Group #20253040 — subgroups

Subgroups — #20253040

Order 6016.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂²×#10625337	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094031	C₂	1	1	yes	maximal, characteristic
3	#15689694	C₂	1	1	yes	maximal, characteristic
4	#15689379	C₂	1	4	yes	maximal, complemented
5	#15689362	C₂	1	4	yes	maximal, complemented
6	#15689159	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689132	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689131	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#536	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
11	C₂×#10625337	—	2	4	no	—
12	C₂×#10625337	C₂²	1	2	yes	—
13	C₂×#10625337	C₂²	1	2	yes	—
14	#13094031	—	2	4	no	—
15	C₂×#10625329	C₂²	1	2	yes	—
16	C₂×#10625329	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	#13094022	—	2	1	no	—
20	#13094011	—	2	1	no	—
21	#13094005	—	2	2	no	—
22	#13094003	C₂²	1	4	yes	—
23	#13093983	C₂²	1	1	yes	characteristic
24	#13093957	C₂²	1	2	yes	—
25	#13093955	C₂²	1	2	yes	—
26	#13093948	C₂²	1	4	yes	—
27	#13093947	C₂²	1	4	yes	—
28	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₄₇	—	2	1	no	—
31	#124×C₄₇	—	2	1	no	—
32	#123×C₄₇	—	2	2	no	—
33	C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	—
34	#117×C₄₇	C₂²	1	1	yes	characteristic
35	#41×C₄₇	—	2	2	no	—
36	#41×C₄₇	—	2	2	no	—
37	C₂×Q₈×C₄₇	—	2	1	no	—
38	C₂×Q₈×C₄₇	—	2	2	no	—
39	C₂×Q₈×C₄₇	—	2	2	no	—
40	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
42	C₂×D₄×C₄₇	D₄	1	2	yes	—
43	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	—	2	1	no	—
45	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×Dic₄₇	—	2	1	no	—
48	C₂×Dic₉₄	—	2	2	no	—
49	C₂×Dic₉₄	—	2	2	no	—
50	C₂×Dic₉₄	—	2	1	no	—
51	C₂×Dic₉₄	C₂³	1	2	yes	—
52	C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₄₇	—	2	1	no	—
55	C₂²⋊C₄×C₄₇	—	2	4	no	—
56	C₄²×C₄₇	—	2	1	no	—
57	C₂²⋊Dic₄₇	—	2	1	no	—
58	C₂²⋊Dic₄₇	—	2	4	no	—
59	#10625337	—	2	4	no	—
60	#10625337	—	2	4	no	—
61	C₄⋊Dic₄₇	C₂³	1	2	yes	—
62	Dic₄₇⋊C₄	—	2	4	no	—
63	C₄×Dic₄₇	—	2	1	no	—
64	#10625329	—	2	2	no	—
65	#10625329	—	2	4	no	—
66	C₂×C₄₇⋊C₈	—	2	4	no	—
67	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
68	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
69	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂³×C₄₇	C₂×D₄	1	2	yes	—
72	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
73	Q₈×C₄₇	—	2	2	no	—
74	Q₈×C₄₇	—	2	2	no	—
75	Q₈×C₄₇	—	2	2	no	—
76	Q₈×C₄₇	—	4	2	no	—
77	D₄×C₄₇	—	2	4	no	—
78	D₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	4	2	no	—
84	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#41	1	2	yes	—
88	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₄₇	—	2	2	no	—
90	C₂×Dic₄₇	—	2	2	no	—
91	C₂×Dic₄₇	—	4	2	no	—
92	Dic₉₄	—	2	2	no	—
93	Dic₉₄	—	2	2	no	—
94	Dic₉₄	—	4	2	no	—
95	C₄₇⋊C₈	—	2	2	no	—
96	C₄₇⋊C₈	C₂×D₄	1	2	yes	—
97	C₄₇⋊C₈	C₂×D₄	1	2	yes	—
98	C₂²×C₄₇	—	2	2	no	—
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	—	2	4	no	—
101	C₂²×C₄₇	#121	1	2	yes	—
102	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₄₇	—	2	2	no	—
104	C₄×C₄₇	—	4	2	no	—
