Group #25249160 — subgroups

Subgroups — #25249160

Order 11008.

1006 conjugacy classes of subgroups; only the 347 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 347 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×#14583297	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583295	C₂	1	2	yes	maximal
3	#19142991	C₂	1	2	yes	maximal
4	#19134750	C₂	1	8	yes	maximal
5	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₃×#498	C₂	1	1	yes	maximal, characteristic
7	C₂×#13063325	C₂²	1	2	yes	—
8	C₂×#13063323	—	2	4	no	—
9	#14583297	—	2	4	no	—
10	#14583295	—	2	8	no	—
11	C₂×#13063257	C₂²	1	4	yes	—
12	C₂×#13063256	C₂²	1	8	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
15	#14582768	—	2	4	no	—
16	#14582768	C₂²	1	8	yes	—
17	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
19	C₄₃×#498	—	2	4	no	—
20	C₂×#106×C₄₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₈₆	—	2	2	no	—
23	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
24	#13063325	—	2	4	no	—
25	#13063325	—	2	4	no	—
26	#13063323	—	4	16	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
29	C₂×Dic₄₃⋊C₄	—	2	16	no	—
30	C₂×C₄×Dic₄₃	D₄	1	4	yes	—
31	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
32	#13063257	—	2	8	no	—
33	#13063256	—	2	16	no	—
34	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
35	C₈×Dic₄₃	Q₈	1	16	yes	complemented
36	C₂²×Q₈×C₄₃	—	2	2	no	—
37	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₄₃	—	2	4	no	—
39	C₄⋊Q₈×C₄₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	#106×C₄₃	—	2	8	no	—
44	C₄×C₈×C₄₃	D₄	1	4	yes	—
45	C₂×Q₈×C₄₃	—	4	4	no	—
46	C₂×Q₈×C₄₃	—	4	8	no	—
47	C₂²×C₄×C₄₃	—	2	4	no	—
48	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
50	C₂²×Dic₄₃	—	2	4	no	—
51	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
52	C₂×Dic₈₆	—	4	4	no	—
53	C₂×Dic₈₆	—	4	8	no	—
54	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₄₃	C₂×Q₈	1	8	yes	—
56	C₄⋊C₄×C₄₃	—	2	4	no	—
57	C₄⋊C₄×C₄₃	—	2	4	no	—
58	C₄⋊C₄×C₄₃	—	4	4	no	—
59	C₄²×C₄₃	C₂×D₄	1	2	yes	—
60	C₄²×C₄₃	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₄₃	—	2	4	no	—
62	C₄⋊Dic₄₃	—	2	4	no	—
63	C₄⋊Dic₄₃	—	4	4	no	—
64	Dic₄₃⋊C₄	—	4	32	no	—
65	C₄×Dic₄₃	SD₁₆	1	16	yes	—
66	C₄×Dic₄₃	C₂×Q₈	1	8	yes	—
67	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
68	C₂×C₄₃⋊C₈	C₂×Q₈	1	8	yes	—
69	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
70	Q₈×C₄₃	—	8	8	no	—
71	C₂×C₄×C₄₃	—	4	4	no	—
72	C₂×C₄×C₄₃	—	4	8	no	—
73	C₂×C₄×C₄₃	#127	1	4	yes	—
74	C₂×C₄×C₄₃	C₂×SD₁₆	1	8	yes	—
75	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
78	C₂×Dic₄₃	—	4	4	no	—
79	C₂×Dic₄₃	—	4	8	no	—
80	C₂×Dic₄₃	C₄⋊Q₈	1	8	yes	—
81	C₂×Dic₄₃	C₂×SD₁₆	1	16	yes	—
82	Dic₈₆	—	8	8	no	—
83	C₈×C₄₃	C₄⋊Q₈	1	8	yes	complemented
84	C₄₃⋊C₈	C₄⋊Q₈	1	8	yes	complemented
85	#33310	—	43	1	no	maximal, Sylow
86	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
87	C₂²×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
88	C₂²×C₄₃	C₂²×SD₁₆	1	4	yes	—
89	C₄×C₄₃	—	8	4	no	—
90	C₄×C₄₃	#492	1	8	yes	—
