Group #25001723 — subgroups

Subgroups — #25001723

Order 11008.

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

normal subgroups only

This group has 264 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	#19135318	C₂	1	1	yes	maximal, characteristic
2	#19134770	C₂	1	1	yes	maximal, characteristic
3	#19132497	C₂	1	2	yes	maximal
4	#19132491	C₂	1	2	yes	maximal
5	C₄₃×#1275	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063261	—	2	1	no	—
7	C₂×#13063261	—	2	1	no	—
8	#14582923	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₃⋊C₈	—	2	1	no	—
11	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
12	#14582653	—	2	1	no	—
13	C₄₃×#431	C₂²	1	2	yes	—
14	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#10621877	—	2	2	no	—
16	C₂²×C₄₃⋊C₈	—	2	1	no	—
17	C₂×C₄⋊Dic₄₃	—	2	1	no	—
18	C₂×C₄×Dic₄₃	—	2	1	no	—
19	#13063261	—	4	2	no	—
20	#13063261	—	4	2	no	—
21	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
23	Dic₄₃⋊C₈	—	4	2	no	—
24	C₄₃⋊#95	Q₈	1	1	yes	characteristic
25	C₄₃⋊#95	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₄₃	—	2	2	no	—
30	C₂²⋊C₈×C₄₃	—	2	2	no	—
31	#97×C₄₃	—	2	2	no	—
32	C₄×C₈×C₄₃	—	2	2	no	—
33	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
36	C₂²×Dic₄₃	—	2	1	no	—
37	C₂²×Dic₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	—	2	1	no	—
39	C₂×C₈×C₄₃	—	2	1	no	—
40	C₂×C₈×C₄₃	—	2	1	no	—
41	C₂×C₈×C₄₃	—	2	1	no	—
42	C₂×C₈×C₄₃	—	4	2	no	—
43	C₂×C₈×C₄₃	D₈	1	2	yes	—
44	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
45	C₄²×C₄₃	#41	1	2	yes	—
46	C₄²×C₄₃	#41	1	2	yes	—
47	C₄⋊Dic₄₃	—	4	2	no	—
48	C₄×Dic₄₃	—	4	1	no	—
49	C₄×Dic₄₃	—	4	1	no	—
50	#10621877	—	4	2	no	—
51	#10621877	—	4	2	no	—
52	C₂×C₄₃⋊C₈	—	4	1	no	—
53	C₂×C₄₃⋊C₈	—	4	2	no	—
54	C₂×C₄₃⋊C₈	—	4	1	no	—
55	C₂×C₄₃⋊C₈	—	4	1	no	—
56	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₄₃	—	4	1	no	—
58	C₂×C₄×C₄₃	—	4	1	no	—
59	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#125	1	2	yes	—
62	C₂×C₄×C₄₃	#126	1	2	yes	—
63	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
67	C₂×Dic₄₃	—	2	2	no	—
68	C₂×Dic₄₃	—	4	1	no	—
69	C₂×Dic₄₃	—	4	1	no	—
70	C₂×Dic₄₃	—	4	1	no	—
71	C₂×Dic₄₃	—	4	1	no	—
72	C₂×Dic₄₃	—	4	1	no	—
73	C₈×C₄₃	—	2	2	no	—
74	C₈×C₄₃	—	2	2	no	—
75	C₈×C₄₃	—	8	2	no	—
76	C₄₃⋊C₈	—	8	1	no	—
77	C₄₃⋊C₈	—	8	2	no	—
78	#10687	—	43	1	no	maximal, Sylow
79	C₂²×C₄₃	—	2	1	no	—
80	C₂²×C₄₃	—	2	1	no	—
81	C₂²×C₄₃	#400	1	1	yes	characteristic
82	C₂²×C₄₃	#474	1	1	yes	characteristic
83	C₂²×C₄₃	#475	1	1	yes	characteristic
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	#480	1	1	yes	characteristic
87	C₄×C₄₃	#483	1	1	yes	characteristic
88	C₄×C₄₃	#486	1	1	yes	characteristic
89	C₄×C₄₃	#491	1	1	yes	characteristic
90	Dic₄₃	—	4	2	no	—
91	Dic₄₃	—	8	1	no	—
92	#1940	—	43	1	no	—
93	#1849	—	43	1	no	—
94	#1303	—	43	2	no	—
95	#1301	—	43	2	no	—
