Group #24983150 — subgroups

Subgroups — #24983150

Order 11008.

442 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₄×C₂²⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
2	C₄₃×#1728	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
4	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
5	#14582654	C₄	1	4	yes	—
6	C₂²×M₁₆×C₄₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
8	C₄₃×#336	—	2	2	no	—
9	C₂²×C₄₃⋊C₈	—	2	4	no	—
10	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
11	C₄×C₄₃⋊C₈	—	4	4	no	—
12	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
14	C₂³×C₄×C₄₃	Q₈	1	1	yes	characteristic
15	C₂×M₁₆×C₄₃	—	2	2	no	—
16	C₂×M₁₆×C₄₃	—	4	2	no	—
17	C₂×M₁₆×C₄₃	—	4	2	no	—
18	C₂×M₁₆×C₄₃	C₈	1	4	yes	complemented
19	C₂²×C₈×C₄₃	—	2	2	no	—
20	C₂×C₄²×C₄₃	—	4	2	no	—
21	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₃	D₄	1	2	yes	—
23	C₂⁴×C₄₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	—	2	2	no	—
25	C₂²×C₄×C₄₃	—	2	1	no	—
26	C₂²×C₄×C₄₃	—	2	1	no	—
27	C₂²×C₄×C₄₃	—	2	4	no	—
28	C₂²×C₄×C₄₃	—	2	4	no	—
29	C₂²×C₄×C₄₃	—	4	1	no	—
30	C₂²×C₄×C₄₃	—	4	1	no	—
31	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
37	M₁₆×C₄₃	—	4	4	no	—
38	M₁₆×C₄₃	—	4	4	no	—
39	C₂×C₈×C₄₃	—	2	4	no	—
40	C₂×C₈×C₄₃	—	4	2	no	—
41	C₂×C₈×C₄₃	—	4	2	no	—
42	C₄²×C₄₃	—	2	4	no	—
43	C₄²×C₄₃	—	4	2	no	—
44	C₂×C₄₃⋊C₈	—	2	8	no	—
45	C₂×C₄₃⋊C₈	—	4	4	no	—
46	C₂×C₄₃⋊C₈	—	4	4	no	—
47	C₂³×C₄₃	—	2	2	no	—
48	C₂³×C₄₃	—	2	2	no	—
49	C₂³×C₄₃	—	2	1	no	—
50	C₂³×C₄₃	—	2	1	no	—
51	C₂³×C₄₃	#95	1	1	yes	characteristic
52	C₂³×C₄₃	C₄⋊C₈	1	2	yes	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	—	2	1	no	—
58	C₂×C₄×C₄₃	—	2	1	no	—
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	2	no	—
63	C₂×C₄×C₄₃	—	4	4	no	—
64	C₂×C₄×C₄₃	—	4	4	no	—
65	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
68	C₂×C₄×C₄₃	#99	1	2	yes	—
69	C₂×C₄×C₄₃	#100	1	1	yes	characteristic
70	C₂×C₄×C₄₃	#101	1	1	yes	characteristic
71	C₂×C₄×C₄₃	#104	1	4	yes	—
72	C₂×C₄×C₄₃	#108	1	4	yes	—
73	C₈×C₄₃	—	4	4	no	—
74	C₄₃⋊C₈	—	4	8	no	—
75	#7026	—	43	1	no	maximal, Sylow
76	C₂²×C₄₃	—	2	2	no	—
77	C₂²×C₄₃	—	2	4	no	—
78	C₂²×C₄₃	—	4	1	no	—
79	C₂²×C₄₃	—	4	1	no	—
80	C₂²×C₄₃	—	4	2	no	—
81	C₂²×C₄₃	#336	1	1	yes	characteristic
82	C₂²×C₄₃	#339	1	2	yes	—
83	C₂²×C₄₃	#341	1	2	yes	—
84	C₂²×C₄₃	#342	1	1	yes	characteristic
85	C₂²×C₄₃	#343	1	1	yes	characteristic
86	C₄×C₄₃	—	2	1	no	—
87	C₄×C₄₃	—	2	1	no	—
88	C₄×C₄₃	—	2	4	no	—
89	C₄×C₄₃	—	2	4	no	—
90	C₄×C₄₃	#323	1	2	yes	—
91	C₄×C₄₃	#324	1	2	yes	—
92	C₄×C₂²⋊C₈	—	43	2	no	—
93	#1728	D₄₃	1	1	yes	characteristic
94	C₂×C₄₃	—	4	2	no	—