91	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
92	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
93	Dic₄₃	—	8	4	no	—
94	Dic₄₃	#498	1	16	yes	complemented
95	C₂×#498	—	43	1	no	—
96	C₂×#498	—	43	2	no	—
97	C₂×#498	D₄₃	1	1	yes	characteristic
98	#2868	—	43	2	no	—
99	#1844	—	43	8	no	—
100	C₄²×C₈	—	43	1	no	—
101	C₂×C₄₃	#2868	1	1	yes	characteristic
102	C₂×C₄₃	C₂×#492	1	2	yes	—
103	C₂×C₄₃	C₂×#498	1	4	yes	—
104	C₂×C₄⋊Q₈	—	43	2	no	—
105	C₂×C₄⋊Q₈	—	86	4	no	—
106	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
107	#498	—	2	4	no	—
108	#498	—	86	8	no	—
109	#498	—	86	4	no	—
110	C₂×#106	—	43	4	no	—
111	C₂×#106	—	43	8	no	—
112	C₂×#106	C₂×D₄₃	1	4	yes	—
113	C₂×C₄×C₈	—	43	4	no	—
114	C₂×C₄×C₈	—	43	1	no	—
115	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
116	#384	—	43	8	no	—
117	#384	—	86	4	no	—
118	C₄³	—	43	1	no	—
119	C₄₃	#33310	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	2	2	no	—
121	C₂²×Q₈	—	86	2	no	—
122	C₂²×C₈	—	43	2	no	—
123	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	86	4	no	—
127	C₄⋊Q₈	—	86	4	no	—
128	C₄⋊Q₈	—	172	16	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	43	4	no	—
131	C₂×C₄⋊C₄	—	86	16	no	—
132	C₂×C₄⋊C₄	—	86	2	no	—
133	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
134	C₂×C₄²	—	43	4	no	—
135	C₂×C₄²	—	43	2	no	—
136	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
137	#106	—	2	8	no	—
138	#106	—	86	16	no	—
139	#106	—	86	8	no	—
140	C₄×C₈	—	43	16	no	—
141	C₄×C₈	—	43	4	no	—
142	C₄×C₈	C₂²⋊D₄₃	1	4	yes	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	4	8	no	—
145	C₂×Q₈	—	172	4	no	—
146	C₂×Q₈	—	172	8	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	43	4	no	—
149	C₂²×C₄	—	86	4	no	—
150	C₂²×C₄	D₄×D₄₃	1	2	yes	—
151	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
152	C₂×C₈	—	43	4	no	—
153	C₂×C₈	—	43	8	no	—
154	C₂×C₈	Q₈×D₄₃	1	8	yes	—
155	C₂×C₈	C₂×C₂²⋊D₄₃	1	4	yes	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	86	4	no	—
160	C₄⋊C₄	—	86	4	no	—
161	C₄⋊C₄	—	172	32	no	—
162	C₄⋊C₄	—	172	4	no	—
163	C₄²	—	43	16	no	—
164	C₄²	—	43	8	no	—
165	C₄²	D₄×D₄₃	1	2	yes	—
166	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
167	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
168	Q₈	—	8	8	no	—
169	Q₈	—	344	8	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	43	16	no	—
173	C₂×C₄	—	43	8	no	—
174	C₂×C₄	—	172	8	no	—
175	C₂×C₄	—	172	4	no	—
176	C₂×C₄	#13063321	1	4	yes	—
177	C₂×C₄	SD₁₆×D₄₃	1	8	yes	—
178	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₈×D₄₃	1	4	yes	—
181	C₈	—	43	8	no	—
182	C₈	#13063323	1	8	yes	complemented
183	C₂²	C₂×#13063321	1	1	yes	characteristic
184	C₂²	C₄⋊Q₈×D₄₃	1	2	yes	—
185	C₂²	C₂×SD₁₆×D₄₃	1	4	yes	—
186	C₄	—	8	4	no	—
187	C₄	—	43	16	no	—
188	C₄	—	344	4	no	—
189	C₄	#14583270	1	8	yes	—
190	C₄	C₄⋊Q₈×D₄₃	1	2	yes	—
191	C₄	C₂×#13063323	1	2	yes	—
192	C₂	#19142991	1	1	yes	characteristic
193	C₂	C₂×#14583270	1	2	yes	—
194	C₂	#498×D₄₃	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583297

