Group #25010875 — subgroups

Subgroups — #25010875

Order 11008.

212 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 63 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	#19136847	C₂	1	1	yes	maximal, characteristic
2	#19136500	C₂	1	1	yes	maximal, characteristic
3	#19133966	C₂	1	1	yes	maximal, characteristic
4	#19133962	C₂	1	1	yes	maximal, characteristic
5	#19132668	C₂	1	1	yes	maximal, characteristic
6	#19132645	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1362	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063271	—	2	1	no	—
9	C₂×#13063271	—	2	1	no	—
10	#14583279	C₂²	1	1	yes	characteristic
11	#14583185	—	2	1	no	—
12	#14583181	—	2	1	no	—
13	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
14	Dic₄₃⋊C₁₆	—	2	1	no	—
15	#14582674	—	2	1	no	—
16	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	#14582652	C₂²	1	1	yes	characteristic
18	#14582645	—	2	1	no	—
19	#14582644	—	2	1	no	—
20	C₄₃×#494	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	#13063325	—	2	1	no	—
26	C₄×Dic₈₆	D₄	1	1	yes	characteristic
27	#13063271	—	4	2	no	—
28	#13063271	—	4	2	no	—
29	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
30	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
31	#13063257	—	4	1	no	—
32	#13063256	—	4	1	no	—
33	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
34	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
35	Dic₄₃⋊C₈	—	2	1	no	—
36	#13063250	—	4	1	no	—
37	#13063248	—	4	1	no	—
38	C₈×Dic₄₃	—	2	1	no	—
39	C₂×Q₁₆×C₄₃	—	2	1	no	—
40	C₂×Q₁₆×C₄₃	—	2	1	no	—
41	C₄⋊Q₈×C₄₃	—	2	1	no	—
42	C₂×C₁₆×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	—	4	1	no	—
47	C₂×Dic₈₆	D₈	1	1	yes	characteristic
48	Q₁₆×C₄₃	—	4	1	no	—
49	Q₁₆×C₄₃	—	4	1	no	—
50	Q₁₆×C₄₃	—	8	1	no	—
51	C₂×C₈×C₄₃	—	4	1	no	—
52	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₄₃	—	4	1	no	—
55	C₄⋊C₄×C₄₃	—	4	1	no	—
56	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊Dic₄₃	—	4	1	no	—
58	C₄⋊Dic₄₃	—	4	1	no	—
59	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
60	Dic₄₃⋊C₄	—	2	1	no	—
61	C₄×Dic₄₃	—	2	1	no	—
62	C₂×C₄₃⋊C₈	—	2	1	no	—
63	C₂×C₄₃⋊C₈	—	4	1	no	—
64	Dic₁₇₂	—	4	1	no	—
65	Dic₁₇₂	—	4	1	no	—
66	Dic₁₇₂	—	8	1	no	—
67	C₁₆×C₄₃	—	4	1	no	—
68	C₄₃⋊C₁₆	—	2	1	no	—
69	C₄₃⋊C₁₆	SD₁₆	1	2	yes	—
70	Q₈×C₄₃	—	8	1	no	—
71	Q₈×C₄₃	—	8	1	no	—
72	C₂×C₄×C₄₃	—	8	1	no	—
73	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
76	C₂×Dic₄₃	—	2	1	no	—
77	C₂×Dic₄₃	—	2	1	no	—
78	C₂×Dic₄₃	—	8	1	no	—
79	Dic₈₆	—	2	1	no	—
80	Dic₈₆	—	2	1	no	—
81	Dic₈₆	—	8	1	no	—
82	Dic₈₆	—	8	1	no	—
83	C₈×C₄₃	—	2	1	no	—
84	C₈×C₄₃	—	8	1	no	—
85	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₄₃⋊C₈	—	4	1	no	—
88	C₄₃⋊C₈	—	8	1	no	—
89	#12443	—	43	1	no	maximal, Sylow
90	C₂²×C₄₃	#466	1	1	yes	characteristic
91	C₄×C₄₃	—	2	1	no	—
92	C₄×C₄₃	—	16	1	no	—
93	C₄×C₄₃	#460	1	1	yes	characteristic
94	C₄×C₄₃	#472	1	1	yes	characteristic
95	C₄×C₄₃	#510	1	1	yes	characteristic
96	C₄×C₄₃	#510	1	1	yes	characteristic
97	Dic₄₃	—	4	1	no	—
98	Dic₄₃	—	4	1	no	—
