Group #25001650 — subgroups

Subgroups — #25001650

Order 11008.

442 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 165 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	#19135556	C₂	1	2	yes	maximal
2	#19135397	C₂	1	1	yes	maximal, characteristic
3	#19134755	C₂	1	2	yes	maximal
4	#19134755	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	C₂²	1	1	yes	characteristic
7	C₂×#13063259	—	2	2	no	—
8	C₂×#13063259	—	2	4	no	—
9	C₂×#13063250	—	2	2	no	—
10	#14582769	C₂²	1	2	yes	—
11	#14582769	C₂²	1	1	yes	characteristic
12	#14582662	—	2	4	no	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₈₆	—	2	1	no	—
16	#13063325	—	2	2	no	—
17	#13063325	—	2	1	no	—
18	#13063325	Q₈	1	2	yes	—
19	C₂×C₄⋊Dic₄₃	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	—	2	1	no	—
21	C₂×C₄⋊Dic₄₃	—	2	2	no	—
22	C₂×C₄⋊Dic₄₃	—	2	2	no	—
23	C₂×C₄⋊Dic₄₃	—	2	2	no	—
24	C₂×C₄⋊Dic₄₃	—	2	2	no	—
25	#13063259	—	4	8	no	—
26	#13063259	—	4	4	no	—
27	#13063250	—	4	4	no	—
28	C₂²×C₈×C₄₃	D₄	1	2	yes	—
29	C₂²×C₈×C₄₃	D₄	1	4	yes	—
30	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₃	—	2	2	no	—
32	C₄×C₈×C₄₃	—	2	2	no	—
33	C₄×C₈×C₄₃	—	2	2	no	—
34	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
36	C₂²×Dic₄₃	—	4	1	no	—
37	C₂²×Dic₄₃	—	4	1	no	—
38	C₂²×Dic₄₃	—	4	2	no	—
39	C₂×Dic₈₆	—	2	2	no	—
40	C₂×Dic₈₆	—	4	2	no	—
41	C₂×Dic₈₆	—	4	2	no	—
42	C₂×Dic₈₆	—	4	1	no	—
43	C₂×C₈×C₄₃	—	2	2	no	—
44	C₂×C₈×C₄₃	—	2	4	no	—
45	C₂×C₈×C₄₃	—	2	4	no	—
46	C₂×C₈×C₄₃	—	2	2	no	—
47	C₂×C₈×C₄₃	D₈	1	4	yes	—
48	C₂×C₈×C₄₃	Q₁₆	1	8	yes	—
49	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₄₃	#41	1	1	yes	characteristic
51	C₄²×C₄₃	#41	1	2	yes	—
52	C₄⋊Dic₄₃	—	2	4	no	—
53	C₄⋊Dic₄₃	—	4	2	no	—
54	C₄⋊Dic₄₃	—	4	2	no	—
55	C₄⋊Dic₄₃	—	4	4	no	—
56	C₄⋊Dic₄₃	—	4	4	no	—
57	C₄⋊Dic₄₃	—	4	4	no	—
58	C₄⋊Dic₄₃	—	4	4	no	—
59	C₂³×C₄₃	#127	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#123	1	2	yes	—
64	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
66	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
67	C₂×C₄×C₄₃	#137	1	2	yes	—
68	C₂×Dic₄₃	—	4	2	no	—
69	C₂×Dic₄₃	—	8	1	no	—
70	C₂×Dic₄₃	—	8	1	no	—
71	C₂×Dic₄₃	—	8	1	no	—
72	C₂×Dic₄₃	—	8	2	no	—
73	C₂×Dic₄₃	—	8	2	no	—
74	C₂×Dic₄₃	—	8	2	no	—
75	C₂×Dic₄₃	—	8	2	no	—
76	Dic₈₆	—	4	4	no	—
77	Dic₈₆	—	8	2	no	—
78	C₈×C₄₃	—	2	4	no	—
79	C₈×C₄₃	—	2	8	no	—
80	#10674	—	43	1	no	maximal, Sylow
81	C₂²×C₄₃	#399	1	1	yes	characteristic
82	C₂²×C₄₃	#493	1	1	yes	characteristic
83	C₂²×C₄₃	#494	1	2	yes	—
84	C₂²×C₄₃	#496	1	2	yes	—
85	C₂²×C₄₃	#497	1	1	yes	characteristic
86	C₄×C₄₃	#474	1	2	yes	—
87	C₄×C₄₃	#477	1	4	yes	—
88	C₄×C₄₃	#480	1	2	yes	—
89	C₄×C₄₃	#484	1	4	yes	—
90	Dic₄₃	—	8	2	no	—
91	Dic₄₃	—	16	1	no	—
92	Dic₄₃	—	16	2	no	—
93	#1990	—	43	2	no	—
94	#1956	—	43	1	no	—
