Group #20251438 — subgroups

Subgroups — #20251438

Order 6016.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13094012	C₂	1	1	yes	maximal, characteristic
2	#15689058	C₂	1	2	yes	maximal, complemented
3	#15689049	C₂	1	2	yes	maximal, complemented
4	#15689045	C₂	1	4	yes	maximal, complemented
5	#15689043	C₂	1	2	yes	maximal, complemented
6	#15689040	C₂	1	2	yes	maximal
7	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
10	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
11	#13094012	—	2	4	no	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
15	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
16	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
17	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
18	C₄₇⋊#95	C₂²	1	2	yes	—
19	C₄₇⋊#95	C₂²	1	2	yes	—
20	C₄₇⋊#95	C₂²	1	4	yes	—
21	#125×C₄₇	—	2	4	no	—
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂³×D₄₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₇	C₂³	1	4	yes	—
34	C₂²×Dic₄₇	C₂³	1	2	yes	—
35	C₂×D₁₈₈	—	2	2	no	—
36	C₂×D₁₈₈	—	4	2	no	—
37	C₂×Dic₉₄	—	2	2	no	—
38	C₂×Dic₉₄	—	4	2	no	—
39	C₄⋊C₄×C₄₇	—	2	4	no	—
40	C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₄²×C₄₇	D₄	1	4	yes	—
44	(C₂×D₄₇)⋊C₄	—	2	4	no	—
45	(C₂×D₄₇)⋊C₄	—	2	4	no	—
46	(C₂×D₄₇)⋊C₄	—	4	4	no	—
47	(C₂×D₄₇)⋊C₄	—	4	4	no	—
48	C₄⋊Dic₄₇	—	4	2	no	—
49	Dic₄₇⋊C₄	—	4	4	no	—
50	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂²×D₄₇	—	2	2	no	—
52	C₂²×D₄₇	—	2	2	no	—
53	C₂²×D₄₇	—	2	4	no	—
54	C₂²×D₄₇	—	2	2	no	—
55	C₂²×D₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂×Dic₄₇	—	2	2	no	—
69	C₂×Dic₄₇	—	2	4	no	—
70	C₂×Dic₄₇	—	2	4	no	—
71	C₂×Dic₄₇	—	2	4	no	—
72	C₂×Dic₄₇	—	2	4	no	—
73	D₁₈₈	—	4	4	no	—
74	Dic₉₄	—	4	4	no	—
75	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
77	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
79	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
80	C₂×D₄₇	—	4	2	no	—
81	C₂×D₄₇	—	4	2	no	—
82	C₂×D₄₇	—	4	2	no	—
83	C₂×D₄₇	—	4	4	no	—
84	C₂×D₄₇	—	4	4	no	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	—	2	2	no	—
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	—	4	2	no	—
89	C₄×C₄₇	—	4	2	no	—
90	Dic₄₇	—	4	2	no	—
91	Dic₄₇	—	4	4	no	—
92	#2820	—	47	1	no	maximal, Sylow
93	C₂×C₄₇	#535	1	1	yes	characteristic
94	C₂×C₄₇	#536	1	1	yes	characteristic
95	C₂×C₄₇	#537	1	1	yes	characteristic
96	C₂×C₄₇	#562	1	2	yes	—
97	C₂×C₄₇	#563	1	2	yes	—
98	D₄₇	—	8	2	no	—
99	C₂×#125	D₄₇	1	1	yes	characteristic, complemented
100	C₂×#124	—	47	1	no	—
101	#399	—	47	2	no	—
102	#397	—	47	2	no	—
103	#396	—	47	4	no	—
104	#395	—	47	2	no	—
105	#394	—	47	2	no	—
106	#382	—	47	1	no	—
107	C₄₇	#2820	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	47	1	no	—
109	C₂²×D₄	—	47	1	no	—
110	#125	—	2	4	no	—
111	#124	—	94	4	no	—
112	C₂×C₄⋊C₄	—	47	4	no	—
113	C₂×C₄⋊C₄	—	47	2	no	—
114	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
115	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
116	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	—	47	2	no	—
119	C₂×C₂²⋊C₄	—	47	2	no	—
120	C₂×C₂²⋊C₄	—	47	4	no	—
121	C₂×C₂²⋊C₄	—	47	4	no	—
122	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
123	#95	—	47	2	no	—
124	#95	—	47	2	no	—
125	#95	—	47	4	no	—
126	C₂⁴	—	47	2	no	—
127	C₂×Q₈	—	94	2	no	—
128	C₂×Q₈	—	188	2	no	—
129	C₂×D₄	—	94	2	no	—
130	C₂×D₄	—	188	2	no	—
131	C₂²×C₄	—	47	4	no	—
132	C₂²×C₄	—	47	2	no	—
133	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
134	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
135	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	188	4	no	—
143	C₄⋊C₄	—	188	2	no	—
144	C₂²⋊C₄	—	94	4	no	—
145	C₂²⋊C₄	—	94	4	no	—
146	C₂²⋊C₄	—	188	4	no	—
147	C₂²⋊C₄	—	188	4	no	—
148	C₄²	C₂²⋊D₄₇	1	4	yes	—
149	C₂³	—	94	2	no	—
150	C₂³	—	94	2	no	—
151	C₂³	—	94	4	no	—
152	C₂³	—	94	2	no	—
153	C₂³	—	94	4	no	—
154	C₂³	C₂³×D₄₇	1	1	yes	characteristic
155	Q₈	—	188	4	no	—
156	D₄	—	188	4	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	94	2	no	—
167	C₂×C₄	—	94	4	no	—
168	C₂×C₄	—	94	4	no	—
169	C₂×C₄	—	94	4	no	—
170	C₂×C₄	—	94	4	no	—
171	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
174	C₂²	—	188	4	no	—
175	C₂²	—	188	4	no	—
176	C₂²	—	188	2	no	—
177	C₂²	—	188	2	no	—
178	C₂²	—	188	2	no	—
179	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂²	#13094070	1	1	yes	characteristic
181	C₂²	#13094070	1	2	yes	—
182	C₂²	#13094071	1	1	yes	characteristic
183	C₂²	#13094072	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	188	4	no	—
190	C₄	—	188	2	no	—
191	C₂	—	376	2	no	—
192	C₂	#15689687	1	1	yes	characteristic
193	C₂	#15689697	1	1	yes	characteristic
194	C₂	#15689701	1	1	yes	characteristic
195	C₂	#15689871	1	2	yes	—
196	C₂	#15689877	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094012

