Group #25249174 — subgroups

Subgroups — #25249174

Order 11008.

862 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 299 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×#14583302	C₂	1	2	yes	maximal
2	C₂×#14583297	C₂	1	1	yes	maximal, characteristic
3	#19142972	C₂	1	2	yes	maximal
4	#19134756	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	#14583302	—	2	8	no	—
9	#14583297	—	2	4	no	—
10	C₂×#13063258	C₂²	1	8	yes	—
11	C₂×#13063257	C₂²	1	4	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
14	#14582806	—	2	4	no	—
15	#14582804	—	2	4	no	—
16	#14582764	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	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
28	C₂×C₄×Dic₄₃	D₄	1	4	yes	—
29	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
30	#13063258	—	2	16	no	—
31	#13063257	—	2	8	no	—
32	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
33	C₈×Dic₄₃	Q₈	1	16	yes	complemented
34	C₄₃⋊#95	—	2	16	no	—
35	C₂²×Q₈×C₄₃	—	2	2	no	—
36	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₄₃	—	2	4	no	—
38	C₄⋊Q₈×C₄₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	#106×C₄₃	—	2	8	no	—
43	C₄×C₈×C₄₃	D₄	1	4	yes	—
44	C₂×Q₈×C₄₃	—	4	4	no	—
45	C₂×Q₈×C₄₃	—	4	8	no	—
46	C₂²×C₄×C₄₃	—	2	4	no	—
47	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₄₃	—	2	4	no	—
50	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
51	C₂×Dic₈₆	—	4	4	no	—
52	C₂×Dic₈₆	—	4	8	no	—
53	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
54	C₂×C₈×C₄₃	C₂×Q₈	1	8	yes	—
55	C₄⋊C₄×C₄₃	—	2	4	no	—
56	C₄⋊C₄×C₄₃	—	2	4	no	—
57	C₄⋊C₄×C₄₃	—	4	4	no	—
58	C₄²×C₄₃	C₂×D₄	1	2	yes	—
59	C₄²×C₄₃	C₂×D₄	1	2	yes	—
60	C₄⋊Dic₄₃	—	2	4	no	—
61	C₄⋊Dic₄₃	—	2	4	no	—
62	C₄⋊Dic₄₃	—	4	4	no	—
63	C₄×Dic₄₃	—	2	8	no	—
64	C₄×Dic₄₃	C₂×Q₈	1	8	yes	—
65	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
66	C₂×C₄₃⋊C₈	C₂×Q₈	1	8	yes	—
67	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
68	Q₈×C₄₃	—	8	8	no	—
69	C₂×C₄×C₄₃	—	4	4	no	—
70	C₂×C₄×C₄₃	—	4	8	no	—
71	C₂×C₄×C₄₃	#127	1	4	yes	—
72	C₂×C₄×C₄₃	#135	1	8	yes	—
73	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
76	C₂×Dic₄₃	—	2	8	no	—
77	C₂×Dic₄₃	—	4	4	no	—
78	C₂×Dic₄₃	—	4	8	no	—
79	C₂×Dic₄₃	C₄⋊Q₈	1	8	yes	—
80	Dic₈₆	—	8	8	no	—
81	C₈×C₄₃	C₄⋊Q₈	1	8	yes	—
82	C₄₃⋊C₈	C₄⋊Q₈	1	8	yes	—
83	#33313	—	43	1	no	maximal, Sylow
84	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
85	C₂²×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
86	C₂²×C₄₃	C₂×#135	1	4	yes	—
87	C₄×C₄₃	—	8	4	no	—
88	C₄×C₄₃	#495	1	8	yes	—
89	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
90	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
91	Dic₄₃	—	2	8	no	—
92	Dic₄₃	—	8	4	no	—
93	C₂×#500	—	43	2	no	—
94	C₂×#498	—	43	1	no	—
