Group #25000759 — subgroups

Subgroups — #25000759

Order 11008.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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	#19145989	C₂	1	1	yes	maximal, characteristic
2	#19133323	C₂	1	2	yes	maximal
3	#19133209	C₂	1	2	yes	maximal
4	#19132506	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1849	C₂	1	1	yes	maximal, characteristic
6	#14583456	C₂²	1	2	yes	—
7	#14583185	—	2	2	no	—
8	#14583157	—	2	2	no	—
9	#14583110	—	2	2	no	—
10	#14582956	C₂²	1	2	yes	—
11	#14582647	—	2	2	no	—
12	#14582646	—	2	2	no	—
13	#14582640	—	2	4	no	—
14	C₂×#108×C₄₃	—	2	2	no	—
15	C₄₃×#423	C₂²	1	1	yes	characteristic
16	C₄₃×#423	C₂²	1	1	yes	characteristic
17	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₇₂	—	2	2	no	—
19	#13063325	D₄	1	2	yes	—
20	#13063325	D₄	1	2	yes	—
21	#13063317	—	2	2	no	—
22	#13063304	—	2	2	no	—
23	#13063304	—	2	2	no	—
24	C₄×Dic₈₆	—	2	2	no	—
25	C₄×Dic₈₆	—	2	2	no	—
26	#13063284	—	2	2	no	—
27	#13063257	—	2	2	no	—
28	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
29	#13063250	—	2	4	no	—
30	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
31	#13063236	—	2	2	no	—
32	C₂×M₁₆×C₄₃	—	2	1	no	—
33	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
34	C₂²×C₈×C₄₃	—	2	1	no	—
35	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	#108×C₄₃	—	4	4	no	—
37	C₄⋊C₈×C₄₃	—	2	2	no	—
38	C₄⋊C₈×C₄₃	—	2	2	no	—
39	#97×C₄₃	—	2	1	no	—
40	#97×C₄₃	D₄	1	2	yes	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	#41	1	2	yes	—
43	C₂×Dic₈₆	—	2	2	no	—
44	C₂×Dic₈₆	—	2	2	no	—
45	M₁₆×C₄₃	—	4	2	no	—
46	M₁₆×C₄₃	—	4	2	no	—
47	C₂×C₈×C₄₃	—	2	2	no	—
48	C₂×C₈×C₄₃	—	2	2	no	—
49	C₂×C₈×C₄₃	—	2	2	no	—
50	C₂×C₈×C₄₃	—	2	2	no	—
51	C₂×C₈×C₄₃	—	4	1	no	—
52	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₄₃	C₂×D₄	1	2	yes	—
54	C₄²×C₄₃	#41	1	1	yes	characteristic
55	C₂²⋊Dic₄₃	—	4	2	no	—
56	C₄⋊Dic₄₃	—	2	2	no	—
57	C₄⋊Dic₄₃	—	2	2	no	—
58	C₄⋊Dic₄₃	—	2	2	no	—
59	C₄⋊Dic₄₃	—	2	2	no	—
60	Dic₄₃⋊C₄	—	4	2	no	—
61	Dic₄₃⋊C₄	—	4	2	no	—
62	C₄×Dic₄₃	—	4	2	no	—
63	C₂×C₄₃⋊C₈	—	4	2	no	—
64	Dic₁₇₂	—	8	2	no	—
65	C₂³×C₄₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	—	4	2	no	—
69	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
70	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
71	C₂×C₄×C₄₃	#121	1	2	yes	—
72	C₂×C₄×C₄₃	#123	1	2	yes	—
73	C₂×Dic₄₃	—	4	2	no	—
74	C₂×Dic₄₃	—	4	2	no	—
75	Dic₈₆	—	4	2	no	—
76	Dic₈₆	—	4	2	no	—
77	Dic₈₆	—	8	2	no	—
78	C₈×C₄₃	—	2	4	no	—
79	C₈×C₄₃	—	4	2	no	—
80	C₈×C₄₃	—	8	1	no	—
81	C₄₃⋊C₈	—	8	2	no	—
82	#10528	—	43	1	no	maximal, Sylow
83	C₂²×C₄₃	—	2	2	no	—
84	C₂²×C₄₃	—	2	1	no	—
85	C₂²×C₄₃	#394	1	1	yes	characteristic