95	C₂×C₄₃	#1278	1	2	yes	—
96	C₂×C₄₃	#1280	1	1	yes	characteristic
97	C₂×C₄₃	#1297	1	1	yes	characteristic
98	C₂×C₄₃	#1298	1	1	yes	characteristic
99	C₂×C₄₃	#1299	1	2	yes	—
100	C₂²×M₁₆	Dic₄₃	1	2	yes	—
101	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
102	C₂×C₂²⋊C₈	—	86	2	no	—
103	C₂×C₄×C₈	—	86	2	no	—
104	#336	—	2	2	no	—
105	#336	—	43	4	no	—
106	C₄₃	#7026	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₄	Dic₈₆	1	1	yes	characteristic
108	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
109	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	4	2	no	—
112	C₂×M₁₆	—	4	2	no	—
113	C₂×M₁₆	C₄₃⋊C₈	1	4	yes	complemented
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	86	4	no	—
116	C₂×C₄²	—	4	2	no	—
117	C₂×C₄²	C₄×D₄₃	1	2	yes	—
118	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
119	C₂²⋊C₈	—	86	8	no	—
120	C₄×C₈	—	172	4	no	—
121	C₂⁴	C₄⋊Dic₄₃	1	1	yes	characteristic
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	4	1	no	—
128	C₂²×C₄	—	4	1	no	—
129	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
130	C₂²×C₄	#10621877	1	2	yes	—
131	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
132	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
133	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
134	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
135	M₁₆	—	4	4	no	—
136	M₁₆	—	4	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	86	8	no	—
141	C₂×C₈	—	172	4	no	—
142	C₂×C₈	—	172	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	—	2	1	no	—
148	C₂³	—	2	1	no	—
149	C₂³	C₄₃⋊#95	1	1	yes	characteristic
150	C₂³	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₄×C₄₃⋊C₈	1	1	yes	characteristic
164	C₂×C₄	#13063236	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
166	C₂×C₄	#13063240	1	2	yes	—
167	C₂×C₄	#13063242	1	1	yes	characteristic
168	C₂×C₄	#13063244	1	1	yes	characteristic
169	C₂×C₄	#13063251	1	4	yes	—
170	C₂×C₄	#13063260	1	4	yes	—
171	C₈	—	4	4	no	—
172	C₈	—	172	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	4	2	no	—
178	C₂²	#14582654	1	1	yes	characteristic
179	C₂²	#14582660	1	2	yes	—
180	C₂²	#14582663	1	2	yes	—
181	C₂²	#14582666	1	1	yes	characteristic
182	C₂²	#14582668	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	#14582623	1	2	yes	—
188	C₄	#14582625	1	2	yes	—
189	C₂	—	4	2	no	—
190	C₂	#19132433	1	2	yes	—
191	C₂	#19132438	1	1	yes	characteristic
192	C₂	#19132480	1	1	yes	characteristic
193	C₂	#19132482	1	1	yes	characteristic
194	C₂	#19132484	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₂²⋊(C₄₃⋊C₈)

