Group #20244521 — subgroups

Subgroups — #20244521

Order 6016.

326 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13093944	C₂	1	2	yes	maximal
2	#15689025	C₂	1	1	yes	maximal, characteristic
3	#15688885	C₂	1	2	yes	maximal, complemented
4	#15688878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
10	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
11	#13093944	—	2	4	no	—
12	C₂²×Q₈×C₄₇	—	2	1	no	—
13	C₄⋊Q₈×C₄₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂³×D₄₇	—	2	1	no	—
18	C₂×Q₈×C₄₇	—	2	4	no	—
19	C₂×Q₈×C₄₇	—	4	2	no	—
20	C₂×Q₈×C₄₇	—	4	2	no	—
21	C₂²×C₄×C₄₇	—	2	2	no	—
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	1	no	—
26	C₂×D₁₈₈	—	2	2	no	—
27	C₂×D₁₈₈	—	2	2	no	—
28	C₂×D₁₈₈	—	2	1	no	—
29	C₂×D₁₈₈	Q₈	1	2	yes	complemented
30	C₄⋊C₄×C₄₇	—	2	4	no	—
31	C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₄⋊C₄×C₄₇	D₄	1	4	yes	—
33	C₄²×C₄₇	—	2	2	no	—
34	(C₂×D₄₇)⋊C₄	—	4	2	no	—
35	(C₂×D₄₇)⋊C₄	—	4	2	no	—
36	C₄⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂×C₄₇⋊C₈	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₄₇	—	2	2	no	—
42	C₂²×D₄₇	—	4	1	no	—
43	C₂²×D₄₇	—	4	1	no	—
44	C₂²×D₄₇	—	4	1	no	—
45	C₂²×D₄₇	—	4	1	no	—
46	C₂²×D₄₇	—	4	1	no	—
47	C₂²×D₄₇	—	4	1	no	—
48	Q₈×C₄₇	—	4	8	no	—
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	4	2	no	—
55	C₂×C₄×C₄₇	—	4	2	no	—
56	C₂×C₄×C₄₇	D₈	1	2	yes	—
57	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#41	1	2	yes	—
61	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂×Dic₄₇	—	4	1	no	—
63	C₂×Dic₄₇	—	4	1	no	—
64	C₂×Dic₄₇	—	4	1	no	—
65	D₁₈₈	—	2	4	no	—
66	D₁₈₈	—	4	2	no	—
67	C₄₇⋊C₈	—	4	2	no	—
68	C₂²×C₄₇	#120	1	1	yes	characteristic
69	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₄₇	#124	1	1	yes	characteristic
71	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
72	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂²×C₄₇	#137	1	1	yes	characteristic
75	C₂×D₄₇	—	4	2	no	—
76	C₂×D₄₇	—	4	2	no	—
77	C₂×D₄₇	—	4	2	no	—
78	C₂×D₄₇	—	8	1	no	—
79	C₂×D₄₇	—	8	1	no	—
80	C₂×D₄₇	—	8	1	no	—
81	C₂×D₄₇	—	8	1	no	—
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	4	4	no	—
85	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
86	C₄×C₄₇	#124	1	2	yes	—
87	Dic₄₇	—	8	1	no	—
88	#2023	—	47	1	no	maximal, Sylow
89	C₂×C₄₇	#393	1	1	yes	characteristic
90	C₂×C₄₇	#447	1	1	yes	characteristic
91	C₂×C₄₇	#448	1	1	yes	characteristic
92	C₂×C₄₇	#474	1	1	yes	characteristic
93	C₂×C₄₇	#476	1	1	yes	characteristic
94	C₂×C₄₇	#486	1	1	yes	characteristic
95	C₂×C₄₇	#489	1	1	yes	characteristic
96	D₄₇	—	8	2	no	—
97	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
98	C₂×#102	—	47	2	no	—
99	#390	—	47	1	no	—
100	#340	—	47	2	no	—
101	#336	—	47	1	no	—
102	C₄₇	#2023	1	1	yes	characteristic, complemented, Sylow
103	C₂²×Q₈	—	2	1	no	—
104	C₂²×D₄	—	47	1	no	—
105	C₂²×C₈	—	47	2	no	—
106	C₄⋊Q₈	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	47	1	no	—
109	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
110	C₂×C₂²⋊C₄	—	94	1	no	—
111	C₂×C₂²⋊C₄	—	94	1	no	—
112	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
113	#102	—	94	4	no	—
114	C₂⁴	—	94	1	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	2	no	—
117	C₂×Q₈	—	4	2	no	—
118	C₂×D₄	—	47	2	no	—
119	C₂×D₄	—	94	2	no	—
120	C₂×D₄	—	94	1	no	—
121	C₂×D₄	—	94	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	94	1	no	—
124	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
127	C₂×C₈	—	94	2	no	—
128	C₂×C₈	—	94	2	no	—
129	C₂×C₈	—	94	2	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	94	2	no	—
133	C₄⋊C₄	C₂²⋊D₄₇	1	4	yes	—
134	C₂²⋊C₄	—	188	2	no	—
135	C₂²⋊C₄	—	188	2	no	—
136	C₄²	—	2	2	no	—
137	C₂³	—	94	2	no	—
138	C₂³	—	188	1	no	—
139	C₂³	—	188	1	no	—
140	C₂³	—	188	1	no	—
141	C₂³	—	188	1	no	—
142	C₂³	—	188	1	no	—
143	C₂³	—	188	1	no	—
144	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
145	Q₈	—	4	8	no	—
146	D₄	—	94	4	no	—
147	D₄	—	188	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	188	1	no	—
156	C₂×C₄	—	188	1	no	—
157	C₂×C₄	—	188	1	no	—
158	C₂×C₄	#10625334	1	2	yes	—
159	C₂×C₄	#10625336	1	2	yes	—
160	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
161	C₂×C₄	#10625354	1	2	yes	—
162	C₂×C₄	#10625354	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
164	C₈	—	188	2	no	—
165	C₂²	—	376	1	no	—
166	C₂²	—	376	1	no	—
167	C₂²	—	376	1	no	—
168	C₂²	—	376	1	no	—
169	C₂²	—	188	2	no	—
170	C₂²	—	188	2	no	—
171	C₂²	—	188	2	no	—
172	C₂²	#13093993	1	1	yes	characteristic
173	C₂²	#13094000	1	1	yes	characteristic
174	C₂²	#13094013	1	1	yes	characteristic
175	C₂²	C₂×#10625334	1	1	yes	characteristic
176	C₂²	C₂×#10625336	1	1	yes	characteristic
177	C₂²	#13094050	1	1	yes	characteristic
178	C₂²	#13094055	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	4	no	—
182	C₄	—	376	1	no	—
183	C₄	#13094000	1	2	yes	—
184	C₄	#13094013	1	2	yes	—
185	C₂	—	376	2	no	—
186	C₂	#15689034	1	1	yes	characteristic
187	C₂	#15689257	1	1	yes	characteristic
188	C₂	#15689262	1	1	yes	characteristic
189	C₂	#15689400	1	1	yes	characteristic
190	C₂	#15689410	1	1	yes	characteristic
191	C₂	#15689471	1	1	yes	characteristic
192	C₂	#15689485	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093944