86	C₄×C₄₃	—	2	2	no	—
87	C₄×C₄₃	—	2	2	no	—
88	C₄×C₄₃	—	2	1	no	—
89	C₄×C₄₃	#453	1	1	yes	characteristic
90	C₄×C₄₃	#456	1	1	yes	characteristic
91	Dic₄₃	—	8	2	no	—
92	Dic₄₃	—	8	2	no	—
93	#3255	—	43	1	no	—
94	#1849	D₄₃	1	1	yes	characteristic
95	#1559	—	43	2	no	—
96	#1531	—	43	2	no	—
97	#1307	—	43	1	no	—
98	C₂×C₄₃	—	4	1	no	—
99	C₂×C₄₃	#2039	1	1	yes	characteristic
100	C₂×C₄₃	#2049	1	2	yes	—
101	#533	—	43	2	no	—
102	#475	—	86	2	no	—
103	#470	—	86	2	no	—
104	#462	—	86	2	no	—
105	#432	—	43	2	no	—
106	C₂×#108	—	2	2	no	—
107	#423	C₂×D₄₃	1	1	yes	characteristic
108	#423	C₂×D₄₃	1	1	yes	characteristic
109	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
110	#333	—	86	2	no	—
111	#332	—	86	2	no	—
112	#330	—	86	4	no	—
113	C₄₃	#10528	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₁₆	—	86	2	no	—
115	C₂×M₁₆	—	2	1	no	—
116	C₂×M₁₆	C₂²⋊D₄₃	1	2	yes	—
117	C₂²×C₈	—	2	1	no	—
118	C₄⋊Q₈	—	43	2	no	—
119	C₄⋊Q₈	—	43	2	no	—
120	#125	—	86	2	no	—
121	C₂²⋊Q₈	—	86	2	no	—
122	C₂²⋊Q₈	—	86	2	no	—
123	C₄×Q₈	—	86	2	no	—
124	C₄×Q₈	—	86	2	no	—
125	#117	—	86	2	no	—
126	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
127	#108	—	4	4	no	—
128	#106	—	86	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	86	2	no	—
132	#103	—	86	4	no	—
133	C₂²⋊C₈	—	86	2	no	—
134	#97	—	2	1	no	—
135	#97	—	86	2	no	—
136	#97	D₁₇₂	1	2	yes	—
137	C₂×Q₈	—	86	2	no	—
138	C₂×Q₈	—	86	2	no	—
139	C₂²×C₄	#10621902	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
141	Q₁₆	—	344	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	172	2	no	—
150	C₄⋊C₄	—	86	2	no	—
151	C₄⋊C₄	—	86	2	no	—
152	C₄⋊C₄	—	86	2	no	—
153	C₄⋊C₄	—	86	2	no	—
154	C₄⋊C₄	—	172	2	no	—
155	C₄⋊C₄	—	172	2	no	—
156	C₂²⋊C₄	—	172	2	no	—
157	C₄²	—	172	2	no	—
158	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
159	C₄²	#10621898	1	1	yes	characteristic
160	C₄²	D₄×D₄₃	1	2	yes	—
161	C₂³	#13063314	1	1	yes	characteristic
162	Q₈	—	344	2	no	—
163	Q₈	—	172	2	no	—
164	Q₈	—	172	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	172	2	no	—
169	C₂×C₄	—	172	2	no	—
170	C₂×C₄	#13063293	1	1	yes	characteristic
171	C₂×C₄	#13063299	1	1	yes	characteristic
172	C₂×C₄	#13063300	1	2	yes	—
173	C₂×C₄	#13063310	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	2	no	—
176	C₈	—	8	1	no	—
177	C₈	—	344	2	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	2	no	—
180	C₂²	#14582816	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	344	2	no	—
185	C₄	—	344	2	no	—
186	C₄	#14583066	1	1	yes	characteristic
187	C₄	#14583081	1	1	yes	characteristic
188	C₂	—	4	1	no	—
189	C₂	#19135819	1	1	yes	characteristic
190	C₂	#19135877	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145989