C₂

yes

maximal, characteristic

C₂×#14583295

C₂

yes

maximal

#19142991

C₂

yes

maximal

#19134750

C₂

yes

maximal

C₄×C₈×Dic₄₃

C₂

yes

maximal, characteristic

C₂×C₄₃×#498

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×#13063256

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₃

C₂²

yes

—

#14582768

—

#14582768

C₂²

yes

—

C₄²×Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

—

C₄₃×#498

—

C₂×#106×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

#13063325

—

#13063325

—

#13063323

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂³

yes

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

D₄

yes

—

C₂×C₄×Dic₄₃

C₂³

yes

—

#13063257

—

#13063256

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₈×Dic₄₃

Q₈

yes

complemented

C₂²×Q₈×C₄₃

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#106×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

SD₁₆

yes

—

C₄×Dic₄₃

C₂×Q₈

yes

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂×C₄₃⋊C₈

C₂×Q₈

yes

—

C₂³×C₄₃

C₂²×D₄

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₄⋊Q₈

yes

—

C₂×Dic₄₃

C₂×SD₁₆

yes

—

Dic₈₆

—

C₈×C₄₃

C₄⋊Q₈

yes

complemented

C₄₃⋊C₈

C₄⋊Q₈

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₄₃

C₂²×SD₁₆

yes

—

C₄×C₄₃

—

C₄×C₄₃

#492

yes

—

C₄×C₄₃

C₂×C₄⋊Q₈

yes

—

C₄×C₄₃

C₂×C₄⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

yes

—

—

D₄₃

yes

characteristic

#2868

—

#1844

—

100

C₄²×C₈

—

101

C₂×C₄₃

#2868

yes

characteristic

102

C₂×C₄₃

C₂×#492

yes

—

103

C₂×C₄₃

C₂×#498

yes

—

104

C₂×C₄⋊Q₈

—

105

C₂×C₄⋊Q₈

—

106

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

107

#498

—

108

#498

—

109

#498

—

110

C₂×#106

—

111

C₂×#106

—

112

C₂×#106

C₂×D₄₃

yes

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

116

#384

—

117

#384

—

118

C₄³

—

119

C₄₃

#33310

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²×D₄₃

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

172

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

C₂²×D₄₃

yes

—

134

C₂×C₄²

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²×D₄₃

yes

characteristic

137

#106

—

138

#106

—

139

#106

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

C₂²⋊D₄₃

yes

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

172

—

146

C₂×Q₈

—

172

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₄₃

yes

—

151

C₂²×C₄

C₂³×D₄₃

yes

characteristic

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

Q₈×D₄₃

yes

—

155

C₂×C₈

C₂×C₂²⋊D₄₃

yes

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

172

—

162

C₄⋊C₄

—

172

—

163

C₄²

—

164

C₄²

—

165

C₄²

D₄×D₄₃

yes

—

166

C₄²

C₂×C₂²⋊D₄₃

yes

—

167

C₂³

C₂×D₄×D₄₃

yes

characteristic

168

Q₈

—

169

Q₈

—

344

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

172

—

175

C₂×C₄

—

172

—

176

C₂×C₄

#13063321

yes

—

177

C₂×C₄

SD₁₆×D₄₃

yes

—

178

C₂×C₄

C₂×D₄×D₄₃

yes

characteristic

179

C₂×C₄

C₂²×C₂²⋊D₄₃

yes

characteristic

180

C₂×C₄

C₂×Q₈×D₄₃

yes

—

181

C₈

—

182

C₈

#13063323

yes

complemented

183

C₂²

C₂×#13063321

yes

characteristic

184

C₂²

C₄⋊Q₈×D₄₃

yes

—

185

C₂²

C₂×SD₁₆×D₄₃

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

344

—

189

C₄

#14583270

yes

—

190

C₄

C₄⋊Q₈×D₄₃

yes

—

191

C₄

C₂×#13063323

yes

—

192

C₂

#19142991

yes

characteristic

193

C₂

C₂×#14583270

yes

—

194

C₂

#498×D₄₃

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#498	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₂×#498	D₄₃
hypocenter	C₄₃	#33310
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

C₂×D₄×D₄₃

commutator, cosocle

C₂²×C₄×C₄₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₂×C₄₃×#498

C₂

socle, prime radical

C₂³×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
35. C₈×Dic₄₃	Q₈	semidirect
83. C₈×C₄₃	C₄⋊Q₈	semidirect
84. C₄₃⋊C₈	C₄⋊Q₈	semidirect
94. Dic₄₃	#498	semidirect
119. C₄₃	#33310	semidirect
182. C₈	#13063323	semidirect

H ≅ G/N

type

35. C₈×Dic₄₃

Q₈

semidirect

83. C₈×C₄₃

C₄⋊Q₈

semidirect

84. C₄₃⋊C₈

C₄⋊Q₈

semidirect

94. Dic₄₃

#498

semidirect

119. C₄₃

#33310

semidirect

182. C₈

#13063323

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14583297	2
C₂×#14583295	2
#19142991	2
#19134750	2
C₄×C₈×Dic₄₃	2
C₂×C₄₃×#498	2
#33310	43

subgroup

index

C₄×C₈×Dic₄₃

C₂×C₄₃×#498

#33310

Subgroups — #25249160

Conjugacy classes of subgroups

Complements

Maximal subgroups