Group #25016428 — subgroups

Subgroups — #25016428

Order 11008.

302 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14582714	C₂	1	2	yes	maximal
2	#19135466	C₂	1	1	yes	maximal, characteristic
3	#19132695	C₂	1	2	yes	maximal
4	#19132679	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
7	C₂×#13063265	C₂²	1	2	yes	—
8	C₂×#13063257	C₂²	1	1	yes	characteristic
9	C₂×#13063250	—	2	1	no	—
10	#14582805	—	2	1	no	—
11	#14582714	—	2	4	no	—
12	C₂×#108×C₄₃	—	2	1	no	—
13	C₂×#108×C₄₃	C₂²	1	2	yes	—
14	C₄₃×#423	—	2	1	no	—
15	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₈₆	—	2	1	no	—
17	C₂×Dic₁₇₂	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	—	2	1	no	—
20	C₂×Dic₁₇₂	D₄	1	2	yes	—
21	C₂×C₄⋊Dic₄₃	—	2	1	no	—
22	#13063265	—	2	2	no	—
23	#13063265	—	2	2	no	—
24	C₂×C₄₃⋊C₁₆	—	2	2	no	—
25	#13063257	—	2	2	no	—
26	#13063250	—	4	2	no	—
27	C₄₃⋊#95	—	4	2	no	—
28	C₂×M₁₆×C₄₃	—	2	2	no	—
29	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
32	#108×C₄₃	—	2	2	no	—
33	#108×C₄₃	—	2	2	no	—
34	#108×C₄₃	—	4	2	no	—
35	C₄⋊C₈×C₄₃	—	4	2	no	—
36	#97×C₄₃	—	2	2	no	—
37	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₄₃	—	4	1	no	—
40	C₂²×Dic₄₃	—	4	1	no	—
41	C₂×Dic₈₆	—	2	2	no	—
42	C₂×Dic₈₆	—	4	1	no	—
43	C₂×Dic₈₆	—	4	1	no	—
44	C₂×Dic₈₆	—	4	1	no	—
45	C₂×Dic₈₆	—	4	1	no	—
46	C₂×Dic₈₆	—	4	1	no	—
47	M₁₆×C₄₃	—	4	2	no	—
48	M₁₆×C₄₃	—	4	2	no	—
49	C₂×C₈×C₄₃	—	2	1	no	—
50	C₂×C₈×C₄₃	—	2	1	no	—
51	C₂×C₈×C₄₃	—	4	2	no	—
52	C₂×C₈×C₄₃	D₈	1	2	yes	—
53	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₄₃	#41	1	2	yes	—
55	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₄₃	#41	1	2	yes	—
57	C₄²×C₄₃	—	2	1	no	—
58	C₄²×C₄₃	—	2	1	no	—
59	C₄⋊Dic₄₃	—	4	1	no	—
60	C₄⋊Dic₄₃	—	4	1	no	—
61	Dic₁₇₂	—	2	4	no	—
62	Dic₁₇₂	—	4	2	no	—
63	C₄₃⋊C₁₆	—	4	2	no	—
64	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	4	2	no	—
67	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
73	C₂×Dic₄₃	—	4	2	no	—
74	C₂×Dic₄₃	—	8	1	no	—
75	C₂×Dic₄₃	—	8	1	no	—
76	C₂×Dic₄₃	—	8	2	no	—
77	C₂×Dic₄₃	—	8	1	no	—
78	Dic₈₆	—	4	2	no	—
79	Dic₈₆	—	4	2	no	—
80	Dic₈₆	—	8	1	no	—
81	Dic₈₆	—	8	1	no	—
82	C₈×C₄₃	—	2	2	no	—
83	C₈×C₄₃	—	8	2	no	—
84	C₈×C₄₃	#123	1	2	yes	—
85	C₈×C₄₃	#126	1	2	yes	—
86	#13488	—	43	1	no	maximal, Sylow
87	C₂²×C₄₃	—	2	2	no	—
88	C₂²×C₄₃	#461	1	1	yes	characteristic
89	C₂²×C₄₃	#481	1	1	yes	characteristic
90	C₂²×C₄₃	#486	1	1	yes	characteristic
91	C₄×C₄₃	—	4	2	no	—
92	C₄×C₄₃	#397	1	1	yes	characteristic
93	C₄×C₄₃	#459	1	1	yes	characteristic
94	C₄×C₄₃	#480	1	1	yes	characteristic
95	C₄×C₄₃	#489	1	1	yes	characteristic
96	Dic₄₃	—	8	2	no	—
97	Dic₄₃	—	16	1	no	—
98	C₂×#362	—	43	2	no	—
99	#1971	—	43	1	no	—
100	#1849	D₄₃	1	1	yes	characteristic
101	#1383	—	43	2	no	—
102	#1377	—	43	1	no	—
103	C₂×C₄₃	—	2	1	no	—
104	C₂×C₄₃	—	2	1	no	—
