—
44	C₄⋊Dic₄₇	—	2	8	no	—
45	C₄×Dic₄₇	—	2	2	no	—
46	C₄×Dic₄₇	—	2	4	no	—
47	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²×D₄₇	—	2	1	no	—
49	C₂²×D₄₇	—	2	2	no	—
50	C₂²×D₄₇	—	2	1	no	—
51	C₂²×D₄₇	C₂×D₄	1	2	yes	—
52	C₂²×D₄₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	2	8	no	—
55	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
59	C₂×C₄×C₄₇	#41	1	4	yes	—
60	C₂×C₄×C₄₇	#41	1	2	yes	—
61	C₂×Dic₄₇	—	2	1	no	—
62	C₂×Dic₄₇	—	2	2	no	—
63	C₂×Dic₄₇	—	2	2	no	—
64	C₂×Dic₄₇	—	2	1	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	4	no	—
67	C₂×Dic₄₇	—	2	4	no	—
68	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
69	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
70	C₄×D₄₇	—	2	8	no	—
71	C₄×D₄₇	—	2	16	no	—
72	Dic₉₄	—	4	8	no	—
73	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
76	C₂²×C₄₇	C₂×#41	1	2	yes	—
77	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
78	C₂×D₄₇	—	2	2	no	—
79	C₂×D₄₇	—	2	4	no	—
80	C₂×D₄₇	—	2	2	no	—
81	C₂×D₄₇	—	2	4	no	—
82	C₂×D₄₇	C₄⋊Q₈	1	4	yes	complemented
83	C₄×C₄₇	—	4	4	no	—
84	C₄×C₄₇	#121	1	4	yes	—
85	C₄×C₄₇	C₂²⋊Q₈	1	8	yes	complemented
86	Dic₄₇	—	2	2	no	—
87	Dic₄₇	—	2	4	no	—
88	Dic₄₇	—	4	4	no	—
89	#2857	—	47	1	no	maximal, Sylow
90	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
91	C₂×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
92	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄₇	#532	1	1	yes	characteristic
94	C₂×C₄₇	#533	1	2	yes	—
95	D₄₇	—	2	4	no	—
96	C₂×C₄⋊Q₈	—	47	1	no	—
97	C₂×C₂²⋊Q₈	—	47	2	no	—
98	C₂²×C₄²	—	47	1	no	—
99	#391	—	47	2	no	—
100	#389	—	47	4	no	—
101	#384	D₄₇	1	1	yes	characteristic, complemented
102	#380	—	47	4	no	—
103	C₄₇	#2857	1	1	yes	characteristic, complemented, Sylow
104	C₂²×Q₈	—	47	2	no	—
105	C₂³×C₄	—	47	1	no	—
106	C₂³×C₄	—	47	2	no	—
107	C₄⋊Q₈	—	94	4	no	—
108	C₂²⋊Q₈	—	94	8	no	—
109	C₂×C₄⋊C₄	—	47	4	no	—
110	C₂×C₄⋊C₄	—	47	4	no	—
111	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
112	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
113	C₂×C₂²⋊C₄	—	47	4	no	—
114	C₂×C₄²	—	47	2	no	—
115	C₂×C₄²	—	47	1	no	—
116	C₂×C₄²	—	94	4	no	—
117	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
118	#95	—	47	8	no	—
119	C₂⁴	—	47	1	no	—
120	C₂×Q₈	—	94	4	no	—
121	C₂×Q₈	—	94	4	no	—
122	C₂×Q₈	—	94	4	no	—
123	C₂²×C₄	—	47	2	no	—
124	C₂²×C₄	—	47	1	no	—
125	C₂²×C₄	—	47	4	no	—
126	C₂²×C₄	—	47	8	no	—
127	C₂²×C₄	—	47	4	no	—
128	C₂²×C₄	—	94	2	no	—
129	C₂²×C₄	—	94	4	no	—
130	C₂²×C₄	—	94	4	no	—
131	C₂²×C₄	—	94	2	no	—
132	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
133	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	94	8	no	—
138	C₄⋊C₄	—	94	8	no	—
139	C₂²⋊C₄	—	94	8	no	—
140	C₄²	—	94	2	no	—
141	C₄²	—	94	4	no	—
142	C₄²	D₁₈₈	1	4	yes	—
143	C₂³	—	47	2	no	—
144	C₂³	—	47	4	no	—
145	C₂³	—	94	1	no	—
146	C₂³	—	94	2	no	—
147	C₂³	—	94	1	no	—
148	C₂³	C₂³×D₄₇	1	1	yes	characteristic
149	Q₈	—	188	8	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	47	2	no	—
153	C₂×C₄	—	47	4	no	—
154	C₂×C₄	—	94	1	no	—
155	C₂×C₄	—	94	2	no	—
156	C₂×C₄	—	94	2	no	—
157	C₂×C₄	—	94	1	no	—
158	C₂×C₄	—	94	8	no	—
159	C₂×C₄	—	94	16	no	—
160	C₂×C₄	—	94	4	no	—
161	C₂×C₄	—	94	4	no	—
162	C₂×C₄	—	94	4	no	—
163	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
164	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
165	C₂×C₄	D₄×D₄₇	1	2	yes	—
166	C₂×C₄	#10625352	1	4	yes	—
167	C₂×C₄	Q₈×D₄₇	1	4	yes	—
168	C₂×C₄	#10625354	1	2	yes	—
169	C₂²	—	47	4	no	—
170	C₂²	—	94	2	no	—
171	C₂²	—	94	4	no	—
172	C₂²	—	94	2	no	—
173	C₂²	—	94	4	no	—
174	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
175	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
176	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
177	C₂²	C₂×#10625352	1	2	yes	—
178	C₂²	C₂×#10625354	1	1	yes	characteristic
179	C₄	—	4	4	no	—
180	C₄	—	94	2	no	—
181	C₄	—	94	4	no	—
182	C₄	—	188	4	no	—
183	C₄	#13093999	1	4	yes	—
184	C₄	#13094004	1	8	yes	complemented
185	C₂	—	94	4	no	—
186	C₂	C₂×#13093999	1	1	yes	characteristic
187	C₂	C₂×#13094004	1	2	yes	—
188	C₂	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
189	C₂	#15689671	1	1	yes	characteristic
190	C₂	#15689678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal, characteristic, complemented