105	C₄×C₄₇	—	4	2	no	—
106	C₄×C₄₇	#121	1	2	yes	—
107	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
109	Dic₄₇	—	4	2	no	—
110	Dic₄₇	—	4	2	no	—
111	#3059	—	47	1	no	maximal, Sylow
112	C₂×C₄₇	—	2	2	no	—
113	C₂×C₄₇	—	4	2	no	—
114	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
115	C₂×C₄₇	#578	1	2	yes	—
116	C₂²×Q₁₆	—	47	1	no	—
117	C₂×#131	—	47	1	no	—
118	#536	—	47	1	no	—
119	#536	D₄₇	1	1	yes	characteristic
120	#470	—	47	4	no	—
121	#467	—	47	4	no	—
122	#427	—	47	1	no	—
123	#419	—	47	1	no	—
124	#419	—	47	1	no	—
125	C₄₇	#3059	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
127	C₂²×Q₈	—	47	1	no	—
128	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
129	C₂×Q₁₆	—	47	2	no	—
130	C₂×Q₁₆	—	47	2	no	—
131	C₂×Q₁₆	—	94	4	no	—
132	#131	—	94	4	no	—
133	C₂×M₁₆	—	47	1	no	—
134	C₂×M₁₆	—	47	2	no	—
135	C₂²×C₈	—	47	1	no	—
136	C₂²×C₈	—	47	2	no	—
137	C₄⋊Q₈	—	2	1	no	—
138	C₄⋊Q₈	—	94	1	no	—
139	#124	—	2	1	no	—
140	#124	—	94	1	no	—
141	#123	—	2	2	no	—
142	#123	—	94	2	no	—
143	C₂²⋊Q₈	—	47	4	no	—
144	C₂²⋊Q₈	C₂×D₄₇	1	4	yes	—
145	#117	—	47	1	no	—
146	#117	C₂×D₄₇	1	1	yes	characteristic
147	#107	—	47	2	no	—
148	#106	—	47	2	no	—
149	#103	—	47	4	no	—
150	#103	—	47	4	no	—
151	#41	—	2	2	no	—
152	#41	—	2	2	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×Q₈	—	47	2	no	—
157	C₂×Q₈	—	94	1	no	—
158	C₂×Q₈	—	94	2	no	—
159	C₂×Q₈	—	94	2	no	—
160	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
161	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
162	C₂×D₄	C₂²⋊D₄₇	1	2	yes	—
163	C₂×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	94	1	no	—
166	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
167	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	Q₁₆	—	94	4	no	—
169	Q₁₆	—	94	4	no	—
170	M₁₆	—	94	4	no	—
171	M₁₆	—	94	2	no	—
172	C₂×C₈	—	47	4	no	—
173	C₂×C₈	—	47	1	no	—
174	C₂×C₈	—	47	1	no	—
175	C₂×C₈	—	47	2	no	—
176	C₂×C₈	—	94	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	47	2	no	—
179	C₄⋊C₄	—	94	4	no	—
180	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	94	1	no	—
184	C₂²⋊C₄	—	94	4	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	94	1	no	—
187	C₂³	C₂×C₂²⋊D₄₇	1	2	yes	—
188	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
189	Q₈	—	2	2	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	4	2	no	—
193	Q₈	—	94	2	no	—
194	Q₈	—	94	2	no	—
195	Q₈	—	188	2	no	—
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	94	2	no	—
204	C₂×C₄	—	94	2	no	—
205	C₂×C₄	—	188	2	no	—
206	C₂×C₄	D₄×D₄₇	1	2	yes	—
207	C₂×C₄	#10625352	1	2	yes	—
208	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
211	C₈	—	47	2	no	—
212	C₈	—	47	2	no	—
213	C₈	—	94	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	#13093996	1	2	yes	—
218	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	188	2	no	—
223	C₄	—	188	2	no	—
224	C₄	#13093996	1	2	yes	—
225	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
226	C₄	C₂×#10625352	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	4	2	no	—
229	C₂	C₂×#13093996	1	1	yes	characteristic
230	C₂	#15689938	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625337

