Group #20255146 — subgroups

Subgroups — #20255146

Order 6016.

504 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13094043	C₂	1	2	yes	maximal, complemented
2	#15689793	C₂	1	2	yes	maximal, complemented
3	#15689356	C₂	1	2	yes	maximal, complemented
4	#15689320	C₂	1	2	yes	maximal, complemented
5	#15689265	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625352	C₂²	1	4	yes	—
10	#13094043	—	2	8	no	—
11	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
12	C₂×#10625335	C₂²	1	4	yes	—
13	C₂×C₈×D₄₇	C₂²	1	2	yes	—
14	#13094020	—	2	2	no	—
15	#13093996	—	2	4	no	—
16	#13093995	C₂²	1	4	yes	—
17	C₄×Dic₉₄	C₂²	1	2	yes	—
18	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
19	D₄×Dic₄₇	—	2	2	no	—
20	#13093958	C₂²	1	1	yes	characteristic
21	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
22	#13093949	C₂²	1	2	yes	—
23	#13093945	C₂²	1	4	yes	—
24	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
25	C₂×D₈×C₄₇	C₂²	1	2	yes	—
26	C₂×D₈×C₄₇	C₂²	1	2	yes	—
27	#127×C₄₇	C₂²	1	2	yes	—
28	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×D₄×C₄₇	—	2	2	no	—
30	C₂×D₄×C₄₇	—	2	2	no	—
31	C₂×D₄×C₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊D₄₇	—	2	4	no	—
33	C₂×C₂²⋊D₄₇	—	2	4	no	—
34	C₂²×Dic₄₇	—	2	4	no	—
35	#10625352	—	2	8	no	—
36	#10625352	—	4	4	no	—
37	C₂×D₁₈₈	D₄	1	1	yes	characteristic
38	C₂×C₄×D₄₇	C₂³	1	2	yes	—
39	C₂×Dic₉₄	C₂³	1	2	yes	—
40	D₈×C₄₇	—	2	4	no	—
41	D₈×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₂²⋊Dic₄₇	—	2	4	no	—
46	#10625335	—	2	8	no	—
47	(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
48	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
49	C₄⋊Dic₄₇	C₂³	1	2	yes	—
50	Dic₄₇⋊C₄	—	2	2	no	—
51	C₄×Dic₄₇	—	2	2	no	—
52	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
53	Dic₁₈₈	D₄	1	4	yes	complemented
54	C₈×D₄₇	—	2	4	no	—
55	C₂³×C₄₇	—	2	4	no	—
56	C₂²×D₄₇	C₂×D₄	1	2	yes	—
57	D₄×C₄₇	—	2	4	no	—
58	D₄×C₄₇	—	2	4	no	—
59	D₄×C₄₇	—	4	4	no	—
60	D₄×C₄₇	—	8	2	no	—
61	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²⋊D₄₇	—	4	8	no	—
65	C₂²⋊D₄₇	—	8	4	no	—
66	C₂×Dic₄₇	—	2	2	no	—
67	C₂×Dic₄₇	—	2	2	no	—
68	C₂×Dic₄₇	—	4	8	no	—
69	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
70	D₁₈₈	—	2	2	no	—
71	C₄×D₄₇	—	2	2	no	—
72	C₄×D₄₇	—	2	2	no	—
73	Dic₉₄	—	2	2	no	—
74	Dic₉₄	C₂×D₄	1	4	yes	—
75	C₈×C₄₇	—	2	1	no	—
76	C₈×C₄₇	C₂×D₄	1	2	yes	—
77	C₄₇⋊C₈	—	2	2	no	—
78	C₂²×C₄₇	—	4	4	no	—
79	C₂²×C₄₇	—	4	4	no	—
80	C₂²×C₄₇	—	4	4	no	—
81	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×D₄₇	—	2	2	no	—
83	C₂×D₄₇	—	2	2	no	—
84	C₂×D₄₇	—	2	2	no	—
85	C₄×C₄₇	—	2	1	no	—
86	C₄×C₄₇	#135	1	2	yes	complemented
87	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
89	Dic₄₇	—	2	4	no	—
90	Dic₄₇	—	2	2	no	—
91	Dic₄₇	—	4	2	no	—
92	#3287	—	47	1	no	maximal, Sylow
