Group #25010751 — subgroups

Subgroups — #25010751

Order 11008.

218 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 65 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	#19136744	C₂	1	1	yes	maximal, characteristic
2	#19136646	C₂	1	1	yes	maximal, characteristic
3	#19134193	C₂	1	1	yes	maximal, characteristic
4	#19133401	C₂	1	1	yes	maximal, characteristic
5	#19132658	C₂	1	1	yes	maximal, characteristic
6	#19132643	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1365	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₄₄	—	2	1	no	—
9	#14583295	—	2	1	no	—
10	#14583279	C₂²	1	1	yes	characteristic
11	#14583249	—	2	1	no	—
12	#14583023	—	2	1	no	—
13	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
14	#14582724	—	2	1	no	—
15	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	#14582704	—	2	1	no	—
17	#14582702	—	2	1	no	—
18	#14582629	—	2	1	no	—
19	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄₃×#500	C₂²	1	1	yes	characteristic
21	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₄₃×#335	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₂	—	2	1	no	—
24	C₂×Dic₁₇₂	—	2	1	no	—
25	C₂×Dic₁₇₂	D₄	1	1	yes	characteristic
26	#13063325	—	2	1	no	—
27	C₄×Dic₈₆	—	2	1	no	—
28	Dic₃₄₄	—	4	2	no	—
29	C₂×C₄₃⋊C₁₆	—	2	1	no	—
30	#13063259	D₄	1	1	yes	characteristic
31	#13063256	—	4	1	no	—
32	#13063256	—	4	1	no	—
33	Dic₄₃⋊C₈	—	4	1	no	—
34	#13063250	—	2	1	no	—
35	#13063249	—	4	2	no	—
36	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
37	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
38	C₈×Dic₄₃	—	4	1	no	—
39	C₄⋊Q₈×C₄₃	—	2	1	no	—
40	C₂×C₁₆×C₄₃	—	2	1	no	—
41	#107×C₄₃	—	2	1	no	—
42	#107×C₄₃	—	2	1	no	—
43	C₄⋊C₈×C₄₃	—	2	1	no	—
44	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₄₃	—	4	1	no	—
46	C₂×Dic₈₆	—	2	1	no	—
47	C₂×Dic₈₆	—	4	1	no	—
48	C₂×C₈×C₄₃	—	4	1	no	—
49	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
51	C₄⋊C₄×C₄₃	—	4	1	no	—
52	C₄⋊C₄×C₄₃	—	4	1	no	—
53	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Dic₄₃	—	2	1	no	—
55	C₄⋊Dic₄₃	—	4	1	no	—
56	C₄⋊Dic₄₃	—	4	1	no	—
57	Dic₄₃⋊C₄	—	4	1	no	—
58	C₄×Dic₄₃	—	4	1	no	—
59	C₂×C₄₃⋊C₈	—	2	1	no	—
60	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
61	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
62	Dic₁₇₂	—	2	1	no	—
63	Dic₁₇₂	—	4	1	no	—
64	Dic₁₇₂	—	4	1	no	—
65	Dic₁₇₂	—	8	1	no	—
66	Dic₁₇₂	SD₁₆	1	2	yes	—
67	C₁₆×C₄₃	—	4	1	no	—
68	C₄₃⋊C₁₆	—	4	1	no	—
69	Q₈×C₄₃	—	8	2	no	—
70	C₂×C₄×C₄₃	—	8	1	no	—
71	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×Dic₄₃	—	4	1	no	—
75	C₂×Dic₄₃	—	4	1	no	—
76	C₂×Dic₄₃	—	8	1	no	—
77	Dic₈₆	—	2	2	no	—
78	Dic₈₆	—	4	1	no	—
79	Dic₈₆	—	8	1	no	—
80	Dic₈₆	—	8	1	no	—
81	C₈×C₄₃	—	2	1	no	—
82	C₈×C₄₃	—	8	1	no	—
83	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₈×C₄₃	#135	1	1	yes	characteristic
85	C₄₃⋊C₈	—	2	1	no	—
86	C₄₃⋊C₈	—	4	1	no	—
87	C₄₃⋊C₈	Q₃₂	1	2	yes	complemented
88	#12421	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	#459	1	1	yes	characteristic
90	C₄×C₄₃	—	2	1	no	—
91	C₄×C₄₃	—	16	1	no	—
92	C₄×C₄₃	#465	1	1	yes	characteristic
93	C₄×C₄₃	#471	1	1	yes	characteristic