C₂

yes

maximal, characteristic

#15689058

C₂

yes

maximal, complemented

#15689049

C₂

yes

maximal, complemented

#15689045

C₂

yes

maximal, complemented

#15689043

C₂

yes

maximal, complemented

#15689040

C₂

yes

maximal

#15688989

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂²

yes

characteristic

#13094012

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

#125×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

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×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#563

yes

—

D₄₇

—

C₂×#125

D₄₇

yes

characteristic, complemented

100

C₂×#124

—

101

#399

—

102

#397

—

103

#396

—

104

#395

—

105

#394

—

106

#382

—

107

C₄₇

#2820

yes

characteristic, complemented, Sylow

108

C₂²×Q₈

—

109

C₂²×D₄

—

110

#125

—

111

#124

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

115

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

116

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

117

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

118

C₂×C₂²⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₂²⋊C₄

—

122

C₂×C₄²

C₂×D₄₇

yes

characteristic

123

#95

—

124

#95

—

125

#95

—

126

C₂⁴

—

127

C₂×Q₈

—

128

C₂×Q₈

—

188

—

129

C₂×D₄

—

130

C₂×D₄

—

188

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²×D₄₇

yes

—

134

C₂²×C₄

C₂²×D₄₇

yes

—

135

C₂²×C₄

C₂²×D₄₇

yes

characteristic

136

C₂²×C₄

C₂²×D₄₇

yes

characteristic

137

C₂²×C₄

C₂²×D₄₇

yes

characteristic

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

188

—

143

C₄⋊C₄

—

188

—

144

C₂²⋊C₄

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

188

—

147

C₂²⋊C₄

—

188

—

148

C₄²

C₂²⋊D₄₇

yes

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

C₂³×D₄₇

yes

characteristic

155

Q₈

—

188

—

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

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

172

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

173

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

174

C₂²

—

188

—

175

C₂²

—

188

—

176

C₂²

—

188

—

177

C₂²

—

188

—

178

C₂²

—

188

—

179

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

180

C₂²

#13094070

yes

characteristic

181

C₂²

#13094070

yes

—

182

C₂²

#13094071

yes

characteristic

183

C₂²

#13094072

yes

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

188

—

190

C₄

—

188

—

191

C₂

—

376

—

192

C₂

#15689687

yes

characteristic

193

C₂

#15689697

yes

characteristic

194

C₂

#15689701

yes

characteristic

195

C₂

#15689871

yes

—

196

C₂

#15689877

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#125×C₄₇	C₂
hypercenter	C₂×#125	D₄₇
hypocenter	C₄₇	#2820
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×#125×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. #15689058	C₂	semidirect
3. #15689049	C₂	semidirect
4. #15689045	C₂	semidirect
5. #15689043	C₂	semidirect
7. #15688989	C₂	semidirect
8. C₂×#125×C₄₇	C₂	semidirect
99. C₂×#125	D₄₇	semidirect
107. C₄₇	#2820	semidirect

H ≅ G/N

type

2. #15689058

C₂

semidirect

3. #15689049

C₂

semidirect

4. #15689045

C₂

semidirect

5. #15689043

C₂

semidirect

7. #15688989

C₂

semidirect

8. C₂×#125×C₄₇

C₂

semidirect

99. C₂×#125

D₄₇

semidirect

107. C₄₇

#2820

semidirect

Subgroups — #20251438

Conjugacy classes of subgroups

Complements