95	C₂×#498	D₄₃	1	1	yes	characteristic
96	#2862	—	43	2	no	—
97	#1846	—	43	8	no	—
98	C₄²×C₈	—	43	1	no	—
99	C₂×C₄₃	#2868	1	1	yes	characteristic
100	C₂×C₄₃	C₂×#495	1	2	yes	—
101	C₂×C₄₃	#3160	1	4	yes	—
102	C₂×C₄⋊Q₈	—	43	2	no	—
103	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
104	#500	—	86	8	no	—
105	#498	—	2	4	no	—
106	#498	—	86	4	no	—
107	C₂×#107	—	43	8	no	—
108	C₂×#106	—	43	4	no	—
109	C₂×#106	C₂×D₄₃	1	4	yes	—
110	C₂×C₄×C₈	—	43	4	no	—
111	C₂×C₄×C₈	—	43	1	no	—
112	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
113	#391	—	86	4	no	—
114	#391	—	86	4	no	—
115	#382	—	43	8	no	—
116	C₄³	—	43	1	no	—
117	C₄₃	#33313	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	2	2	no	—
119	C₂²×Q₈	—	86	2	no	—
120	C₂²×C₈	—	43	2	no	—
121	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	2	4	no	—
124	C₄⋊Q₈	—	86	4	no	—
125	C₄⋊Q₈	—	86	4	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	43	4	no	—
128	C₂×C₄⋊C₄	—	86	2	no	—
129	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
130	C₂×C₄²	—	43	2	no	—
131	C₂×C₄²	—	43	4	no	—
132	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
133	#107	—	86	16	no	—
134	#106	—	2	8	no	—
135	#106	—	86	8	no	—
136	C₄×C₈	—	43	16	no	—
137	C₄×C₈	—	43	4	no	—
138	C₄×C₈	C₂²⋊D₄₃	1	4	yes	—
139	#95	—	86	16	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂×Q₈	—	4	8	no	—
142	C₂×Q₈	—	172	4	no	—
143	C₂×Q₈	—	172	8	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	43	4	no	—
146	C₂²×C₄	—	86	4	no	—
147	C₂²×C₄	D₄×D₄₃	1	2	yes	—
148	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
149	C₂×C₈	—	43	4	no	—
150	C₂×C₈	—	43	8	no	—
151	C₂×C₈	Q₈×D₄₃	1	8	yes	—
152	C₂×C₈	C₂×C₂²⋊D₄₃	1	4	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	86	4	no	—
157	C₄⋊C₄	—	86	4	no	—
158	C₄⋊C₄	—	172	4	no	—
159	C₄²	—	43	8	no	—
160	C₄²	—	86	8	no	—
161	C₄²	D₄×D₄₃	1	2	yes	—
162	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
163	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
164	Q₈	—	8	8	no	—
165	Q₈	—	344	8	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	43	8	no	—
169	C₂×C₄	—	86	8	no	—
170	C₂×C₄	—	172	8	no	—
171	C₂×C₄	—	172	4	no	—
172	C₂×C₄	#13063321	1	4	yes	—
173	C₂×C₄	#13063346	1	8	yes	—
174	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
176	C₂×C₄	C₂×Q₈×D₄₃	1	4	yes	—
177	C₈	—	43	8	no	—
178	C₈	#13063323	1	8	yes	—
179	C₂²	C₂×#13063321	1	1	yes	characteristic
180	C₂²	C₄⋊Q₈×D₄₃	1	2	yes	—
181	C₂²	C₂×#13063346	1	4	yes	—
182	C₄	—	8	4	no	—
183	C₄	—	86	8	no	—
184	C₄	—	344	4	no	—
185	C₄	#14583280	1	8	yes	—
186	C₄	C₄⋊Q₈×D₄₃	1	2	yes	—
187	C₄	C₂×#13063323	1	2	yes	—
188	C₂	#19142991	1	1	yes	characteristic
189	C₂	C₂×#14583280	1	2	yes	—
190	C₂	#19145104	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583302