C₂

yes

maximal, characteristic

#19133323

C₂

yes

maximal

#19133209

C₂

yes

maximal

#19132506

C₂

yes

maximal, characteristic

C₄₃×#1849

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

—

—

—

C₂²

yes

—

#14582647

—

#14582646

—

#14582640

—

C₂×#108×C₄₃

—

C₄₃×#423

C₂²

yes

characteristic

C₄₃×#423

C₂²

yes

characteristic

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

#13063325

D₄

yes

—

#13063325

D₄

yes

—

#13063317

—

#13063304

—

#13063304

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

#13063284

—

#13063257

—

C₄⋊(C₄₃⋊C₈)

—

#13063250

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

D₄

yes

—

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#108×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

#97×C₄₃

—

#97×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

C₂³×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10528

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#394

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

#3255

—

#1849

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

#2039

yes

characteristic

100

C₂×C₄₃

#2049

yes

—

101

#533

—

102

#475

—

103

#470

—

104

#462

—

105

#432

—

106

C₂×#108

—

107

#423

C₂×D₄₃

yes

characteristic

108

#423

C₂×D₄₃

yes

characteristic

109

C₂×#97

C₂×D₄₃

yes

characteristic

110

#333

—

111

#332

—

112

#330

—

113

C₄₃

#10528

yes

characteristic, complemented, Sylow

114

C₂×Q₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂²⋊D₄₃

yes

—

117

C₂²×C₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

#125

—

121

C₂²⋊Q₈

—

122

C₂²⋊Q₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

#117

—

126

C₂×C₄²

C₂²×D₄₃

yes

characteristic

127

#108

—

128

#106

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

C₂²⋊C₈

—

134

#97

—

135

#97

—

136

#97

D₁₇₂

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

#10621902

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

141

Q₁₆

—

344

—

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

172

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

172

—

155

C₄⋊C₄

—

172

—

156

C₂²⋊C₄

—

172

—

157

C₄²

—

172

—

158

C₄²

C₂×D₁₇₂

yes

characteristic

159

C₄²

#10621898

yes

characteristic

160

C₄²

D₄×D₄₃

yes

—

161

C₂³

#13063314

yes

characteristic

162

Q₈

—

344

—

163

Q₈

—

172

—

164

Q₈

—

172

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

172

—

169

C₂×C₄

—

172

—

170

C₂×C₄

#13063293

yes

characteristic

171

C₂×C₄

#13063299

yes

characteristic

172

C₂×C₄

#13063300

yes

—

173

C₂×C₄

#13063310

yes

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

344

—

178

C₂²

—

179

C₂²

—

180

C₂²

#14582816

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

344

—

185

C₄

—

344

—

186

C₄

#14583066

yes

characteristic

187

C₄

#14583081

yes

characteristic

188

C₂

—

189

C₂

#19135819

yes

characteristic

190

C₂

#19135877

yes

—

	subgroup	G/H
center	C₂²	#14582816
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1849	C₂
socle	C₂²×C₄₃	#394
prime radical	C₂³×C₄₃	#124
hypercenter	#1849	D₄₃
hypocenter	C₄₃	#10528
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂²

#14582816

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1849

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
113. C₄₃	#10528	semidirect

H ≅ G/N

type

113. C₄₃

#10528

semidirect

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

subgroup	index
#19145989	2
#19133323	2
#19133209	2
#19132506	2
C₄₃×#1849	2
#10528	43

subgroup

index

C₄₃×#1849

Subgroups — #25000759

Conjugacy classes of subgroups

Complements

Maximal subgroups