Group #20248746 — subgroups

Subgroups — #20248746

Order 6016.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#15689058	C₂	1	2	yes	maximal, complemented
3	#15689034	C₂	1	2	yes	maximal, complemented
4	#15689025	C₂	1	2	yes	maximal
5	#15689021	C₂	1	2	yes	maximal, complemented
6	#15688994	C₂	1	4	yes	maximal, complemented
7	#15688989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
11	C₄×D₁₈₈	—	2	4	no	—
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
15	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
16	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	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	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₄₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂³×D₄₇	C₂³	1	2	yes	—
28	C₂×Q₈×C₄₇	—	2	2	no	—
29	C₂×Q₈×C₄₇	—	2	4	no	—
30	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₄₇	C₂³	1	4	yes	—
36	C₂²×Dic₄₇	C₂³	1	2	yes	—
37	C₂×D₁₈₈	—	2	2	no	—
38	C₂×D₁₈₈	—	2	2	no	—
39	C₂×D₁₈₈	Q₈	1	4	yes	complemented
40	C₂×C₄×D₄₇	—	2	4	no	—
41	C₂×C₄×D₄₇	—	4	4	no	—
42	C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	4	no	—
45	C₄²×C₄₇	D₄	1	4	yes	—
46	(C₂×D₄₇)⋊C₄	—	2	4	no	—
47	(C₂×D₄₇)⋊C₄	—	2	8	no	—
48	(C₂×D₄₇)⋊C₄	—	4	4	no	—
49	C₄⋊Dic₄₇	—	2	2	no	—
50	C₄⋊Dic₄₇	—	2	4	no	—
51	Dic₄₇⋊C₄	—	4	4	no	—
52	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂²×D₄₇	—	2	2	no	—
54	C₂²×D₄₇	—	2	4	no	—
55	C₂²×D₄₇	—	2	2	no	—
56	C₂²×D₄₇	—	2	4	no	—
57	C₂²×D₄₇	C₂×Q₈	1	4	yes	—
58	Q₈×C₄₇	—	2	8	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	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₄₇	#41	1	4	yes	—
71	C₂×C₄×C₄₇	#41	1	2	yes	—
72	C₂×Dic₄₇	—	2	2	no	—
73	C₂×Dic₄₇	—	2	4	no	—
74	C₂×Dic₄₇	—	2	4	no	—
75	C₂×Dic₄₇	—	2	4	no	—
76	C₂×Dic₄₇	—	2	4	no	—
77	D₁₈₈	—	2	8	no	—
78	C₄×D₄₇	—	4	8	no	—
79	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₄₇	C₂×#41	1	2	yes	—
82	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
83	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
84	C₂×D₄₇	—	2	4	no	—
85	C₂×D₄₇	—	2	8	no	—
86	C₂×D₄₇	—	4	2	no	—
87	C₂×D₄₇	—	4	2	no	—
88	C₂×D₄₇	—	4	4	no	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	4	2	no	—
93	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
94	Dic₄₇	—	4	2	no	—
95	Dic₄₇	—	4	4	no	—
96	#2580	—	47	1	no	maximal, Sylow
97	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄₇	#525	1	1	yes	characteristic
99	C₂×C₄₇	#535	1	1	yes	characteristic
100	C₂×C₄₇	#554	1	2	yes	—
101	C₂×C₄₇	#559	1	2	yes	—
102	D₄₇	—	4	4	no	—
103	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic, complemented
104	C₂×C₄×D₄	—	47	1	no	—
105	#399	—	47	2	no	—
106	#393	—	47	2	no	—
107	#390	—	47	2	no	—
108	#389	—	47	2	no	—
109	#385	—	47	4	no	—
110	#382	—	47	1	no	—
111	C₄₇	#2580	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
113	C₂²×D₄	—	47	1	no	—
114	C₂³×C₄	—	47	2	no	—
115	C₄×Q₈	—	2	4	no	—
116	C₄×D₄	—	94	4	no	—
117	C₂×C₄⋊C₄	—	47	1	no	—
118	C₂×C₄⋊C₄	—	47	2	no	—
119	C₂×C₄⋊C₄	—	47	4	no	—
120	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	—	47	2	no	—
123	C₂×C₂²⋊C₄	—	47	4	no	—
124	C₂×C₂²⋊C₄	—	47	4	no	—
125	C₂×C₄²	C₂×D₄₇	1	2	yes	—
126	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
127	#95	—	47	2	no	—
128	#95	—	47	2	no	—
129	#95	—	47	4	no	—
130	C₂⁴	—	47	2	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×D₄	—	47	4	no	—
134	C₂×D₄	—	94	2	no	—
135	C₂×D₄	—	94	2	no	—
136	C₂²×C₄	—	47	4	no	—
137	C₂²×C₄	—	47	2	no	—
138	C₂²×C₄	—	94	4	no	—
139	C₂²×C₄	—	188	4	no	—
140	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
141	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
142	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	94	4	no	—
148	C₄⋊C₄	—	94	2	no	—
149	C₄⋊C₄	—	188	4	no	—
150	C₂²⋊C₄	—	94	8	no	—
151	C₂²⋊C₄	—	94	4	no	—
152	C₂²⋊C₄	—	188	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	C₂²⋊D₄₇	1	4	yes	—
155	C₂³	—	47	4	no	—
156	C₂³	—	94	2	no	—
157	C₂³	—	94	4	no	—
158	C₂³	—	94	2	no	—
159	C₂³	—	94	4	no	—
160	C₂³	C₂³×D₄₇	1	1	yes	characteristic
161	Q₈	—	2	8	no	—
162	D₄	—	94	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	94	2	no	—
171	C₂×C₄	—	94	4	no	—
172	C₂×C₄	—	94	4	no	—
173	C₂×C₄	—	94	4	no	—
174	C₂×C₄	—	94	4	no	—
175	C₂×C₄	—	188	8	no	—
176	C₂×C₄	#10625350	1	4	yes	—
177	C₂×C₄	Q₈×D₄₇	1	2	yes	—
178	C₂×C₄	#10625354	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
180	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
182	C₂²	—	94	8	no	—
183	C₂²	—	94	4	no	—
184	C₂²	—	188	4	no	—
185	C₂²	—	188	2	no	—
186	C₂²	—	188	2	no	—
187	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
189	C₂²	C₂×#10625350	1	2	yes	—
190	C₂²	C₂×#10625354	1	1	yes	characteristic
191	C₂²	#13094070	1	2	yes	—
192	C₄	—	2	4	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	188	4	no	—
197	C₄	—	188	2	no	—
198	C₄	#13094000	1	4	yes	—
199	C₂	—	188	4	no	—
200	C₂	C₂×#13094000	1	1	yes	characteristic
201	C₂	#15689634	1	1	yes	characteristic
202	C₂	#15689687	1	1	yes	characteristic
203	C₂	#15689833	1	2	yes	—
204	C₂	#15689858	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₁₈₈

