Group #25000693 — subgroups

Subgroups — #25000693

Order 11008.

344 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 81 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	#19134714	C₂	1	1	yes	maximal, characteristic
2	#19133295	C₂	1	2	yes	maximal
3	#19133107	C₂	1	2	yes	maximal
4	#19132465	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#3249	C₂	1	1	yes	maximal, characteristic
6	#14582956	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
8	#14582745	C₂²	1	1	yes	characteristic
9	#14582653	—	2	2	no	—
10	#14582645	—	2	2	no	—
11	#14582645	—	2	2	no	—
12	#14582636	—	2	4	no	—
13	C₄₃×#533	C₂²	1	2	yes	—
14	C₄₃×#477	—	2	2	no	—
15	C₄₃×#467	—	2	2	no	—
16	C₄₃×#462	—	2	2	no	—
17	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄₃⋊C₈	—	2	1	no	—
19	C₂×C₄×Dic₄₃	—	2	1	no	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
23	#13063236	—	2	2	no	—
24	C₄₃⋊#95	—	2	1	no	—
25	C₄₃⋊#95	D₄	1	2	yes	—
26	C₂×Q₁₆×C₄₃	—	2	2	no	—
27	C₂²×C₈×C₄₃	D₄	1	2	yes	—
28	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
30	#125×C₄₃	—	2	2	no	—
31	C₂²⋊Q₈×C₄₃	—	2	2	no	—
32	C₂²⋊Q₈×C₄₃	—	2	2	no	—
33	C₄×Q₈×C₄₃	—	2	2	no	—
34	C₄×Q₈×C₄₃	—	2	2	no	—
35	#117×C₄₃	—	2	2	no	—
36	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	#107×C₄₃	—	2	2	no	—
38	C₄⋊C₈×C₄₃	—	2	2	no	—
39	#103×C₄₃	—	2	4	no	—
40	C₂×Q₈×C₄₃	—	2	2	no	—
41	C₂×Q₈×C₄₃	—	2	2	no	—
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
45	C₂²×Dic₄₃	—	2	2	no	—
46	Q₁₆×C₄₃	—	8	2	no	—
47	C₂×C₈×C₄₃	—	2	2	no	—
48	C₂×C₈×C₄₃	—	4	2	no	—
49	C₄⋊C₄×C₄₃	—	2	2	no	—
50	C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₄⋊C₄×C₄₃	—	4	2	no	—
54	C₄⋊C₄×C₄₃	—	4	2	no	—
55	C₂²⋊C₄×C₄₃	—	4	2	no	—
56	C₄²×C₄₃	—	4	2	no	—
57	C₄²×C₄₃	C₂×D₄	1	2	yes	—
58	C₄²×C₄₃	#41	1	2	yes	—
59	C₄×Dic₄₃	—	4	1	no	—
60	C₄×Dic₄₃	—	4	1	no	—
61	C₂×C₄₃⋊C₈	—	4	2	no	—
62	C₂×C₄₃⋊C₈	—	4	1	no	—
63	C₂×C₄₃⋊C₈	—	4	2	no	—
64	C₂³×C₄₃	#121	1	1	yes	characteristic
65	Q₈×C₄₃	—	4	2	no	—
66	Q₈×C₄₃	—	4	2	no	—
67	Q₈×C₄₃	—	8	2	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	1	no	—
71	C₂×C₄×C₄₃	—	4	1	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	—	4	2	no	—
74	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
75	C₂×C₄×C₄₃	#121	1	2	yes	—
76	C₂×C₄×C₄₃	#123	1	2	yes	—
77	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
78	C₂×Dic₄₃	—	4	2	no	—
79	C₂×Dic₄₃	—	4	2	no	—
80	C₂×Dic₄₃	—	4	2	no	—
81	C₈×C₄₃	—	4	2	no	—
82	C₄₃⋊C₈	—	8	2	no	—
83	C₄₃⋊C₈	—	8	1	no	—
84	#10517	—	43	1	no	maximal, Sylow
85	C₂²×C₄₃	—	2	1	no	—
86	C₂²×C₄₃	—	2	1	no	—
87	C₂²×C₄₃	#394	1	1	yes	characteristic
88	C₂²×C₄₃	#473	1	2	yes	—
89	C₄×C₄₃	—	2	2	no	—
90	C₄×C₄₃	—	2	1	no	—
91	C₄×C₄₃	—	2	2	no	—
92	C₄×C₄₃	—	8	2	no	—
93	C₄×C₄₃	—	8	2	no	—
94	C₄×C₄₃	#454	1	1	yes	characteristic
95	C₄×C₄₃	#455	1	1	yes	characteristic