C₂

yes

maximal

C₄₃×#1728

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₃⋊C₈)

—

C₂×C₄×C₄₃⋊C₈

—

#14582654

C₄

yes

—

C₂²×M₁₆×C₄₃

C₄

yes

—

C₂²×C₄²×C₄₃

C₂²

yes

characteristic

C₄₃×#336

—

C₂²×C₄₃⋊C₈

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂³×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₃

D₄

yes

characteristic

C₂³×C₄×C₄₃

Q₈

yes

characteristic

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

C₈

yes

complemented

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂⁴×C₄₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₄²

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₄⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₈

yes

—

C₂²×C₄×C₄₃

M₁₆

yes

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

#95

yes

characteristic

C₂³×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×C₈

yes

characteristic

C₂×C₄×C₄₃

#97

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₈

yes

—

C₂×C₄×C₄₃

#99

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

#108

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#7026

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#336

yes

characteristic

C₂²×C₄₃

#339

yes

—

C₂²×C₄₃

#341

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#323

yes

—

C₄×C₄₃

#324

yes

—

C₄×C₂²⋊C₈

—

#1728

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

#1278

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

100

C₂²×M₁₆

Dic₄₃

yes

—

101

C₂²×C₄²

C₂×D₄₃

yes

characteristic

102

C₂×C₂²⋊C₈

—

103

C₂×C₄×C₈

—

104

#336

—

105

#336

—

106

C₄₃

#7026

yes

characteristic, complemented, Sylow

107

C₂³×C₄

Dic₈₆

yes

characteristic

108

C₂³×C₄

D₁₇₂

yes

characteristic

109

C₂³×C₄

C₂×Dic₄₃

yes

characteristic

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

113

C₂×M₁₆

C₄₃⋊C₈

yes

complemented

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂×C₄²

—

117

C₂×C₄²

C₄×D₄₃

yes

—

118

C₂×C₄²

C₂²⋊D₄₃

yes

—

119

C₂²⋊C₈

—

120

C₄×C₈

—

172

—

121

C₂⁴

C₄⋊Dic₄₃

yes

characteristic

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

C₂×C₄₃⋊C₈

yes

—

130

C₂²×C₄

#10621877

yes

—

131

C₂²×C₄

C₄×Dic₄₃

yes

characteristic

132

C₂²×C₄

Dic₄₃⋊C₄

yes

—

133

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

—

134

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

172

—

142

C₂×C₈

—

172

—

143

C₄²

—

144

C₄²

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

C₄₃⋊#95

yes

characteristic

150

C₂³

C₄⋊(C₄₃⋊C₈)

yes

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

C₄×C₄₃⋊C₈

yes

characteristic

164

C₂×C₄

#13063236

yes

characteristic

165

C₂×C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

166

C₂×C₄

#13063240

yes

—

167

C₂×C₄

#13063242

yes

characteristic

168

C₂×C₄

#13063244

yes

characteristic

169

C₂×C₄

#13063251

yes

—

170

C₂×C₄

#13063260

yes

—

171

C₈

—

172

C₈

—

172

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#14582654

yes

characteristic

179

C₂²

#14582660

yes

—

180

C₂²

#14582663

yes

—

181

C₂²

#14582666

yes

characteristic

182

C₂²

#14582668

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

#14582623

yes

—

188

C₄

#14582625

yes

—

189

C₂

—

190

C₂

#19132433

yes

—

191

C₂

#19132438

yes

characteristic

192

C₂

#19132480

yes

characteristic

193

C₂

#19132482

yes

characteristic

194

C₂

#19132484

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₃
Fitting	C₄₃×#1728	C₂
socle	C₂³×C₄₃	#95
prime radical	C₂⁴×C₄₃	C₄⋊C₄
cosocle	C₂²×C₄²×C₄₃	C₂²
hypercenter	#1728	D₄₃
hypocenter	C₄₃	#7026
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₄₃

commutator

C₂×C₄×C₄₃

C₄×C₈

Frattini

C₂²×C₄²

C₂×D₄₃

Fitting

C₄₃×#1728

C₂

socle

C₂³×C₄₃

#95

prime radical

C₂⁴×C₄₃

C₄⋊C₄

cosocle

C₂²×C₄²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

N	H ≅ G/N	type
18. C₂×M₁₆×C₄₃	C₈	semidirect
106. C₄₃	#7026	semidirect
113. C₂×M₁₆	C₄₃⋊C₈	semidirect

H ≅ G/N

type

18. C₂×M₁₆×C₄₃

C₈

semidirect

106. C₄₃

#7026

semidirect

113. C₂×M₁₆

C₄₃⋊C₈

semidirect

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

subgroup	index
C₄×C₂²⋊(C₄₃⋊C₈)	2
C₄₃×#1728	2
#7026	43

subgroup

index

C₄×C₂²⋊(C₄₃⋊C₈)

C₄₃×#1728

#7026

Subgroups — #24983150

Conjugacy classes of subgroups

Complements

Maximal subgroups