93	C₂×C₄₇	—	8	4	no	—
94	C₂×C₄₇	D₄²	1	1	yes	characteristic
95	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
96	C₂×C₄₇	#576	1	1	yes	characteristic
97	D₄₇	—	4	2	no	—
98	C₂×#135	—	47	2	no	—
99	#550	—	47	2	no	—
100	#493	D₄₇	1	1	yes	characteristic, complemented
101	#466	—	47	2	no	—
102	#460	—	47	2	no	—
103	#449	—	47	4	no	—
104	C₄×Q₁₆	—	47	1	no	—
105	C₈×D₄	—	47	1	no	—
106	C₄₇	#3287	1	1	yes	characteristic, complemented, Sylow
107	C₂×#41	—	47	4	no	—
108	#135	—	94	8	no	—
109	C₂×Q₁₆	—	47	1	no	—
110	C₂×SD₁₆	—	47	4	no	—
111	C₂×D₈	C₂×D₄₇	1	2	yes	—
112	C₂×D₈	C₂×D₄₇	1	2	yes	—
113	C₂²×C₈	—	47	2	no	—
114	#127	—	94	2	no	—
115	#127	C₂×D₄₇	1	2	yes	—
116	#121	—	47	4	no	—
117	#121	—	94	4	no	—
118	C₄×Q₈	—	47	2	no	—
119	C₄×D₄	—	47	1	no	—
120	C₄×D₄	—	94	2	no	—
121	#107	—	47	1	no	—
122	C₄⋊C₈	—	47	1	no	—
123	#103	—	47	2	no	—
124	#102	—	47	4	no	—
125	C₂²⋊C₈	—	47	2	no	—
126	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
127	#41	—	94	8	no	—
128	#41	—	188	4	no	—
129	C₂×Q₈	—	47	2	no	—
130	C₂×D₄	—	2	2	no	—
131	C₂×D₄	—	2	2	no	—
132	C₂×D₄	—	47	1	no	—
133	C₂×D₄	—	94	4	no	—
134	C₂×D₄	—	94	4	no	—
135	C₂×D₄	C₂²×D₄₇	1	4	yes	—
136	C₂²×C₄	—	47	2	no	—
137	C₂²×C₄	—	94	4	no	—
138	Q₁₆	—	47	4	no	—
139	SD₁₆	—	94	8	no	—
140	D₈	—	2	4	no	—
141	D₈	—	4	2	no	—
142	C₂×C₈	—	47	2	no	—
143	C₂×C₈	—	94	4	no	—
144	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
145	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
146	C₄⋊C₄	—	47	1	no	—
147	C₄⋊C₄	—	47	2	no	—
148	C₄⋊C₄	—	94	2	no	—
149	C₂²⋊C₄	—	47	2	no	—
150	C₂²⋊C₄	—	94	4	no	—
151	C₄²	—	94	2	no	—
152	C₄²	C₂²×D₄₇	1	1	yes	characteristic
153	C₂³	—	2	4	no	—
154	C₂³	—	47	2	no	—
155	Q₈	—	47	4	no	—
156	Q₈	—	94	2	no	—
157	D₄	—	2	4	no	—
158	D₄	—	2	4	no	—
159	D₄	—	4	4	no	—
160	D₄	—	8	2	no	—
161	D₄	—	94	2	no	—
162	D₄	—	376	4	no	—
163	D₄	—	188	8	no	—
164	C₂×C₄	—	47	2	no	—
165	C₂×C₄	—	94	2	no	—
166	C₂×C₄	—	94	2	no	—
167	C₂×C₄	—	94	2	no	—
168	C₂×C₄	—	94	2	no	—
169	C₂×C₄	—	188	8	no	—
170	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
171	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
172	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
173	C₈	—	2	1	no	—
174	C₈	—	94	2	no	—
175	C₈	D₄×D₄₇	1	2	yes	—
176	C₂²	—	4	4	no	—
177	C₂²	—	4	4	no	—
178	C₂²	—	4	4	no	—
179	C₂²	—	94	2	no	—
180	C₂²	—	94	2	no	—
181	C₂²	—	94	2	no	—
182	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	94	2	no	—
185	C₄	—	94	4	no	—
186	C₄	—	188	2	no	—
187	C₄	#13094043	1	2	yes	complemented
188	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
189	C₄	#13094069	1	1	yes	characteristic
190	C₂	—	8	4	no	—
191	C₂	—	188	2	no	—
192	C₂	#15689758	1	1	yes	characteristic
193	C₂	C₂×#13094043	1	1	yes	characteristic
194	C₂	#15689924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094043

