Group #25015082 — subgroups

Subgroups — #25015082

Order 11008.

462 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 143 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	#19136351	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582715	C₂	1	4	yes	maximal
3	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063265	C₂²	1	2	yes	—
6	C₂×#13063261	C₂²	1	2	yes	—
7	C₄×#10621877	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
9	#14582715	—	2	8	no	—
10	#14582664	—	2	1	no	—
11	C₂²×Q₁₆×C₄₃	—	2	1	no	—
12	C₂²×Q₁₆×C₄₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₄₃	—	2	1	no	—
14	C₄₃×#494	—	2	2	no	—
15	C₄₃×#494	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621877	—	2	2	no	—
18	#13063265	—	2	4	no	—
19	C₂×C₄₃⋊C₁₆	—	2	2	no	—
20	#13063261	—	2	2	no	—
21	#13063261	D₄	1	4	yes	—
22	#13063236	—	4	1	no	—
23	C₄×C₄₃⋊C₈	—	4	1	no	—
24	C₂²×Q₈×C₄₃	—	2	2	no	—
25	C₂×Q₁₆×C₄₃	—	2	4	no	—
26	C₂×Q₁₆×C₄₃	—	2	4	no	—
27	C₂×Q₁₆×C₄₃	—	2	2	no	—
28	C₂×Q₁₆×C₄₃	—	2	4	no	—
29	C₂×Q₁₆×C₄₃	—	4	4	no	—
30	C₂×Q₁₆×C₄₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₃	—	2	4	no	—
34	C₄⋊Q₈×C₄₃	—	4	1	no	—
35	C₄⋊Q₈×C₄₃	—	4	1	no	—
36	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
37	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₃	D₄	1	2	yes	—
40	C₂×Q₈×C₄₃	—	2	4	no	—
41	C₂×Q₈×C₄₃	—	4	2	no	—
42	C₂×Q₈×C₄₃	—	4	4	no	—
43	C₂×Q₈×C₄₃	—	4	4	no	—
44	C₂²×C₄×C₄₃	—	4	2	no	—
45	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₄₃	—	2	8	no	—
48	Q₁₆×C₄₃	—	4	8	no	—
49	Q₁₆×C₄₃	—	4	2	no	—
50	Q₁₆×C₄₃	—	4	2	no	—
51	C₂×C₈×C₄₃	—	2	2	no	—
52	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₃	#41	1	2	yes	—
57	C₄⋊C₄×C₄₃	—	4	4	no	—
58	C₄⋊C₄×C₄₃	—	4	4	no	—
59	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₄₃	D₈	1	1	yes	characteristic
61	C₄²×C₄₃	D₈	1	1	yes	characteristic
62	#10621877	—	2	4	no	—
63	#10621877	—	4	2	no	—
64	C₂×C₄₃⋊C₈	—	4	2	no	—
65	C₄₃⋊C₁₆	—	4	2	no	—
66	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₄₃	—	4	4	no	—
68	Q₈×C₄₃	—	4	4	no	—
69	Q₈×C₄₃	—	8	4	no	—
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	4	4	no	—
72	C₂×C₄×C₄₃	—	8	4	no	—
73	C₂×C₄×C₄₃	#102	1	2	yes	—
74	C₂×C₄×C₄₃	#102	1	2	yes	—
75	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#121	1	2	yes	—
77	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₈×C₄₃	—	2	2	no	—
80	C₈×C₄₃	C₄×D₄	1	2	yes	—
81	C₈×C₄₃	#124	1	1	yes	characteristic
82	C₈×C₄₃	#127	1	1	yes	characteristic
83	C₄₃⋊C₈	—	4	4	no	—
84	#13256	—	43	1	no	maximal, Sylow
85	C₂²×C₄₃	—	2	2	no	—
86	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
87	C₂²×C₄₃	#460	1	2	yes	—
88	C₄×C₄₃	—	2	2	no	—
89	C₄×C₄₃	—	8	4	no	—
90	C₄×C₄₃	#362	1	4	yes	complemented
91	C₄×C₄₃	#390	1	1	yes	characteristic
92	C₄×C₄₃	#418	1	1	yes	characteristic
93	C₄×C₄₃	#459	1	2	yes	—
94	C₂×#494	D₄₃	1	1	yes	characteristic
95	#2150	—	43	1	no	—
96	C₂×#362	—	43	4	no	—
97	C₄×#108	—	43	1	no	—
98	C₂×C₄₃	—	2	2	no	—
99	C₂×C₄₃	#1982	1	1	yes	characteristic
100	C₂×C₄₃	C₂×#362	1	2	yes	—
101	C₂²×Q₁₆	—	2	1	no	—
102	C₂²×Q₁₆	C₂×D₄₃	1	2	yes	—
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂×M₃₂	—	43	2	no	—
105	#494	—	2	2	no	—
106	#494	Dic₄₃	1	4	yes	—
107	C₂×#108	—	43	2	no	—
108	C₄×M₁₆	—	86	1	no	—
109	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
110	C₄⋊C₁₆	—	86	2	no	—
111	#362	—	86	8	no	—
112	#341	—	86	1	no	—
113	C₄₃	#13256	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	2	2	no	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	2	4	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	4	4	no	—
120	C₂×Q₁₆	C₂×Dic₄₃	1	4	yes	—
121	C₂×M₁₆	—	86	2	no	—
122	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	1	no	—
126	C₄⋊Q₈	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
129	M₃₂	—	86	4	no	—
130	C₂×C₁₆	—	86	2	no	—
131	#108	—	43	4	no	—
132	#108	—	86	2	no	—
133	#97	—	172	1	no	—
134	C₄×C₈	—	172	1	no	—
135	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
136	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	#10621883	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	4	8	no	—
146	Q₁₆	—	4	2	no	—
147	Q₁₆	—	4	2	no	—
148	M₁₆	—	86	4	no	—
149	M₁₆	—	172	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	172	2	no	—
152	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
153	C₂×C₈	D₄×D₄₃	1	2	yes	—
154	C₂×C₈	#10621900	1	2	yes	—
155	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄²	#10621882	1	1	yes	characteristic
160	C₄²	#10621882	1	1	yes	characteristic
161	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
162	C₁₆	—	172	2	no	—
163	C₂³	C₂×#10621883	1	1	yes	characteristic
164	Q₈	—	4	4	no	—
165	Q₈	—	4	4	no	—
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	#13063246	1	2	yes	—
171	C₂×C₄	#13063246	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
173	C₂×C₄	#13063296	1	2	yes	—
174	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
175	C₂×C₄	#13063351	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	172	4	no	—
178	C₈	D₄×Dic₄₃	1	2	yes	—
179	C₈	#13063312	1	1	yes	characteristic
180	C₈	#13063321	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#13063246	1	1	yes	characteristic
183	C₂²	#14583096	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	#14582715	1	4	yes	complemented
187	C₄	#14582802	1	1	yes	characteristic
188	C₄	#14582904	1	1	yes	characteristic
189	C₄	#14583091	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	#19135523	1	1	yes	characteristic
192	C₂	C₂×#14582715	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19136351