94	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
95	C₄×C₄₃	#510	1	1	yes	characteristic
96	Dic₄₃	—	4	2	no	—
97	Dic₄₃	—	8	1	no	—
98	Dic₄₃	—	16	1	no	—
99	#2227	—	43	1	no	—
100	#2209	—	43	1	no	—
101	#1707	—	43	1	no	—
102	C₈×Q₁₆	—	43	1	no	—
103	#1367	—	43	1	no	—
104	#1365	D₄₃	1	1	yes	characteristic
105	#1362	—	43	1	no	—
106	C₂×C₄₃	#1665	1	1	yes	characteristic
107	C₂×C₄₃	#2209	1	1	yes	characteristic
108	C₂×C₄₃	#2212	1	1	yes	characteristic
109	C₂×Q₃₂	—	86	1	no	—
110	#500	C₂×D₄₃	1	1	yes	characteristic
111	#498	—	86	1	no	—
112	#494	—	43	1	no	—
113	#487	—	86	1	no	—
114	C₈×Q₈	—	86	1	no	—
115	C₄×Q₁₆	—	43	1	no	—
116	#366	—	86	1	no	—
117	C₄⋊C₁₆	—	43	1	no	—
118	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
119	#358	—	86	1	no	—
120	#358	—	86	1	no	—
121	#335	C₂×D₄₃	1	1	yes	characteristic
122	Q₈⋊C₈	—	86	1	no	—
123	C₈²	—	43	1	no	—
124	C₄₃	#12421	1	1	yes	characteristic, complemented, Sylow
125	C₂×Q₁₆	—	43	1	no	—
126	C₂×Q₁₆	—	86	1	no	—
127	C₂×Q₁₆	—	86	1	no	—
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	—	86	1	no	—
130	C₄×Q₈	—	86	1	no	—
131	Q₃₂	—	172	2	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	86	1	no	—
134	#107	—	2	1	no	—
135	#107	—	2	1	no	—
136	#107	—	43	1	no	—
137	#106	—	172	1	no	—
138	#106	—	172	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	—	172	1	no	—
141	#103	—	86	1	no	—
142	#103	—	172	2	no	—
143	C₄×C₈	—	43	1	no	—
144	C₄×C₈	—	43	1	no	—
145	C₄×C₈	—	172	1	no	—
146	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
147	C₂×Q₈	—	4	1	no	—
148	C₂×Q₈	—	86	1	no	—
149	C₂×Q₈	—	172	1	no	—
150	Q₁₆	—	43	2	no	—
151	Q₁₆	—	86	1	no	—
152	Q₁₆	—	344	1	no	—
153	Q₁₆	—	172	1	no	—
154	Q₁₆	—	172	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	—	43	1	no	—
157	C₂×C₈	—	43	1	no	—
158	C₂×C₈	—	86	1	no	—
159	C₂×C₈	#10621898	1	1	yes	characteristic
160	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄⋊C₄	—	86	1	no	—
164	C₄⋊C₄	—	172	1	no	—
165	C₄⋊C₄	—	172	1	no	—
166	C₄⋊C₄	—	172	1	no	—
167	C₄²	—	172	1	no	—
168	C₄²	D₄×D₄₃	1	1	yes	characteristic
169	C₁₆	—	4	1	no	—
170	C₁₆	—	172	1	no	—
171	Q₈	—	8	2	no	—
172	Q₈	—	86	2	no	—
173	Q₈	—	344	1	no	—
174	Q₈	—	344	1	no	—
175	Q₈	—	172	1	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	344	1	no	—
178	C₂×C₄	—	172	1	no	—
179	C₂×C₄	—	172	1	no	—
180	C₂×C₄	#13063298	1	1	yes	characteristic
181	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
182	C₂×C₄	#13063352	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	8	1	no	—
185	C₈	—	43	2	no	—
186	C₈	—	86	1	no	—
187	C₈	—	172	1	no	—
188	C₈	SD₁₆×D₄₃	1	1	yes	characteristic
189	C₈	#13063348	1	1	yes	characteristic
190	C₂²	#14583093	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	16	1	no	—
193	C₄	—	344	1	no	—
194	C₄	—	172	2	no	—
195	C₄	—	688	1	no	—
196	C₄	#14583131	1	1	yes	characteristic
197	C₄	#14583165	1	1	yes	characteristic
198	C₄	Q₃₂×D₄₃	1	1	yes	characteristic
199	C₄	#14583338	1	1	yes	characteristic
200	C₂	#19133939	1	1	yes	characteristic
201	C₂	#19136644	1	1	yes	characteristic
202	C₂	#19136663	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136744