C₂

yes

maximal, characteristic

#15689058

C₂

yes

maximal, complemented

#15689034

C₂

yes

maximal, complemented

#15689025

C₂

yes

maximal

#15689021

C₂

yes

maximal, complemented

#15688994

C₂

yes

maximal, complemented

#15688989

C₂

yes

maximal, characteristic, complemented

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂²×C₄×D₄₇

C₂²

yes

—

C₄×D₁₈₈

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊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

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

Q₈

yes

complemented

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×Q₈

yes

—

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

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

—

D₁₈₈

—

C₄×D₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

#2580

—

maximal, Sylow

C₂×C₄₇

C₂×C₂²⋊Q₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

100

C₂×C₄₇

#554

yes

—

101

C₂×C₄₇

#559

yes

—

102

D₄₇

—

103

C₂×C₄×Q₈

D₄₇

yes

characteristic, complemented

104

C₂×C₄×D₄

—

105

#399

—

106

#393

—

107

#390

—

108

#389

—

109

#385

—

110

#382

—

111

C₄₇

#2580

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

C₂×D₄₇

yes

characteristic

113

C₂²×D₄

—

114

C₂³×C₄

—

115

C₄×Q₈

—

116

C₄×D₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

121

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

122

C₂×C₂²⋊C₄

—

123

C₂×C₂²⋊C₄

—

124

C₂×C₂²⋊C₄

—

125

C₂×C₄²

C₂×D₄₇

yes

—

126

C₂×C₄²

C₂×D₄₇

yes

characteristic

127

#95

—

128

#95

—

129

#95

—

130

C₂⁴

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

188

—

140

C₂²×C₄

C₂²×D₄₇

yes

—

141

C₂²×C₄

C₂²×D₄₇

yes

—

142

C₂²×C₄

C₂²×D₄₇

yes

characteristic

143

C₂²×C₄

C₂²×D₄₇

yes

characteristic

144

C₂²×C₄

C₂²×D₄₇

yes

characteristic

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

188

—

150

C₂²⋊C₄

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

188

—

153

C₄²

—

154

C₄²

C₂²⋊D₄₇

yes

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

C₂³×D₄₇

yes

characteristic

161

Q₈

—

162

D₄

—

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

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

188

—

176

C₂×C₄

#10625350

yes

—

177

C₂×C₄

Q₈×D₄₇

yes

—

178

C₂×C₄

#10625354

yes

—

179

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

180

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

181

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

188

—

185

C₂²

—

188

—

186

C₂²

—

188

—

187

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

188

C₂²

C₂×Q₈×D₄₇

yes

characteristic

189

C₂²

C₂×#10625350

yes

—

190

C₂²

C₂×#10625354

yes

characteristic

191

C₂²

#13094070

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

188

—

197

C₄

—

188

—

198

C₄

#13094000

yes

—

199

C₂

—

188

—

200

C₂

C₂×#13094000

yes

characteristic

201

C₂

#15689634

yes

characteristic

202

C₂

#15689687

yes

characteristic

203

C₂

#15689833

yes

—

204

C₂

#15689858

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄×Q₈×C₄₇	C₂
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	#2580
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄×Q₈×C₄₇

C₂

hypercenter

C₂×C₄×Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
2. #15689058	C₂	semidirect
3. #15689034	C₂	semidirect
5. #15689021	C₂	semidirect
6. #15688994	C₂	semidirect
7. #15688989	C₂	semidirect
8. C₂×C₄×Q₈×C₄₇	C₂	semidirect
39. C₂×D₁₈₈	Q₈	semidirect
103. C₂×C₄×Q₈	D₄₇	semidirect
111. C₄₇	#2580	semidirect

H ≅ G/N

type

2. #15689058

C₂

semidirect

3. #15689034

C₂

semidirect

5. #15689021

C₂

semidirect

6. #15688994

C₂

semidirect

7. #15688989

C₂

semidirect

8. C₂×C₄×Q₈×C₄₇

C₂

semidirect

39. C₂×D₁₈₈

Q₈

semidirect

103. C₂×C₄×Q₈

D₄₇

semidirect

111. C₄₇

#2580

semidirect

Subgroups — #20248746

Conjugacy classes of subgroups

Complements