99	Dic₄₃	—	16	1	no	—
100	#2247	—	43	1	no	—
101	#2183	—	43	1	no	—
102	#1670	—	43	1	no	—
103	#1670	—	43	1	no	—
104	#1372	—	43	1	no	—
105	#1362	—	43	1	no	—
106	#1362	D₄₃	1	1	yes	characteristic
107	C₂×C₄₃	#1668	1	1	yes	characteristic
108	C₂×C₄₃	#2219	1	1	yes	characteristic
109	C₂×C₄₃	#2222	1	1	yes	characteristic
110	C₂×Q₃₂	—	86	1	no	—
111	C₂×Q₃₂	—	86	1	no	—
112	#494	—	43	1	no	—
113	#494	C₂×D₄₃	1	1	yes	characteristic
114	#475	—	86	1	no	—
115	#475	—	86	1	no	—
116	C₈×Q₈	—	43	1	no	—
117	C₄⋊C₁₆	—	86	1	no	—
118	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
119	#346	—	86	1	no	—
120	C₄×C₁₆	—	43	1	no	—
121	#335	—	43	1	no	—
122	#335	C₂×D₄₃	1	1	yes	characteristic
123	#332	—	86	1	no	—
124	#332	—	86	1	no	—
125	C₄₃	#12443	1	1	yes	characteristic, complemented, Sylow
126	C₂×Q₁₆	—	2	1	no	—
127	C₂×Q₁₆	—	2	1	no	—
128	C₂×Q₁₆	—	86	1	no	—
129	C₂×Q₁₆	—	86	1	no	—
130	C₄⋊Q₈	—	2	1	no	—
131	C₄⋊Q₈	—	86	1	no	—
132	C₄×Q₈	—	43	1	no	—
133	Q₃₂	—	172	2	no	—
134	Q₃₂	—	172	2	no	—
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	—	43	1	no	—
137	C₂×C₁₆	—	43	1	no	—
138	#106	—	172	1	no	—
139	#106	—	172	1	no	—
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	43	1	no	—
142	C₄⋊C₈	—	86	1	no	—
143	C₄⋊C₈	—	86	1	no	—
144	#103	—	172	1	no	—
145	#103	—	172	1	no	—
146	C₄×C₈	—	86	1	no	—
147	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
148	C₂×Q₈	—	4	1	no	—
149	C₂×Q₈	—	43	1	no	—
150	C₂×Q₈	—	172	1	no	—
151	Q₁₆	—	4	1	no	—
152	Q₁₆	—	4	1	no	—
153	Q₁₆	—	8	1	no	—
154	Q₁₆	—	344	1	no	—
155	Q₁₆	—	172	1	no	—
156	Q₁₆	—	172	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	86	1	no	—
159	C₂×C₈	—	172	1	no	—
160	C₂×C₈	#10621898	1	1	yes	characteristic
161	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
162	C₄⋊C₄	—	4	1	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	43	1	no	—
165	C₄⋊C₄	—	86	1	no	—
166	C₄⋊C₄	—	172	1	no	—
167	C₄⋊C₄	—	172	1	no	—
168	C₄²	—	86	1	no	—
169	C₄²	D₄×D₄₃	1	1	yes	characteristic
170	C₁₆	—	4	1	no	—
171	C₁₆	—	43	2	no	—
172	C₁₆	—	86	1	no	—
173	Q₈	—	8	1	no	—
174	Q₈	—	8	1	no	—
175	Q₈	—	86	1	no	—
176	Q₈	—	86	1	no	—
177	Q₈	—	344	1	no	—
178	Q₈	—	344	1	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	86	1	no	—
181	C₂×C₄	—	86	1	no	—
182	C₂×C₄	—	344	1	no	—
183	C₂×C₄	#13063298	1	1	yes	characteristic
184	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
185	C₂×C₄	#13063346	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	8	1	no	—
188	C₈	—	344	1	no	—
189	C₈	—	172	1	no	—
190	C₈	SD₁₆×D₄₃	1	1	yes	characteristic
191	C₈	#13063349	1	1	yes	characteristic
192	C₂²	#14583134	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	16	1	no	—
195	C₄	—	172	1	no	—
196	C₄	—	172	1	no	—
197	C₄	—	688	1	no	—
198	C₄	#14583098	1	1	yes	characteristic
199	C₄	#14583168	1	1	yes	characteristic
200	C₄	#14583338	1	1	yes	characteristic
201	C₄	#14583341	1	1	yes	characteristic
202	C₂	#19133954	1	1	yes	characteristic
203	C₂	#19136702	1	1	yes	characteristic
204	C₂	#19136719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136847