C₂

yes

maximal, complemented

#15689793

C₂

yes

maximal, complemented

#15689356

C₂

yes

maximal, complemented

#15689320

C₂

yes

maximal, complemented

#15689265

C₂

yes

maximal, complemented

C₄×Dic₁₈₈

C₂

yes

maximal, characteristic, complemented

C₈×D₁₈₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#493

C₂

yes

maximal, characteristic, complemented

C₂×#10625352

C₂²

yes

—

#13094043

—

C₂×Dic₁₈₈

C₂²

yes

characteristic

C₂×#10625335

C₂²

yes

—

C₂×C₈×D₄₇

C₂²

yes

—

#13094020

—

#13093996

—

#13093995

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

C₄×D₁₈₈

C₂²

yes

characteristic

D₄×Dic₄₇

—

#13093958

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093949

C₂²

yes

—

#13093945

C₂²

yes

—

(C₂×D₄₇)⋊C₈

C₂²

yes

—

C₂×D₈×C₄₇

C₂²

yes

—

C₂×D₈×C₄₇

C₂²

yes

—

#127×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂³

yes

—

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

#10625352

—

#10625352

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

C₂³

yes

—

C₂×Dic₉₄

C₂³

yes

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

#10625335

—

(C₂×D₄₇)⋊C₄

D₄

yes

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

C₂³

yes

—

Dic₁₈₈

D₄

yes

complemented

C₈×D₄₇

—

C₂³×C₄₇

—

C₂²×D₄₇

C₂×D₄

yes

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

C₂×D₄

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#135

yes

complemented

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#3287

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

D₄²

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂×#135

—

#550

—

100

#493

D₄₇

yes

characteristic, complemented

101

#466

—

102

#460

—

103

#449

—

104

C₄×Q₁₆

—

105

C₈×D₄

—

106

C₄₇

#3287

yes

characteristic, complemented, Sylow

107

C₂×#41

—

108

#135

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×D₈

C₂×D₄₇

yes

—

112

C₂×D₈

C₂×D₄₇

yes

—

113

C₂²×C₈

—

114

#127

—

115

#127

C₂×D₄₇

yes

—

116

#121

—

117

#121

—

118

C₄×Q₈

—

119

C₄×D₄

—

120

C₄×D₄

—

121

#107

—

122

C₄⋊C₈

—

123

#103

—

124

#102

—

125

C₂²⋊C₈

—

126

C₄×C₈

C₂×D₄₇

yes

characteristic

127

#41

—

128

#41

—

188

—

129

C₂×Q₈

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

C₂²×D₄₇

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

Q₁₆

—

139

SD₁₆

—

140

D₈

—

141

D₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

C₂²×D₄₇

yes

characteristic

145

C₂×C₈

C₂²×D₄₇

yes

characteristic

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₂²⋊C₄

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₄²

C₂²×D₄₇

yes

characteristic

153

C₂³

—

154

C₂³

—

155

Q₈

—

156

Q₈

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

376

—

163

D₄

—

188

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

188

—

170

C₂×C₄

D₄×D₄₇

yes

characteristic

171

C₂×C₄

D₄×D₄₇

yes

characteristic

172

C₂×C₄

C₂³×D₄₇

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

D₄×D₄₇

yes

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₂×D₄×D₄₇

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

188

—

187

C₄

#13094043

yes

complemented

188

C₄

C₂×D₄×D₄₇

yes

characteristic

189

C₄

#13094069

yes

characteristic

190

C₂

—

191

C₂

—

188

—

192

C₂

#15689758

yes

characteristic

193

C₂

C₂×#13094043

yes

characteristic

194

C₂

#15689924

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#493	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#493	D₄₇
hypocenter	C₄₇	#3287
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#493

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. C₂×#13094043	C₂	semidirect
2. #15689793	C₂	semidirect
3. #15689356	C₂	semidirect
4. #15689320	C₂	semidirect
5. #15689265	C₂	semidirect
6. C₄×Dic₁₈₈	C₂	semidirect
7. C₈×D₁₈₈	C₂	semidirect
8. C₄₇×#493	C₂	semidirect
53. Dic₁₈₈	D₄	semidirect
86. C₄×C₄₇	#135	semidirect
100. #493	D₄₇	semidirect
106. C₄₇	#3287	semidirect
187. C₄	#13094043	semidirect

H ≅ G/N

type

1. C₂×#13094043

C₂

semidirect

2. #15689793

C₂

semidirect

3. #15689356

C₂

semidirect

4. #15689320

C₂

semidirect

5. #15689265

C₂

semidirect

6. C₄×Dic₁₈₈

C₂

semidirect

7. C₈×D₁₈₈

C₂

semidirect

8. C₄₇×#493

C₂

semidirect

53. Dic₁₈₈

D₄

semidirect

86. C₄×C₄₇

#135

semidirect

100. #493

D₄₇

semidirect

106. C₄₇

#3287

semidirect

187. C₄

#13094043

semidirect

Subgroups — #20255146

Conjugacy classes of subgroups

Complements