Group #20252871 — subgroups

Subgroups — #20252871

Order 6016.

480 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13094055	C₂	1	2	yes	maximal
2	C₄×#10625350	C₂	1	1	yes	maximal, characteristic
3	#15689330	C₂	1	4	yes	maximal, complemented
4	#15689325	C₂	1	4	yes	maximal, complemented
5	#15689147	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689132	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625350	C₂²	1	1	yes	characteristic
9	#13094055	—	2	8	no	—
10	C₂×#10625337	C₂²	1	4	yes	—
11	C₂×#10625336	C₂²	1	4	yes	—
12	C₂×#10625329	C₂²	1	2	yes	—
13	C₄×Dic₉₄	C₂²	1	2	yes	—
14	C₄×D₁₈₈	C₂²	1	2	yes	—
15	C₄×C₂²⋊D₄₇	—	2	2	no	—
16	#13093983	C₂²	1	1	yes	characteristic
17	#13093981	—	2	1	no	—
18	C₄²×D₄₇	—	2	1	no	—
19	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
20	#13093948	C₂²	1	8	yes	—
21	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
22	C₄⋊Q₈×C₄₇	—	2	1	no	—
23	C₄⋊Q₈×C₄₇	—	2	1	no	—
24	C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₄₇	—	2	2	no	—
28	C₂×Q₈×C₄₇	—	2	4	no	—
29	C₂×Q₈×C₄₇	—	2	4	no	—
30	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	D₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₄₇	—	2	1	no	—
35	#10625350	—	2	2	no	—
36	#10625350	D₄	1	4	yes	—
37	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₄₇	—	2	1	no	—
39	C₂×C₄×D₄₇	—	2	2	no	—
40	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₂³	1	2	yes	—
46	C₂²⋊Dic₄₇	—	2	1	no	—
47	#10625337	—	2	8	no	—
48	#10625336	—	2	8	no	—
49	(C₂×D₄₇)⋊C₄	—	2	2	no	—
50	C₄⋊Dic₄₇	C₂³	1	2	yes	—
51	Dic₄₇⋊C₄	—	2	2	no	—
52	C₄×Dic₄₇	—	2	1	no	—
53	C₄×Dic₄₇	—	2	2	no	—
54	#10625329	—	2	4	no	—
55	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
56	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×D₄₇	—	2	1	no	—
58	Q₈×C₄₇	—	2	4	no	—
59	Q₈×C₄₇	—	2	4	no	—
60	Q₈×C₄₇	—	4	4	no	—
61	C₂×C₄×C₄₇	—	2	4	no	—
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	—	4	4	no	—
64	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₄₇	#41	1	2	yes	—
69	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₄₇	—	2	4	no	—
71	C₂×Dic₄₇	—	2	1	no	—
72	C₂×Dic₄₇	—	2	2	no	—
73	D₁₈₈	—	2	1	no	—
74	D₁₈₈	C₂×D₄	1	2	yes	—
75	C₄×D₄₇	—	2	4	no	—
76	C₄×D₄₇	—	4	4	no	—
77	Dic₉₄	—	2	1	no	—
78	Dic₉₄	C₂×D₄	1	2	yes	—
79	C₄₇⋊C₈	—	2	4	no	—
80	C₂²×C₄₇	—	2	2	no	—
81	C₂²×C₄₇	#121	1	2	yes	—
82	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₄₇	—	2	2	no	—
84	C₂×D₄₇	—	4	2	no	—
85	C₄×C₄₇	—	2	2	no	—
86	C₄×C₄₇	—	4	4	no	—
87	C₄×C₄₇	#121	1	2	yes	—
88	C₄×C₄₇	#137	1	4	yes	complemented
89	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
91	Dic₄₇	—	2	2	no	—
92	Dic₄₇	—	4	2	no	—
93	#3039	—	47	1	no	maximal, Sylow
94	C₂×C₄₇	—	2	2	no	—
95	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
96	C₂×C₄₇	C₂×#137	1	2	yes	—
97	D₄₇	—	4	2	no	—
98	C₂×#137	—	47	2	no	—
99	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
100	C₄×#41	—	47	1	no	—
101	#462	—	47	4	no	—
102	#461	—	47	4	no	—
103	#423	—	47	1	no	—
104	#419	—	47	2	no	—
105	C₄₇	#3039	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	—	47	1	no	—
107	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
108	#137	—	94	8	no	—
109	C₂×Q₁₆	—	47	4	no	—
110	C₂×SD₁₆	—	47	4	no	—
111	C₂×M₁₆	—	47	2	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
115	C₄×Q₈	—	47	2	no	—
116	C₄×D₄	—	47	2	no	—
117	C₄×D₄	—	94	2	no	—
118	#117	—	47	1	no	—
119	#117	—	94	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	94	1	no	—
122	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
123	C₄⋊C₈	—	47	4	no	—
124	#103	—	47	8	no	—
125	#41	—	47	4	no	—
126	#41	—	94	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	—	47	1	no	—
131	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
132	C₂×D₄	—	47	1	no	—
133	C₂×D₄	—	94	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	94	1	no	—
136	C₂²×C₄	—	94	2	no	—
137	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
138	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	Q₁₆	—	94	8	no	—
140	SD₁₆	—	94	8	no	—
141	M₁₆	—	94	4	no	—
142	C₂×C₈	—	47	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	47	2	no	—
146	C₄⋊C₄	—	94	2	no	—
147	C₂²⋊C₄	—	94	1	no	—
148	C₂²⋊C₄	—	94	2	no	—
149	C₄²	—	94	1	no	—
150	C₄²	—	94	2	no	—
151	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
152	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
153	C₄²	C₂²×D₄₇	1	2	yes	—
154	C₂³	—	94	1	no	—
155	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
156	Q₈	—	2	4	no	—
157	Q₈	—	2	4	no	—
158	Q₈	—	4	4	no	—
159	Q₈	—	47	2	no	—
160	Q₈	—	94	1	no	—
161	D₄	—	47	2	no	—
162	D₄	—	94	4	no	—
163	D₄	—	94	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	94	1	no	—
168	C₂×C₄	—	94	4	no	—
169	C₂×C₄	—	94	2	no	—
170	C₂×C₄	—	188	4	no	—
171	C₂×C₄	D₄×D₄₇	1	2	yes	—
172	C₂×C₄	#10625352	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
176	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
177	C₈	—	94	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	94	2	no	—
180	C₂²	—	188	2	no	—
181	C₂²	#13093995	1	2	yes	—
182	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	94	2	no	—
186	C₄	—	188	2	no	—
187	C₄	#13093995	1	2	yes	—
188	C₄	#13094055	1	4	yes	complemented
189	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
190	C₄	C₂×#10625352	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	188	2	no	—
193	C₂	C₂×#13093995	1	1	yes	characteristic
194	C₂	C₂×#13094055	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094055