C₂

yes

maximal

C₂×#14583297

C₂

yes

maximal, characteristic

#19142972

C₂

yes

maximal

#19134756

C₂

yes

maximal

C₄×C₈×Dic₄₃

C₂

yes

maximal, characteristic

C₂×C₄₃×#498

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

—

#14583302

—

#14583297

—

C₂×#13063258

C₂²

yes

—

C₂×#13063257

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₃

C₂²

yes

—

#14582806

—

#14582804

—

#14582764

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

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂³

yes

—

C₂×C₄×Dic₄₃

D₄

yes

—

C₂×C₄×Dic₄₃

C₂³

yes

—

#13063258

—

#13063257

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₈×Dic₄₃

Q₈

yes

complemented

C₄₃⋊#95

—

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

—

C₄×Dic₄₃

—

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

#135

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

—

C₂×Dic₄₃

C₄⋊Q₈

yes

—

Dic₈₆

—

C₈×C₄₃

C₄⋊Q₈

yes

—

C₄₃⋊C₈

C₄⋊Q₈

yes

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₄₃

C₂×#135

yes

—

C₄×C₄₃

—

C₄×C₄₃

#495

yes

—

C₄×C₄₃

C₂×C₄⋊Q₈

yes

—

C₄×C₄₃

C₂×C₄⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

C₂×#500

—

C₂×#498

—

C₂×#498

D₄₃

yes

characteristic

#2862

—

#1846

—

C₄²×C₈

—

C₂×C₄₃

#2868

yes

characteristic

100

C₂×C₄₃

C₂×#495

yes

—

101

C₂×C₄₃

#3160

yes

—

102

C₂×C₄⋊Q₈

—

103

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

104

#500

—

105

#498

—

106

#498

—

107

C₂×#107

—

108

C₂×#106

—

109

C₂×#106

C₂×D₄₃

yes

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

113

#391

—

114

#391

—

115

#382

—

116

C₄³

—

117

C₄₃

#33313

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

119

C₂²×Q₈

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²×D₄₃

yes

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

C₂²×D₄₃

yes

—

130

C₂×C₄²

—

131

C₂×C₄²

—

132

C₂×C₄²

C₂²×D₄₃

yes

characteristic

133

#107

—

134

#106

—

135

#106

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

C₂²⋊D₄₃

yes

—

139

#95

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

172

—

143

C₂×Q₈

—

172

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

D₄×D₄₃

yes

—

148

C₂²×C₄

C₂³×D₄₃

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

Q₈×D₄₃

yes

—

152

C₂×C₈

C₂×C₂²⋊D₄₃

yes

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

172

—

159

C₄²

—

160

C₄²

—

161

C₄²

D₄×D₄₃

yes

—

162

C₄²

C₂×C₂²⋊D₄₃

yes

—

163

C₂³

C₂×D₄×D₄₃

yes

characteristic

164

Q₈

—

165

Q₈

—

344

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

172

—

171

C₂×C₄

—

172

—

172

C₂×C₄

#13063321

yes

—

173

C₂×C₄

#13063346

yes

—

174

C₂×C₄

C₂×D₄×D₄₃

yes

characteristic

175

C₂×C₄

C₂²×C₂²⋊D₄₃

yes

characteristic

176

C₂×C₄

C₂×Q₈×D₄₃

yes

—

177

C₈

—

178

C₈

#13063323

yes

—

179

C₂²

C₂×#13063321

yes

characteristic

180

C₂²

C₄⋊Q₈×D₄₃

yes

—

181

C₂²

C₂×#13063346

yes

—

182

C₄

—

183

C₄

—

184

C₄

—

344

—

185

C₄

#14583280

yes

—

186

C₄

C₄⋊Q₈×D₄₃

yes

—

187

C₄

C₂×#13063323

yes

—

188

C₂

#19142991

yes

characteristic

189

C₂

C₂×#14583280

yes

—

190

C₂

#19145104

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	#33313
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
33. C₈×Dic₄₃	Q₈	semidirect
117. C₄₃	#33313	semidirect

H ≅ G/N

type

33. C₈×Dic₄₃

Q₈

semidirect

117. C₄₃

#33313

semidirect

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

subgroup	index
C₂×#14583302	2
C₂×#14583297	2
#19142972	2
#19134756	2
C₄×C₈×Dic₄₃	2
C₂×C₄₃×#498	2
#33313	43

subgroup

index

C₄×C₈×Dic₄₃

C₂×C₄₃×#498

#33313

Subgroups — #25249174

Conjugacy classes of subgroups

Complements

Maximal subgroups