C₂

yes

maximal, characteristic

#19136646

C₂

yes

maximal, characteristic

#19134193

C₂

yes

maximal, characteristic

#19133401

C₂

yes

maximal, characteristic

#19132658

C₂

yes

maximal, characteristic

#19132643

C₂

yes

maximal, characteristic

C₄₃×#1365

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

—

#14583295

—

#14583279

C₂²

yes

characteristic

#14583249

—

#14583023

—

C₄×Dic₁₇₂

C₂²

yes

characteristic

#14582724

—

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

—

—

—

C₈×C₄₃⋊C₈

C₂²

yes

characteristic

C₄₃×#500

C₂²

yes

characteristic

C₄₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₄₃×#335

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

D₄

yes

characteristic

#13063325

—

C₄×Dic₈₆

—

Dic₃₄₄

—

C₂×C₄₃⋊C₁₆

—

#13063259

D₄

yes

characteristic

#13063256

—

#13063256

—

Dic₄₃⋊C₈

—

#13063250

—

#13063249

—

C₄×C₄₃⋊C₈

D₄

yes

characteristic

C₄×C₄₃⋊C₈

D₄

yes

characteristic

C₈×Dic₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₁₆×C₄₃

—

#107×C₄₃

—

#107×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₈

yes

characteristic

C₂×C₄₃⋊C₈

D₈

yes

characteristic

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

SD₁₆

yes

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂×SD₁₆

yes

characteristic

C₈×C₄₃

#135

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

Q₃₂

yes

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×Q₃₂

yes

characteristic

C₄×C₄₃

#510

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#2227

—

100

#2209

—

101

#1707

—

102

C₈×Q₁₆

—

103

#1367

—

104

#1365

D₄₃

yes

characteristic

105

#1362

—

106

C₂×C₄₃

#1665

yes

characteristic

107

C₂×C₄₃

#2209

yes

characteristic

108

C₂×C₄₃

#2212

yes

characteristic

109

C₂×Q₃₂

—

110

#500

C₂×D₄₃

yes

characteristic

111

#498

—

112

#494

—

113

#487

—

114

C₈×Q₈

—

115

C₄×Q₁₆

—

116

#366

—

117

C₄⋊C₁₆

—

118

C₄⋊C₁₆

C₂×D₄₃

yes

characteristic

119

#358

—

120

#358

—

121

#335

C₂×D₄₃

yes

characteristic

122

Q₈⋊C₈

—

123

C₈²

—

124

C₄₃

#12421

yes

characteristic, complemented, Sylow

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄×Q₈

—

131

Q₃₂

—

172

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

#107

—

135

#107

—

136

#107

—

137

#106

—

172

—

138

#106

—

172

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

172

—

141

#103

—

142

#103

—

172

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

172

—

146

C₄×C₈

C₂²×D₄₃

yes

characteristic

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

172

—

150

Q₁₆

—

151

Q₁₆

—

152

Q₁₆

—

344

—

153

Q₁₆

—

172

—

154

Q₁₆

—

172

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

#10621898

yes

characteristic

160

C₂×C₈

D₄×D₄₃

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

172

—

165

C₄⋊C₄

—

172

—

166

C₄⋊C₄

—

172

—

167

C₄²

—

172

—

168

C₄²

D₄×D₄₃

yes

characteristic

169

C₁₆

—

170

C₁₆

—

172

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

344

—

174

Q₈

—

344

—

175

Q₈

—

172

—

176

C₂×C₄

—

177

C₂×C₄

—

344

—

178

C₂×C₄

—

172

—

179

C₂×C₄

—

172

—

180

C₂×C₄

#13063298

yes

characteristic

181

C₂×C₄

D₈×D₄₃

yes

characteristic

182

C₂×C₄

#13063352

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

172

—

188

C₈

SD₁₆×D₄₃

yes

characteristic

189

C₈

#13063348

yes

characteristic

190

C₂²

#14583093

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

344

—

194

C₄

—

172

—

195

C₄

—

688

—

196

C₄

#14583131

yes

characteristic

197

C₄

#14583165

yes

characteristic

198

C₄

Q₃₂×D₄₃

yes

characteristic

199

C₄

#14583338

yes

characteristic

200

C₂

#19133939

yes

characteristic

201

C₂

#19136644

yes

characteristic

202

C₂

#19136663

yes

characteristic

	subgroup	G/H
center	C₂²	#14583093
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1365	C₂
socle, prime radical	C₂²×C₄₃	#459
hypercenter	#1365	D₄₃
hypocenter	C₄₃	#12421
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1365

subgroup

G/H

center

C₂²

#14583093

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1365

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1365

N	H ≅ G/N	type
87. C₄₃⋊C₈	Q₃₂	semidirect
124. C₄₃	#12421	semidirect

H ≅ G/N

type

87. C₄₃⋊C₈

Q₃₂

semidirect

124. C₄₃

#12421

semidirect

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

subgroup	index
#19136744	2
#19136646	2
#19134193	2
#19133401	2
#19132658	2
#19132643	2
C₄₃×#1365	2
#12421	43

subgroup

index

C₄₃×#1365

Subgroups — #25010751

Conjugacy classes of subgroups

Complements

Maximal subgroups