C₂

yes

maximal, characteristic

C₂×#14582715

C₂

yes

maximal

C₄×#13063261

C₂

yes

maximal, characteristic

C₂×C₄₃×#494

C₂

yes

maximal, characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063261

C₂²

yes

—

C₄×#10621877

—

C₄⋊(C₄₃⋊C₁₆)

—

#14582715

—

#14582664

—

C₂²×Q₁₆×C₄₃

—

C₂²×Q₁₆×C₄₃

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₃

—

C₄₃×#494

—

C₄₃×#494

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

#13063261

—

#13063261

D₄

yes

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₂²×Q₈×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

C₂×C₄

yes

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

SD₁₆

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

D₈

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×SD₁₆

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₈×C₄₃

—

C₈×C₄₃

C₄×D₄

yes

—

C₈×C₄₃

yes

C₈×C₄₃

yes

C₄₃⋊C₈

—

#13256

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#102

yes

characteristic

C₂²×C₄₃

#460

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×#494

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

#1982

yes

characteristic

100

C₂×C₄₃

C₂×#362

yes

—

101

C₂²×Q₁₆

—

102

C₂²×Q₁₆

C₂×D₄₃

yes

—

103

C₂×C₄⋊Q₈

—

104

C₂×M₃₂

—

105

#494

—

106

#494

Dic₄₃

yes

—

107

C₂×#108

—

108

C₄×M₁₆

—

109

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

110

C₄⋊C₁₆

—

111

#362

—

112

#341

—

113

C₄₃

#13256

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

C₂×Dic₄₃

yes

—

121

C₂×M₁₆

—

122

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

123

C₂²×C₈

C₂²×D₄₃

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

133

#97

—

172

—

134

C₄×C₈

—

172

—

135

C₄×C₈

C₂×Dic₄₃

yes

—

136

C₄×C₈

C₂²⋊D₄₃

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

#10621883

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

—

147

Q₁₆

—

148

M₁₆

—

149

M₁₆

—

172

—

150

C₂×C₈

—

151

C₂×C₈

—

172

—

152

C₂×C₈

C₂²⋊Dic₄₃

yes

—

153

C₂×C₈

D₄×D₄₃

yes

—

154

C₂×C₈

#10621900

yes

—

155

C₂×C₈

C₂²×Dic₄₃

yes

characteristic

156

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

#10621882

yes

characteristic

160

C₄²

#10621882

yes

characteristic

161

C₄²

C₂²⋊Dic₄₃

yes

—

162

C₁₆

—

172

—

163

C₂³

C₂×#10621883

yes

characteristic

164

Q₈

—

165

Q₈

—

166

Q₈

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#13063246

yes

—

171

C₂×C₄

#13063246

yes

—

172

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

173

C₂×C₄

#13063296

yes

—

174

C₂×C₄

C₂×#10621882

yes

characteristic

175

C₂×C₄

#13063351

yes

—

176

C₈

—

177

C₈

—

172

—

178

C₈

D₄×Dic₄₃

yes

—

179

C₈

#13063312

yes

characteristic

180

C₈

#13063321

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#13063246

yes

characteristic

183

C₂²

#14583096

yes

—

184

C₄

—

185

C₄

—

186

C₄

#14582715

yes

complemented

187

C₄

#14582802

yes

characteristic

188

C₄

#14582904

yes

characteristic

189

C₄

#14583091

yes

—

190

C₂

—

191

C₂

#19135523

yes

characteristic

192

C₂

C₂×#14582715

yes

—

	subgroup	G/H
center	C₂²	C₂×#13063246
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#494	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#494	D₄₃
hypocenter	C₄₃	#13256
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

C₂×#13063246

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×#494

C₂

socle

C₂²×C₄₃

C₂×#102

prime radical

C₂³×C₄₃

C₂×SD₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
90. C₄×C₄₃	#362	semidirect
113. C₄₃	#13256	semidirect
186. C₄	#14582715	semidirect

H ≅ G/N

type

90. C₄×C₄₃

#362

semidirect

113. C₄₃

#13256

semidirect

186. C₄

#14582715

semidirect

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

subgroup	index
#19136351	2
C₂×#14582715	2
C₄×#13063261	2
C₂×C₄₃×#494	2
#13256	43

subgroup

index

C₂×C₄₃×#494

Subgroups — #25015082

Conjugacy classes of subgroups

Complements

Maximal subgroups