96	Dic₄₃	—	8	2	no	—
97	#3249	D₄₃	1	1	yes	characteristic
98	#1832	—	43	1	no	—
99	#1552	—	43	2	no	—
100	#1506	—	43	2	no	—
101	#1290	—	43	1	no	—
102	C₂×C₄₃	—	2	2	no	—
103	C₂×C₄₃	#2040	1	1	yes	characteristic
104	C₂×C₄₃	#2047	1	1	yes	characteristic
105	C₂×C₄₃	#2053	1	1	yes	characteristic
106	#533	C₂×D₄₃	1	2	yes	—
107	#477	—	2	2	no	—
108	#467	—	2	2	no	—
109	#462	—	2	2	no	—
110	#432	—	43	2	no	—
111	C₂×C₄⋊C₈	—	43	1	no	—
112	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
113	#376	—	43	1	no	—
114	#336	—	86	2	no	—
115	#332	—	86	2	no	—
116	#332	—	86	2	no	—
117	#328	—	86	4	no	—
118	C₄₃	#10517	1	1	yes	characteristic, complemented, Sylow
119	C₂×Q₁₆	—	2	2	no	—
120	C₂²×C₈	—	86	1	no	—
121	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
122	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
123	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
124	#125	—	2	2	no	—
125	C₂²⋊Q₈	—	2	2	no	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₄×Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	#117	—	2	2	no	—
130	C₂×C₄²	—	86	1	no	—
131	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
132	#107	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	86	2	no	—
135	C₄⋊C₈	—	86	2	no	—
136	#103	—	2	4	no	—
137	C₂²⋊C₈	—	86	2	no	—
138	#97	—	86	2	no	—
139	#95	—	43	2	no	—
140	#95	—	86	1	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	—	86	2	no	—
144	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
145	C₂²×C₄	#10621900	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
147	Q₁₆	—	8	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	172	2	no	—
151	C₂×C₈	—	172	1	no	—
152	C₂×C₈	—	172	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	—	172	1	no	—
162	C₄²	—	172	1	no	—
163	C₄²	#10621900	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
165	C₂³	#13063296	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	Q₈	—	4	2	no	—
168	Q₈	—	8	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	172	2	no	—
176	C₂×C₄	—	172	2	no	—
177	C₂×C₄	—	172	2	no	—
178	C₂×C₄	#13063294	1	1	yes	characteristic
179	C₂×C₄	#13063297	1	2	yes	—
180	C₂×C₄	#13063306	1	2	yes	—
181	C₂×C₄	#13063312	1	1	yes	characteristic
182	C₈	—	4	2	no	—
183	C₈	—	344	2	no	—
184	C₈	—	344	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#14582813	1	1	yes	characteristic
188	C₂²	#14583171	1	2	yes	—
189	C₄	—	2	1	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	8	2	no	—
193	C₄	—	8	2	no	—
194	C₄	—	344	2	no	—
195	C₄	#14583072	1	1	yes	characteristic
196	C₄	#14583077	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#19135821	1	1	yes	characteristic
199	C₂	#19135860	1	1	yes	characteristic
200	C₂	#19135898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134714