105	C₂×C₄₃	#2071	1	1	yes	characteristic
106	C₂×C₄₃	#2239	1	2	yes	—
107	C₂²×Q₁₆	—	43	1	no	—
108	C₂×M₃₂	—	43	2	no	—
109	C₂×#108	—	2	1	no	—
110	C₂×#108	C₂×D₄₃	1	2	yes	—
111	C₂×#106	—	43	1	no	—
112	#423	—	2	1	no	—
113	C₂×#103	—	86	1	no	—
114	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
115	#391	—	86	1	no	—
116	#362	—	86	4	no	—
117	C₄₃	#13488	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	86	1	no	—
119	C₂×Q₁₆	—	43	2	no	—
120	C₂×Q₁₆	—	86	2	no	—
121	C₂×Q₁₆	—	86	1	no	—
122	C₂×Q₁₆	—	86	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	86	1	no	—
127	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
128	M₃₂	—	86	2	no	—
129	M₃₂	—	86	2	no	—
130	C₂×C₁₆	—	86	2	no	—
131	#108	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	4	2	no	—
134	#106	—	86	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	#103	—	172	2	no	—
137	#97	—	2	2	no	—
138	#95	—	172	2	no	—
139	C₂×Q₈	—	86	2	no	—
140	C₂×Q₈	—	172	1	no	—
141	C₂×Q₈	—	172	1	no	—
142	C₂×Q₈	—	172	1	no	—
143	C₂×Q₈	—	172	1	no	—
144	C₂×Q₈	—	172	1	no	—
145	C₂²×C₄	—	172	1	no	—
146	C₂²×C₄	—	172	1	no	—
147	C₂²×C₄	#10621884	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
149	Q₁₆	—	86	4	no	—
150	Q₁₆	—	172	2	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	#10621882	1	2	yes	—
157	C₂×C₈	#10621898	1	2	yes	—
158	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
159	C₂×C₈	#10621900	1	1	yes	characteristic
160	C₂×C₈	#10621902	1	2	yes	—
161	C₄⋊C₄	—	172	1	no	—
162	C₄⋊C₄	—	172	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	172	2	no	—
166	C₂³	C₂×#10621884	1	1	yes	characteristic
167	Q₈	—	344	1	no	—
168	Q₈	—	344	1	no	—
169	Q₈	—	172	2	no	—
170	Q₈	—	172	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	344	1	no	—
174	C₂×C₄	—	344	1	no	—
175	C₂×C₄	—	344	2	no	—
176	C₂×C₄	—	344	1	no	—
177	C₂×C₄	—	172	2	no	—
178	C₂×C₄	#13063296	1	1	yes	characteristic
179	C₂×C₄	#13063308	1	1	yes	characteristic
180	C₂×C₄	#13063314	1	1	yes	characteristic
181	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
182	C₂×C₄	#13063351	1	1	yes	characteristic
183	C₂×C₄	#13063356	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	8	2	no	—
186	C₈	#13063310	1	2	yes	—
187	C₈	#13063320	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#14583101	1	1	yes	characteristic
190	C₂²	#14583217	1	1	yes	characteristic
191	C₂²	#14583247	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	344	2	no	—
194	C₄	—	688	1	no	—
195	C₄	#14582825	1	1	yes	characteristic
196	C₄	#14583091	1	1	yes	characteristic
197	C₄	#14583212	1	1	yes	characteristic
198	C₄	#14583261	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#19135999	1	1	yes	characteristic
202	C₂	#19136814	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582714