C₂

yes

maximal

C₄×#10625350

C₂

yes

maximal, characteristic

#15689330

C₂

yes

maximal, complemented

#15689325

C₂

yes

maximal, complemented

#15689147

C₂

yes

maximal, characteristic, complemented

#15689132

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

characteristic

#13094055

—

C₂×#10625337

C₂²

yes

—

C₂×#10625336

C₂²

yes

—

C₂×#10625329

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

—

C₄×D₁₈₈

C₂²

yes

—

C₄×C₂²⋊D₄₇

—

#13093983

C₂²

yes

characteristic

#13093981

—

C₄²×D₄₇

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093948

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

D₄

yes

—

C₂×D₁₈₈

C₂³

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

—

#10625337

—

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×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

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

C₂×D₄

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

C₄×C₄₇

#137

yes

complemented

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#3039

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#121

yes

characteristic

C₂×C₄₇

C₂×#137

yes

—

D₄₇

—

C₂×#137

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic, complemented

100

C₄×#41

—

101

#462

—

102

#461

—

103

#423

—

104

#419

—

105

C₄₇

#3039

yes

characteristic, complemented, Sylow

106

C₂×#41

—

107

C₂²×Q₈

C₂×D₄₇

yes

—

108

#137

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×M₁₆

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×D₄₇

yes

—

115

C₄×Q₈

—

116

C₄×D₄

—

117

C₄×D₄

—

118

#117

—

119

#117

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂×D₄₇

yes

characteristic

123

C₄⋊C₈

—

124

#103

—

125

#41

—

126

#41

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

C₂²×D₄₇

yes

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊D₄₇

yes

—

138

C₂²×C₄

C₂²×D₄₇

yes

characteristic

139

Q₁₆

—

140

SD₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂²⋊D₄₇

yes

characteristic

152

C₄²

C₂²⋊D₄₇

yes

characteristic

153

C₄²

C₂²×D₄₇

yes

—

154

C₂³

—

155

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

D₄

—

162

D₄

—

163

D₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

188

—

171

C₂×C₄

D₄×D₄₇

yes

—

172

C₂×C₄

#10625352

yes

—

173

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

174

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

175

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

176

C₂×C₄

C₂³×D₄₇

yes

characteristic

177

C₈

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

188

—

181

C₂²

#13093995

yes

—

182

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

188

—

187

C₄

#13093995

yes

—

188

C₄

#13094055

yes

complemented

189

C₄

C₂×D₄×D₄₇

yes

characteristic

190

C₄

C₂×#10625352

yes

characteristic

191

C₂

—

192

C₂

—

188

—

193

C₂

C₂×#13093995

yes

characteristic

194

C₂

C₂×#13094055

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#3039
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

#3039

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #15689330	C₂	semidirect
4. #15689325	C₂	semidirect
5. #15689147	C₂	semidirect
6. #15689132	C₂	semidirect
7. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
88. C₄×C₄₇	#137	semidirect
99. C₂×C₄⋊Q₈	D₄₇	semidirect
105. C₄₇	#3039	semidirect
188. C₄	#13094055	semidirect

H ≅ G/N

type

3. #15689330

C₂

semidirect

4. #15689325

C₂

semidirect

5. #15689147

C₂

semidirect

6. #15689132

C₂

semidirect

7. C₂×C₄⋊Q₈×C₄₇

C₂

semidirect

88. C₄×C₄₇

#137

semidirect

99. C₂×C₄⋊Q₈

D₄₇

semidirect

105. C₄₇

#3039

semidirect

188. C₄

#13094055

semidirect

Subgroups — #20252871

Conjugacy classes of subgroups

Complements