96	#1275	D₄₃	1	1	yes	characteristic
97	C₂×C₄₃	—	2	1	no	—
98	C₂×C₄₃	—	2	1	no	—
99	C₂×C₄₃	#2099	1	1	yes	characteristic
100	C₂×C₄₃	#2102	1	1	yes	characteristic
101	C₂×C₄₃	#2103	1	1	yes	characteristic
102	#431	C₂×D₄₃	1	2	yes	—
103	C₂×#108	—	86	1	no	—
104	C₂×#108	—	86	1	no	—
105	#423	—	43	2	no	—
106	C₂×C₄⋊C₈	—	43	1	no	—
107	C₂×C₄⋊C₈	—	86	1	no	—
108	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	43	1	no	—
110	#336	—	86	1	no	—
111	C₄₃	#10687	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₁₆	—	86	2	no	—
113	C₂²×C₈	—	86	1	no	—
114	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
115	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	86	1	no	—
117	C₂×C₄²	—	86	1	no	—
118	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
119	#108	—	172	2	no	—
120	#108	—	172	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	86	4	no	—
123	C₄⋊C₈	—	86	2	no	—
124	C₄⋊C₈	—	172	2	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	#95	—	43	1	no	—
129	#95	—	43	1	no	—
130	C₂²×C₄	—	86	1	no	—
131	C₂²×C₄	—	86	1	no	—
132	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
133	C₂²×C₄	#10621900	1	1	yes	characteristic
134	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
135	M₁₆	—	172	2	no	—
136	M₁₆	—	172	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	172	1	no	—
143	C₂×C₈	—	172	2	no	—
144	C₂×C₈	—	172	1	no	—
145	C₂×C₈	—	172	1	no	—
146	C₂×C₈	#10621873	1	2	yes	—
147	C₂×C₈	D₃₄₄	1	2	yes	—
148	C₄⋊C₄	—	172	2	no	—
149	C₄²	—	172	1	no	—
150	C₄²	—	172	1	no	—
151	C₄²	#10621898	1	2	yes	—
152	C₄²	#10621900	1	2	yes	—
153	C₂³	#13063305	1	1	yes	characteristic
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	86	2	no	—
157	C₂×C₄	—	172	1	no	—
158	C₂×C₄	—	172	1	no	—
159	C₂×C₄	—	172	1	no	—
160	C₂×C₄	—	172	1	no	—
161	C₂×C₄	—	172	1	no	—
162	C₂×C₄	#13063309	1	1	yes	characteristic
163	C₂×C₄	#13063313	1	1	yes	characteristic
164	C₂×C₄	#13063315	1	2	yes	—
165	C₂×C₄	#13063318	1	2	yes	—
166	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
167	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
168	C₂×C₄	#13063358	1	1	yes	characteristic
169	C₂×C₄	#13063358	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	344	1	no	—
174	C₈	—	344	2	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	#14582839	1	1	yes	characteristic
178	C₂²	#14583180	1	1	yes	characteristic
179	C₂²	#14583185	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	344	1	no	—
183	C₄	—	172	2	no	—
184	C₄	#14583213	1	1	yes	characteristic
185	C₄	#14583230	1	1	yes	characteristic
186	C₄	#14583246	1	1	yes	characteristic
187	C₄	#14583268	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#19136154	1	1	yes	characteristic
191	C₂	#19136171	1	1	yes	characteristic
192	C₂	#19136176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19135318