C₂

yes

maximal

#15689025

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal, complemented

#15688878

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

—

C₂×(C₂×D₄₇)⋊C₄

—

#13093944

—

C₂²×Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

Q₈

yes

complemented

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×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

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

#124

yes

characteristic

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₂²×C₄₇

C₂×SD₁₆

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

#124

yes

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic, complemented

C₂×#102

—

#390

—

100

#340

—

101

#336

—

102

C₄₇

#2023

yes

characteristic, complemented, Sylow

103

C₂²×Q₈

—

104

C₂²×D₄

—

105

C₂²×C₈

—

106

C₄⋊Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

C₂×D₄₇

yes

characteristic

113

#102

—

114

C₂⁴

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

125

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

126

C₂²×C₄

C₂²×D₄₇

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₂²⋊D₄₇

yes

—

134

C₂²⋊C₄

—

188

—

135

C₂²⋊C₄

—

188

—

136

C₄²

—

137

C₂³

—

138

C₂³

—

188

—

139

C₂³

—

188

—

140

C₂³

—

188

—

141

C₂³

—

188

—

142

C₂³

—

188

—

143

C₂³

—

188

—

144

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

188

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

188

—

156

C₂×C₄

—

188

—

157

C₂×C₄

—

188

—

158

C₂×C₄

#10625334

yes

—

159

C₂×C₄

#10625336

yes

—

160

C₂×C₄

Q₈×D₄₇

yes

characteristic

161

C₂×C₄

#10625354

yes

—

162

C₂×C₄

#10625354

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

164

C₈

—

188

—

165

C₂²

—

376

—

166

C₂²

—

376

—

167

C₂²

—

376

—

168

C₂²

—

376

—

169

C₂²

—

188

—

170

C₂²

—

188

—

171

C₂²

—

188

—

172

C₂²

#13093993

yes

characteristic

173

C₂²

#13094000

yes

characteristic

174

C₂²

#13094013

yes

characteristic

175

C₂²

C₂×#10625334

yes

characteristic

176

C₂²

C₂×#10625336

yes

characteristic

177

C₂²

#13094050

yes

characteristic

178

C₂²

#13094055

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

376

—

183

C₄

#13094000

yes

—

184

C₄

#13094013

yes

—

185

C₂

—

376

—

186

C₂

#15689034

yes

characteristic

187

C₂

#15689257

yes

characteristic

188

C₂

#15689262

yes

characteristic

189

C₂

#15689400

yes

characteristic

190

C₂

#15689410

yes

characteristic

191

C₂

#15689471

yes

characteristic

192

C₂

#15689485

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#2023
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

#2023

LCS

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #15688885	C₂	semidirect
4. #15688878	C₂	semidirect
5. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
29. C₂×D₁₈₈	Q₈	semidirect
97. C₂×C₄⋊Q₈	D₄₇	semidirect
102. C₄₇	#2023	semidirect

H ≅ G/N

type

3. #15688885

C₂

semidirect

4. #15688878

C₂

semidirect

5. C₂×C₄⋊Q₈×C₄₇

C₂

semidirect

29. C₂×D₁₈₈

Q₈

semidirect

97. C₂×C₄⋊Q₈

D₄₇

semidirect

102. C₄₇

#2023

semidirect

Subgroups — #20244521

Conjugacy classes of subgroups

Complements