C₂

yes

maximal

#19135466

C₂

yes

maximal, characteristic

#19132695

C₂

yes

maximal

#19132679

C₂

yes

maximal, characteristic

C₄₃×#1849

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

C₂²

yes

characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063257

C₂²

yes

—

—

—

C₂×#108×C₄₃

—

C₂×#108×C₄₃

C₂²

yes

—

C₄₃×#423

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

#13063265

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

#13063257

—

#13063250

—

C₄₃⋊#95

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

#108×C₄₃

—

#108×C₄₃

—

#108×C₄₃

—

C₄⋊C₈×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

SD₁₆

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×D₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#123

yes

—

C₈×C₄₃

#126

yes

—

#13488

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

C₂×#362

—

#1971

—

100

#1849

D₄₃

yes

characteristic

101

#1383

—

102

#1377

—

103

C₂×C₄₃

—

104

C₂×C₄₃

—

105

C₂×C₄₃

#2071

yes

characteristic

106

C₂×C₄₃

#2239

yes

—

107

C₂²×Q₁₆

—

108

C₂×M₃₂

—

109

C₂×#108

—

110

C₂×#108

C₂×D₄₃

yes

—

111

C₂×#106

—

112

#423

—

113

C₂×#103

—

114

C₂×#97

C₂×D₄₃

yes

characteristic

115

#391

—

116

#362

—

117

C₄₃

#13488

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×M₁₆

—

124

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

125

C₂²×C₈

C₂²×D₄₃

yes

characteristic

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

—

133

#108

—

134

#106

—

135

C₄⋊C₈

—

136

#103

—

172

—

137

#97

—

138

#95

—

172

—

139

C₂×Q₈

—

140

C₂×Q₈

—

172

—

141

C₂×Q₈

—

172

—

142

C₂×Q₈

—

172

—

143

C₂×Q₈

—

172

—

144

C₂×Q₈

—

172

—

145

C₂²×C₄

—

172

—

146

C₂²×C₄

—

172

—

147

C₂²×C₄

#10621884

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

172

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#10621882

yes

—

157

C₂×C₈

#10621898

yes

—

158

C₂×C₈

D₄×D₄₃

yes

characteristic

159

C₂×C₈

#10621900

yes

characteristic

160

C₂×C₈

#10621902

yes

—

161

C₄⋊C₄

—

172

—

162

C₄⋊C₄

—

172

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

172

—

166

C₂³

C₂×#10621884

yes

characteristic

167

Q₈

—

344

—

168

Q₈

—

344

—

169

Q₈

—

172

—

170

Q₈

—

172

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

344

—

174

C₂×C₄

—

344

—

175

C₂×C₄

—

344

—

176

C₂×C₄

—

344

—

177

C₂×C₄

—

172

—

178

C₂×C₄

#13063296

yes

characteristic

179

C₂×C₄

#13063308

yes

characteristic

180

C₂×C₄

#13063314

yes

characteristic

181

C₂×C₄

C₂×#10621882

yes

characteristic

182

C₂×C₄

#13063351

yes

characteristic

183

C₂×C₄

#13063356

yes

characteristic

184

C₈

—

185

C₈

—

186

C₈

#13063310

yes

—

187

C₈

#13063320

yes

—

188

C₂²

—

189

C₂²

#14583101

yes

characteristic

190

C₂²

#14583217

yes

characteristic

191

C₂²

#14583247

yes

characteristic

192

C₄

—

193

C₄

—

344

—

194

C₄

—

688

—

195

C₄

#14582825

yes

characteristic

196

C₄

#14583091

yes

characteristic

197

C₄

#14583212

yes

characteristic

198

C₄

#14583261

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#19135999

yes

characteristic

202

C₂

#19136814

yes

—

	subgroup	G/H
center	C₂²	#14583217
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1849	C₂
socle	C₂²×C₄₃	#481
prime radical	C₂³×C₄₃	C₂×SD₁₆
hypercenter	#1849	D₄₃
hypocenter	C₄₃	#13488
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#14583217

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1849

C₂

socle

C₂²×C₄₃

#481

prime radical

C₂³×C₄₃

C₂×SD₁₆

D₄₃

C₄₃

G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
117. C₄₃	#13488	semidirect

H ≅ G/N

type

117. C₄₃

#13488

semidirect

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

subgroup	index
C₂×#14582714	2
#19135466	2
#19132695	2
#19132679	2
C₄₃×#1849	2
#13488	43

subgroup

index

C₄₃×#1849

Subgroups — #25016428

Conjugacy classes of subgroups

Complements

Maximal subgroups