C₂

yes

maximal, characteristic

#19136500

C₂

yes

maximal, characteristic

#19133966

C₂

yes

maximal, characteristic

#19133962

C₂

yes

maximal, characteristic

#19132668

C₂

yes

maximal, characteristic

#19132645

C₂

yes

maximal, characteristic

C₄₃×#1362

C₂

yes

maximal, characteristic

C₂×#13063271

—

C₂×#13063271

—

#14583279

C₂²

yes

characteristic

#14583185

—

#14583181

—

C₈×Dic₈₆

C₂²

yes

characteristic

Dic₄₃⋊C₁₆

—

#14582674

—

C₄×C₄₃⋊C₁₆

C₂²

yes

characteristic

#14582652

C₂²

yes

characteristic

#14582645

—

#14582644

—

C₄₃×#494

C₂²

yes

characteristic

C₄₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₄₃×#335

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

C₄×Dic₈₆

D₄

yes

characteristic

#13063271

—

#13063271

—

C₂×C₄₃⋊C₁₆

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

D₄

yes

characteristic

#13063257

—

#13063256

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

characteristic

Dic₄₃⋊C₈

—

#13063250

—

#13063248

—

C₈×Dic₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₁₆×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₈

yes

characteristic

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

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

D₈

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

SD₁₆

yes

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂×SD₁₆

yes

characteristic

C₈×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

100

#2247

—

101

#2183

—

102

#1670

—

103

#1670

—

104

#1372

—

105

#1362

—

106

#1362

D₄₃

yes

characteristic

107

C₂×C₄₃

#1668

yes

characteristic

108

C₂×C₄₃

#2219

yes

characteristic

109

C₂×C₄₃

#2222

yes

characteristic

110

C₂×Q₃₂

—

111

C₂×Q₃₂

—

112

#494

—

113

#494

C₂×D₄₃

yes

characteristic

114

#475

—

115

#475

—

116

C₈×Q₈

—

117

C₄⋊C₁₆

—

118

C₄⋊C₁₆

C₂×D₄₃

yes

characteristic

119

#346

—

120

C₄×C₁₆

—

121

#335

—

122

#335

C₂×D₄₃

yes

characteristic

123

#332

—

124

#332

—

125

C₄₃

#12443

yes

characteristic, complemented, Sylow

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄×Q₈

—

133

Q₃₂

—

172

—

134

Q₃₂

—

172

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

#106

—

172

—

139

#106

—

172

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

#103

—

172

—

145

#103

—

172

—

146

C₄×C₈

—

147

C₄×C₈

C₂²×D₄₃

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

172

—

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

—

154

Q₁₆

—

344

—

155

Q₁₆

—

172

—

156

Q₁₆

—

172

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

172

—

160

C₂×C₈

#10621898

yes

characteristic

161

C₂×C₈

D₄×D₄₃

yes

characteristic

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

172

—

167

C₄⋊C₄

—

172

—

168

C₄²

—

169

C₄²

D₄×D₄₃

yes

characteristic

170

C₁₆

—

171

C₁₆

—

172

C₁₆

—

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

Q₈

—

177

Q₈

—

344

—

178

Q₈

—

344

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

344

—

183

C₂×C₄

#13063298

yes

characteristic

184

C₂×C₄

D₈×D₄₃

yes

characteristic

185

C₂×C₄

#13063346

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

344

—

189

C₈

—

172

—

190

C₈

SD₁₆×D₄₃

yes

characteristic

191

C₈

#13063349

yes

characteristic

192

C₂²

#14583134

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

172

—

196

C₄

—

172

—

197

C₄

—

688

—

198

C₄

#14583098

yes

characteristic

199

C₄

#14583168

yes

characteristic

200

C₄

#14583338

yes

characteristic

201

C₄

#14583341

yes

characteristic

202

C₂

#19133954

yes

characteristic

203

C₂

#19136702

yes

characteristic

204

C₂

#19136719

yes

characteristic

	subgroup	G/H
center	C₂²	#14583134
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1362	C₂
socle, prime radical	C₂²×C₄₃	#466
hypercenter	#1362	D₄₃
hypocenter	C₄₃	#12443
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1362

subgroup

G/H

center

C₂²

#14583134

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1362

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1362

N	H ≅ G/N	type
125. C₄₃	#12443	semidirect

H ≅ G/N

type

125. C₄₃

#12443

semidirect

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

subgroup	index
#19136847	2
#19136500	2
#19133966	2
#19133962	2
#19132668	2
#19132645	2
C₄₃×#1362	2
#12443	43

subgroup

index

C₄₃×#1362

Subgroups — #25010875

Conjugacy classes of subgroups

Complements

Maximal subgroups