C₂

yes

maximal, characteristic, complemented

C₂×#13094031

C₂

yes

maximal, characteristic

#15689694

C₂

yes

maximal, characteristic

#15689379

C₂

yes

maximal, complemented

#15689362

C₂

yes

maximal, complemented

#15689159

C₂

yes

maximal, characteristic, complemented

#15689132

C₂

yes

maximal, characteristic, complemented

#15689131

C₂

yes

maximal, characteristic, complemented

C₄₇×#536

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂×#10625337

—

C₂×#10625337

C₂²

yes

—

C₂×#10625337

C₂²

yes

—

#13094031

—

C₂×#10625329

C₂²

yes

—

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

—

—

C₂²

yes

—

#13093983

C₂²

yes

characteristic

#13093957

C₂²

yes

—

#13093955

C₂²

yes

—

#13093948

C₂²

yes

—

#13093947

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

characteristic

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

—

#124×C₄₇

—

#123×C₄₇

—

C₂²⋊Q₈×C₄₇

C₂²

yes

—

#117×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

—

C₂×D₄×C₄₇

D₄

yes

—

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

#10625337

—

#10625337

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×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

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×D₄

yes

—

C₄₇⋊C₈

C₂×D₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

100

C₂²×C₄₇

—

101

C₂²×C₄₇

#121

yes

—

102

C₂²×C₄₇

C₂²×D₄

yes

characteristic

103

C₄×C₄₇

—

104

C₄×C₄₇

—

105

C₄×C₄₇

—

106

C₄×C₄₇

#121

yes

—

107

C₄×C₄₇

C₂²×D₄

yes

characteristic

108

C₄×C₄₇

C₂×#41

yes

characteristic

109

Dic₄₇

—

110

Dic₄₇

—

111

#3059

—

maximal, Sylow

112

C₂×C₄₇

—

113

C₂×C₄₇

—

114

C₂×C₄₇

C₂×#121

yes

characteristic

115

C₂×C₄₇

#578

yes

—

116

C₂²×Q₁₆

—

117

C₂×#131

—

118

#536

—

119

#536

D₄₇

yes

characteristic

120

#470

—

121

#467

—

122

#427

—

123

#419

—

124

#419

—

125

C₄₇

#3059

yes

characteristic, complemented, Sylow

126

C₂×#41

C₂×D₄₇

yes

characteristic

127

C₂²×Q₈

—

128

C₂²×Q₈

C₂×D₄₇

yes

characteristic

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

#131

—

133

C₂×M₁₆

—

134

C₂×M₁₆

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

#124

—

140

#124

—

141

#123

—

142

#123

—

143

C₂²⋊Q₈

—

144

C₂²⋊Q₈

C₂×D₄₇

yes

—

145

#117

—

146

#117

C₂×D₄₇

yes

characteristic

147

#107

—

148

#106

—

149

#103

—

150

#103

—

151

#41

—

152

#41

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

161

C₂×Q₈

C₂²×D₄₇

yes

—

162

C₂×D₄

C₂²⋊D₄₇

yes

—

163

C₂×D₄

C₂²⋊D₄₇

yes

characteristic

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

C₂²×D₄₇

yes

—

167

C₂²×C₄

C₂²×D₄₇

yes

characteristic

168

Q₁₆

—

169

Q₁₆

—

170

M₁₆

—

171

M₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

C₂²×D₄₇

yes

—

181

C₂²⋊C₄

—

182

C₂²⋊C₄

—

183

C₂²⋊C₄

—

184

C₂²⋊C₄

—

185

C₄²

—

186

C₄²

—

187

C₂³

C₂×C₂²⋊D₄₇

yes

—

188

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

Q₈

—

194

Q₈

—

195

Q₈

—

188

—

196

D₄

—

197

D₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

188

—

206

C₂×C₄

D₄×D₄₇

yes

—

207

C₂×C₄

#10625352

yes

—

208

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

209

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

210

C₂×C₄

C₂³×D₄₇

yes

characteristic

211

C₈

—

212

C₈

—

213

C₈

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

#13093996

yes

—

218

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

188

—

223

C₄

—

188

—

224

C₄

#13093996

yes

—

225

C₄

C₂×D₄×D₄₇

yes

characteristic

226

C₄

C₂×#10625352

yes

characteristic

227

C₂

—

228

C₂

—

229

C₂

C₂×#13093996

yes

characteristic

230

C₂

#15689938

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#536	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#536	D₄₇
hypocenter	C₄₇	#3059
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #536

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#536

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #536

N	H ≅ G/N	type
1. C₂²×#10625337	C₂	semidirect
4. #15689379	C₂	semidirect
5. #15689362	C₂	semidirect
6. #15689159	C₂	semidirect
7. #15689132	C₂	semidirect
8. #15689131	C₂	semidirect
125. C₄₇	#3059	semidirect

H ≅ G/N

type

1. C₂²×#10625337

C₂

semidirect

4. #15689379

C₂

semidirect

5. #15689362

C₂

semidirect

6. #15689159

C₂

semidirect

7. #15689132

C₂

semidirect

8. #15689131

C₂

semidirect

125. C₄₇

#3059

semidirect

Subgroups — #20253040

Conjugacy classes of subgroups

Complements