95	#1845	—	43	2	no	—
96	#1845	—	43	1	no	—
97	#1273	D₄₃	1	1	yes	characteristic
98	C₂×C₄₃	#1713	1	1	yes	characteristic
99	C₂×C₄₃	#1721	1	1	yes	characteristic
100	C₂×C₄₃	#1723	1	2	yes	—
101	C₂×C₄₃	#2093	1	1	yes	characteristic
102	C₂×C₄₃	#2095	1	2	yes	—
103	C₂×C₄⋊Q₈	—	43	1	no	—
104	C₂×#107	—	86	4	no	—
105	C₂×#107	—	86	2	no	—
106	C₂×#103	—	86	2	no	—
107	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
109	#384	—	43	1	no	—
110	#384	—	43	2	no	—
111	#340	—	86	4	no	—
112	C₄₃	#10674	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	86	1	no	—
114	C₂²×C₈	D₁₇₂	1	2	yes	—
115	C₂²×C₈	D₁₇₂	1	4	yes	—
116	C₄⋊Q₈	—	43	2	no	—
117	C₄⋊Q₈	—	86	2	no	—
118	C₄⋊Q₈	—	86	1	no	—
119	C₂×C₄⋊C₄	—	86	2	no	—
120	C₂×C₄⋊C₄	—	86	2	no	—
121	C₂×C₄⋊C₄	—	86	1	no	—
122	C₂×C₄⋊C₄	—	86	2	no	—
123	C₂×C₄⋊C₄	—	86	2	no	—
124	C₂×C₄⋊C₄	—	86	2	no	—
125	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
126	#107	—	172	8	no	—
127	#107	—	172	4	no	—
128	#103	—	172	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₂×Q₈	—	86	2	no	—
133	C₂×Q₈	—	172	2	no	—
134	C₂×Q₈	—	172	2	no	—
135	C₂×Q₈	—	172	1	no	—
136	C₂²×C₄	—	172	2	no	—
137	C₂²×C₄	—	172	1	no	—
138	C₂²×C₄	—	172	1	no	—
139	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	D₃₄₄	1	4	yes	—
146	C₂×C₈	Dic₁₇₂	1	8	yes	—
147	C₄⋊C₄	—	86	4	no	—
148	C₄⋊C₄	—	172	2	no	—
149	C₄⋊C₄	—	172	4	no	—
150	C₄⋊C₄	—	172	4	no	—
151	C₄⋊C₄	—	172	4	no	—
152	C₄⋊C₄	—	172	4	no	—
153	C₄⋊C₄	—	172	2	no	—
154	C₄²	#10621900	1	1	yes	characteristic
155	C₄²	#10621900	1	2	yes	—
156	C₄²	Q₈×D₄₃	1	1	yes	characteristic
157	C₂³	#13063322	1	1	yes	characteristic
158	Q₈	—	344	2	no	—
159	Q₈	—	172	4	no	—
160	C₂×C₄	—	344	1	no	—
161	C₂×C₄	—	344	1	no	—
162	C₂×C₄	—	344	1	no	—
163	C₂×C₄	—	344	2	no	—
164	C₂×C₄	—	344	2	no	—
165	C₂×C₄	—	344	2	no	—
166	C₂×C₄	—	344	2	no	—
167	C₂×C₄	—	172	2	no	—
168	C₂×C₄	#13063305	1	1	yes	characteristic
169	C₂×C₄	#13063305	1	2	yes	—
170	C₂×C₄	#13063309	1	1	yes	characteristic
171	C₂×C₄	#13063309	1	2	yes	—
172	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
173	C₂×C₄	C₂×Dic₁₇₂	1	4	yes	—
174	C₂×C₄	#13063353	1	4	yes	—
175	C₂×C₄	#13063358	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	#14582836	1	1	yes	characteristic
179	C₂²	#14583276	1	1	yes	characteristic
180	C₂²	#14583279	1	2	yes	—
181	C₂²	#14583288	1	2	yes	—
182	C₂²	#14583293	1	1	yes	characteristic
183	C₄	—	344	2	no	—
184	C₄	—	688	1	no	—
185	C₄	—	688	2	no	—
186	C₄	#14583180	1	2	yes	—
187	C₄	#14583195	1	4	yes	—
188	C₄	#14583213	1	2	yes	—
189	C₄	#14583235	1	4	yes	—
190	C₂	#19134213	1	1	yes	characteristic
191	C₂	#19134261	1	1	yes	characteristic
192	C₂	#19134272	1	2	yes	—
193	C₂	#19136122	1	1	yes	characteristic
194	C₂	#19136132	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135556