C₂

yes

maximal, characteristic

#19134770

C₂

yes

maximal, characteristic

#19132497

C₂

yes

maximal

#19132491

C₂

yes

maximal

C₄₃×#1275

C₂

yes

maximal, characteristic

C₂×#13063261

—

C₂×#13063261

—

#14582923

C₂²

yes

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₈

—

C₄×C₄⋊Dic₄₃

C₂²

yes

characteristic

#14582653

—

C₄₃×#431

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

C₂²×C₄₃⋊C₈

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄×Dic₄₃

—

#13063261

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

C₄₃⋊#95

Q₈

yes

characteristic

C₄₃⋊#95

Q₈

yes

characteristic

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

#41

yes

—

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

#126

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10687

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#2099

yes

characteristic

100

C₂×C₄₃

#2102

yes

characteristic

101

C₂×C₄₃

#2103

yes

characteristic

102

#431

C₂×D₄₃

yes

—

103

C₂×#108

—

104

C₂×#108

—

105

#423

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×#97

C₂×D₄₃

yes

characteristic

109

C₄×C₄⋊C₄

—

110

#336

—

111

C₄₃

#10687

yes

characteristic, complemented, Sylow

112

C₂×M₁₆

—

113

C₂²×C₈

—

114

C₂²×C₈

D₁₇₂

yes

characteristic

115

C₂²×C₈

D₁₇₂

yes

characteristic

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

—

118

C₂×C₄²

C₂²×D₄₃

yes

characteristic

119

#108

—

172

—

120

#108

—

172

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

172

—

125

C₂²⋊C₈

—

126

#97

—

127

C₄×C₈

—

128

#95

—

129

#95

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

133

C₂²×C₄

#10621900

yes

characteristic

134

C₂²×C₄

Q₈×D₄₃

yes

characteristic

135

M₁₆

—

172

—

136

M₁₆

—

172

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

172

—

143

C₂×C₈

—

172

—

144

C₂×C₈

—

172

—

145

C₂×C₈

—

172

—

146

C₂×C₈

#10621873

yes

—

147

C₂×C₈

D₃₄₄

yes

—

148

C₄⋊C₄

—

172

—

149

C₄²

—

172

—

150

C₄²

—

172

—

151

C₄²

#10621898

yes

—

152

C₄²

#10621900

yes

—

153

C₂³

#13063305

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

172

—

158

C₂×C₄

—

172

—

159

C₂×C₄

—

172

—

160

C₂×C₄

—

172

—

161

C₂×C₄

—

172

—

162

C₂×C₄

#13063309

yes

characteristic

163

C₂×C₄

#13063313

yes

characteristic

164

C₂×C₄

#13063315

yes

—

165

C₂×C₄

#13063318

yes

—

166

C₂×C₄

C₂×D₃₄₄

yes

characteristic

167

C₂×C₄

C₂×#10621873

yes

characteristic

168

C₂×C₄

#13063358

yes

characteristic

169

C₂×C₄

#13063358

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

344

—

174

C₈

—

344

—

175

C₂²

—

176

C₂²

—

177

C₂²

#14582839

yes

characteristic

178

C₂²

#14583180

yes

characteristic

179

C₂²

#14583185

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

344

—

183

C₄

—

172

—

184

C₄

#14583213

yes

characteristic

185

C₄

#14583230

yes

characteristic

186

C₄

#14583246

yes

characteristic

187

C₄

#14583268

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

#19136154

yes

characteristic

191

C₂

#19136171

yes

characteristic

192

C₂

#19136176

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063313
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1275	C₂
socle	C₂²×C₄₃	#400
prime radical	C₂³×C₄₃	C₂²⋊Q₈
hypercenter	#1275	D₄₃
hypocenter	C₄₃	#10687
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

subgroup

G/H

center

C₂×C₄

#13063313

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1275

C₂

socle

C₂²×C₄₃

#400

prime radical

C₂³×C₄₃

C₂²⋊Q₈

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1275

N	H ≅ G/N	type
111. C₄₃	#10687	semidirect

H ≅ G/N

type

111. C₄₃

#10687

semidirect

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

subgroup	index
#19135318	2
#19134770	2
#19132497	2
#19132491	2
C₄₃×#1275	2
#10687	43

subgroup

index

C₄₃×#1275

Subgroups — #25001723

Conjugacy classes of subgroups

Complements

Maximal subgroups