C₂

yes

maximal, characteristic

#19133295

C₂

yes

maximal

#19133107

C₂

yes

maximal

#19132465

C₂

yes

maximal, characteristic

C₄₃×#3249

C₂

yes

maximal, characteristic

#14582956

C₂²

yes

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#14582745

C₂²

yes

—

—

—

—

C₄₃×#533

C₂²

yes

—

C₄₃×#477

—

C₄₃×#467

—

C₄₃×#462

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

—

C₂×C₄×Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

C₄₃⋊#95

—

C₄₃⋊#95

D₄

yes

—

C₂×Q₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

D₄

yes

—

#125×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#107×C₄₃

—

C₄⋊C₈×C₄₃

—

#103×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂²×Dic₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#121

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10517

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#394

yes

characteristic

C₂²×C₄₃

#473

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#3249

D₄₃

yes

characteristic

#1832

—

#1552

—

100

#1506

—

101

#1290

—

102

C₂×C₄₃

—

103

C₂×C₄₃

#2040

yes

characteristic

104

C₂×C₄₃

#2047

yes

characteristic

105

C₂×C₄₃

#2053

yes

characteristic

106

#533

C₂×D₄₃

yes

—

107

#477

—

108

#467

—

109

#462

—

110

#432

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄⋊C₈

C₂×D₄₃

yes

characteristic

113

#376

—

114

#336

—

115

#332

—

116

#332

—

117

#328

—

118

C₄₃

#10517

yes

characteristic, complemented, Sylow

119

C₂×Q₁₆

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²⋊D₄₃

yes

—

122

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

123

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

124

#125

—

125

C₂²⋊Q₈

—

126

C₂²⋊Q₈

—

127

C₄×Q₈

—

128

C₄×Q₈

—

129

#117

—

130

C₂×C₄²

—

131

C₂×C₄²

C₂²×D₄₃

yes

characteristic

132

#107

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

#103

—

137

C₂²⋊C₈

—

138

#97

—

139

#95

—

140

#95

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

D₄×D₄₃

yes

characteristic

145

C₂²×C₄

#10621900

yes

characteristic

146

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

147

Q₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

172

—

151

C₂×C₈

—

172

—

152

C₂×C₈

—

172

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₄²

—

172

—

162

C₄²

—

172

—

163

C₄²

#10621900

yes

—

164

C₄²

C₂×C₂²⋊D₄₃

yes

—

165

C₂³

#13063296

yes

characteristic

166

Q₈

—

167

Q₈

—

168

Q₈

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

172

—

176

C₂×C₄

—

172

—

177

C₂×C₄

—

172

—

178

C₂×C₄

#13063294

yes

characteristic

179

C₂×C₄

#13063297

yes

—

180

C₂×C₄

#13063306

yes

—

181

C₂×C₄

#13063312

yes

characteristic

182

C₈

—

183

C₈

—

344

—

184

C₈

—

344

—

185

C₂²

—

186

C₂²

—

187

C₂²

#14582813

yes

characteristic

188

C₂²

#14583171

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

344

—

195

C₄

#14583072

yes

characteristic

196

C₄

#14583077

yes

characteristic

197

C₂

—

198

C₂

#19135821

yes

characteristic

199

C₂

#19135860

yes

characteristic

200

C₂

#19135898

yes

characteristic

	subgroup	G/H
center	C₂²	#14582813
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#3249	C₂
socle	C₂²×C₄₃	#394
prime radical	C₂³×C₄₃	#121
hypercenter	#3249	D₄₃
hypocenter	C₄₃	#10517
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3249

subgroup

G/H

center

C₂²

#14582813

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#3249

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3249

N	H ≅ G/N	type
118. C₄₃	#10517	semidirect

H ≅ G/N

type

118. C₄₃

#10517

semidirect

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

subgroup	index
#19134714	2
#19133295	2
#19133107	2
#19132465	2
C₄₃×#3249	2
#10517	43

subgroup

index

C₄₃×#3249

Subgroups — #25000693

Conjugacy classes of subgroups

Complements

Maximal subgroups