C₂

yes

maximal

#19135397

C₂

yes

maximal, characteristic

#19134755

C₂

yes

maximal

#19134755

C₂

yes

maximal, characteristic

C₄₃×#1273

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

—

—

—

C₂²

yes

—

#14582769

C₂²

yes

characteristic

#14582662

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

#13063325

—

#13063325

—

#13063325

Q₈

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

#13063259

—

#13063250

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

—

C₄²×C₄₃

C₂×Q₈

yes

C₄²×C₄₃

yes

C₄²×C₄₃

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂³×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#123

yes

characteristic

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#496

yes

—

C₂²×C₄₃

#497

yes

characteristic

C₄×C₄₃

#474

yes

—

C₄×C₄₃

#477

yes

—

C₄×C₄₃

#480

yes

—

C₄×C₄₃

#484

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

100

C₂×C₄₃

#1723

yes

—

101

C₂×C₄₃

#2093

yes

characteristic

102

C₂×C₄₃

#2095

yes

—

103

C₂×C₄⋊Q₈

—

104

C₂×#107

—

105

C₂×#107

—

106

C₂×#103

—

107

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

108

C₂×C₄×C₈

C₂×D₄₃

yes

—

109

#384

—

110

#384

—

111

#340

—

112

C₄₃

#10674

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂²×C₈

D₁₇₂

yes

—

115

C₂²×C₈

D₁₇₂

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂²×D₄₃

yes

characteristic

126

#107

—

172

—

127

#107

—

172

—

128

#103

—

172

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

172

—

134

C₂×Q₈

—

172

—

135

C₂×Q₈

—

172

—

136

C₂²×C₄

—

172

—

137

C₂²×C₄

—

172

—

138

C₂²×C₄

—

172

—

139

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

140

C₂²×C₄

C₂×D₁₇₂

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

D₃₄₄

yes

—

146

C₂×C₈

Dic₁₇₂

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

172

—

149

C₄⋊C₄

—

172

—

150

C₄⋊C₄

—

172

—

151

C₄⋊C₄

—

172

—

152

C₄⋊C₄

—

172

—

153

C₄⋊C₄

—

172

—

154

C₄²

#10621900

yes

characteristic

155

C₄²

#10621900

yes

—

156

C₄²

Q₈×D₄₃

yes

characteristic

157

C₂³

#13063322

yes

characteristic

158

Q₈

—

344

—

159

Q₈

—

172

—

160

C₂×C₄

—

344

—

161

C₂×C₄

—

344

—

162

C₂×C₄

—

344

—

163

C₂×C₄

—

344

—

164

C₂×C₄

—

344

—

165

C₂×C₄

—

344

—

166

C₂×C₄

—

344

—

167

C₂×C₄

—

172

—

168

C₂×C₄

#13063305

yes

characteristic

169

C₂×C₄

#13063305

yes

—

170

C₂×C₄

#13063309

yes

characteristic

171

C₂×C₄

#13063309

yes

—

172

C₂×C₄

C₂×D₃₄₄

yes

—

173

C₂×C₄

C₂×Dic₁₇₂

yes

—

174

C₂×C₄

#13063353

yes

—

175

C₂×C₄

#13063358

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#14582836

yes

characteristic

179

C₂²

#14583276

yes

characteristic

180

C₂²

#14583279

yes

—

181

C₂²

#14583288

yes

—

182

C₂²

#14583293

yes

characteristic

183

C₄

—

344

—

184

C₄

—

688

—

185

C₄

—

688

—

186

C₄

#14583180

yes

—

187

C₄

#14583195

yes

—

188

C₄

#14583213

yes

—

189

C₄

#14583235

yes

—

190

C₂

#19134213

yes

characteristic

191

C₂

#19134261

yes

characteristic

192

C₂

#19134272

yes

—

193

C₂

#19136122

yes

characteristic

194

C₂

#19136132

yes

—

	subgroup	G/H
center	C₂³	#13063322
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1273	C₂
socle, prime radical	C₂³×C₄₃	#127
hypercenter	#1273	D₄₃
hypocenter	C₄₃	#10674
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#13063322

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1273

C₂

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
112. C₄₃	#10674	semidirect

H ≅ G/N

type

112. C₄₃

#10674

semidirect

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

subgroup	index
#19135556	2
#19135397	2
#19134755	2
C₄₃×#1273	2
#10674	43

subgroup

index

C₄₃×#1273

Subgroups — #25001650

Conjugacy classes of subgroups

Complements

Maximal subgroups