C₂×#13094004

C₂

yes

maximal

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic

#15689028

C₂

yes

maximal, complemented

#15689022

C₂

yes

maximal, complemented

#15688980

C₂

yes

maximal

C₄₇×#384

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂²×C₄×D₄₇

C₂²

yes

—

#13094024

—

#13094004

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄²×D₄₇

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

Q₈

yes

complemented

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×D₄

yes

—

C₂²×D₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₄⋊Q₈

yes

complemented

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

C₄×C₄₇

C₂²⋊Q₈

yes

complemented

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

—

C₂×C₄₇

C₂×C₄⋊Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

—

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂²×C₄²

—

#391

—

100

#389

—

101

#384

D₄₇

yes

characteristic, complemented

102

#380

—

103

C₄₇

#2857

yes

characteristic, complemented, Sylow

104

C₂²×Q₈

—

105

C₂³×C₄

—

106

C₂³×C₄

—

107

C₄⋊Q₈

—

108

C₂²⋊Q₈

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

112

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

113

C₂×C₂²⋊C₄

—

114

C₂×C₄²

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×D₄₇

yes

characteristic

118

#95

—

119

C₂⁴

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂²×D₄₇

yes

—

133

C₂²×C₄

C₂²×D₄₇

yes

characteristic

134

C₂²×C₄

C₂²×D₄₇

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₂²⋊C₄

—

140

C₄²

—

141

C₄²

—

142

C₄²

D₁₈₈

yes

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

C₂³×D₄₇

yes

characteristic

149

Q₈

—

188

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

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

—

163

C₂×C₄

C₂×D₁₈₈

yes

—

164

C₂×C₄

C₂×D₁₈₈

yes

—

165

C₂×C₄

D₄×D₄₇

yes

—

166

C₂×C₄

#10625352

yes

—

167

C₂×C₄

Q₈×D₄₇

yes

—

168

C₂×C₄

#10625354

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×D₄×D₄₇

yes

characteristic

175

C₂²

C₂²×D₁₈₈

yes

characteristic

176

C₂²

C₂×Q₈×D₄₇

yes

—

177

C₂²

C₂×#10625352

yes

—

178

C₂²

C₂×#10625354

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

188

—

183

C₄

#13093999

yes

—

184

C₄

#13094004

yes

complemented

185

C₂

—

186

C₂

C₂×#13093999

yes

characteristic

187

C₂

C₂×#13094004

yes

—

188

C₂

C₄⋊Q₈×D₄₇

yes

characteristic

189

C₂

#15689671

yes

characteristic

190

C₂

#15689678

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#384	C₂
hypercenter	#384	D₄₇
hypocenter	C₄₇	#2857
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#384

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#13094024	C₂	semidirect
4. #15689028	C₂	semidirect
5. #15689022	C₂	semidirect
7. C₄₇×#384	C₂	semidirect
35. C₂×C₄×D₄₇	Q₈	semidirect
82. C₂×D₄₇	C₄⋊Q₈	semidirect
85. C₄×C₄₇	C₂²⋊Q₈	semidirect
101. #384	D₄₇	semidirect
103. C₄₇	#2857	semidirect
184. C₄	#13094004	semidirect

H ≅ G/N

type

1. C₂×#13094024

C₂

semidirect

4. #15689028

C₂

semidirect

5. #15689022

C₂

semidirect

7. C₄₇×#384

C₂

semidirect

35. C₂×C₄×D₄₇

Q₈

semidirect

82. C₂×D₄₇

C₄⋊Q₈

semidirect

85. C₄×C₄₇

C₂²⋊Q₈

semidirect

101. #384

D₄₇

semidirect

103. C₄₇

#2857

semidirect

184. C₄

#13094004

semidirect

Subgroups — #20251741

Conjugacy classes of subgroups

Complements