Group #25000765 — subgroups

Subgroups — #25000765

Order 11008.

326 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19134735	C₂	1	1	yes	maximal, characteristic
3	#19133326	C₂	1	2	yes	maximal
4	#19133211	C₂	1	2	yes	maximal
5	C₄₃×#1307	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	#14582923	C₂²	1	1	yes	characteristic
11	C₂×#13063236	C₂²	1	1	yes	characteristic
12	#14582663	—	2	2	no	—
13	#14582647	—	2	2	no	—
14	#14582644	—	2	2	no	—
15	#14582640	—	2	4	no	—
16	C₄₃×#432	C₂²	1	2	yes	—
17	C₄₃×#423	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×#10621877	—	2	1	no	—
20	C₂²×C₄₃⋊C₈	—	2	1	no	—
21	#13063325	D₄	1	2	yes	—
22	#13063325	D₄	1	2	yes	—
23	#13063317	—	2	2	no	—
24	#13063304	—	2	2	no	—
25	#13063304	—	2	2	no	—
26	C₄×Dic₈₆	—	2	2	no	—
27	C₄×Dic₈₆	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	#13063257	—	2	2	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	#13063250	—	2	4	no	—
32	#13063236	—	2	1	no	—
33	#13063236	D₄	1	2	yes	—
34	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
35	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₃	—	2	2	no	—
37	C₄⋊C₈×C₄₃	—	2	2	no	—
38	C₂²⋊C₈×C₄₃	—	2	2	no	—
39	#97×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	#41	1	2	yes	—
42	C₂×Dic₈₆	—	2	2	no	—
43	C₂×Dic₈₆	—	2	2	no	—
44	M₁₆×C₄₃	—	4	2	no	—
45	C₂×C₈×C₄₃	—	2	2	no	—
46	C₂×C₈×C₄₃	—	4	2	no	—
47	C₄²×C₄₃	C₂×D₄	1	2	yes	—
48	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₄₃	#41	1	1	yes	characteristic
50	C₂²⋊Dic₄₃	—	4	2	no	—
51	C₄⋊Dic₄₃	—	2	2	no	—
52	C₄⋊Dic₄₃	—	2	2	no	—
53	C₄⋊Dic₄₃	—	2	2	no	—
54	C₄⋊Dic₄₃	—	2	2	no	—
55	Dic₄₃⋊C₄	—	4	2	no	—
56	Dic₄₃⋊C₄	—	4	2	no	—
57	C₄×Dic₄₃	—	4	2	no	—
58	#10621877	—	4	2	no	—
59	C₂×C₄₃⋊C₈	—	2	2	no	—
60	C₂×C₄₃⋊C₈	—	4	1	no	—
61	C₂×C₄₃⋊C₈	—	4	1	no	—
62	C₂×C₄₃⋊C₈	—	4	1	no	—
63	Dic₁₇₂	—	8	2	no	—
64	C₂³×C₄₃	#124	1	1	yes	characteristic
65	C₂×C₄×C₄₃	—	2	1	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	4	2	no	—
68	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#121	1	2	yes	—
70	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
71	C₂×C₄×C₄₃	#123	1	2	yes	—
72	C₂×Dic₄₃	—	4	2	no	—
73	C₂×Dic₄₃	—	4	2	no	—
74	Dic₈₆	—	4	2	no	—
75	Dic₈₆	—	4	2	no	—
76	Dic₈₆	—	8	2	no	—
77	C₈×C₄₃	—	4	2	no	—
78	C₈×C₄₃	—	8	2	no	—
79	C₄₃⋊C₈	—	4	2	no	—
80	C₄₃⋊C₈	—	8	1	no	—
81	#10529	—	43	1	no	maximal, Sylow
82	C₂²×C₄₃	—	2	2	no	—
83	C₂²×C₄₃	—	2	1	no	—
84	C₂²×C₄₃	#394	1	1	yes	characteristic
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	—	2	2	no	—
87	C₄×C₄₃	—	2	1	no	—
88	C₄×C₄₃	#454	1	1	yes	characteristic
89	C₄×C₄₃	#455	1	1	yes	characteristic
90	Dic₄₃	—	8	2	no	—
91	Dic₄₃	—	8	2	no	—
92	#3255	—	43	1	no	—
93	#1838	—	43	1	no	—
94	#1559	—	43	2	no	—
95	#1531	—	43	2	no	—
96	#1307	D₄₃	1	1	yes	characteristic
97	C₂×C₄₃	—	4	1	no	—
98	C₂×C₄₃	#2040	1	1	yes	characteristic
99	C₂×C₄₃	#2049	1	2	yes	—
100	#533	—	43	2	no	—
101	#475	—	86	2	no	—
102	#470	—	86	2	no	—
103	#462	—	86	2	no	—
104	#432	C₂×D₄₃	1	2	yes	—
105	#423	—	43	1	no	—
106	#423	C₂×D₄₃	1	1	yes	characteristic
107	C₂×#97	—	43	1	no	—
108	#341	—	86	2	no	—
109	#333	—	86	2	no	—
110	#332	—	86	2	no	—
111	#330	—	86	4	no	—
112	C₄₃	#10529	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₁₆	—	86	2	no	—
114	C₂×M₁₆	—	86	1	no	—
115	C₂×M₁₆	D₁₇₂	1	2	yes	—
116	C₂²×C₈	—	86	1	no	—
117	C₄⋊Q₈	—	43	2	no	—
118	C₄⋊Q₈	—	43	2	no	—
119	#125	—	86	2	no	—
120	C₂²⋊Q₈	—	86	2	no	—
121	C₂²⋊Q₈	—	86	2	no	—
122	C₄×Q₈	—	86	2	no	—
123	C₄×Q₈	—	86	2	no	—
124	#117	—	86	2	no	—
125	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
126	#106	—	86	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	86	2	no	—
130	#103	—	86	4	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	#97	—	2	2	no	—
133	#97	—	43	2	no	—
134	#97	—	86	1	no	—
135	C₂×Q₈	—	86	2	no	—
136	C₂×Q₈	—	86	2	no	—
137	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
138	C₂²×C₄	#10621898	1	2	yes	—
139	Q₁₆	—	344	2	no	—
140	M₁₆	—	4	2	no	—
141	M₁₆	—	172	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	86	2	no	—
145	C₂×C₈	—	172	1	no	—
146	C₂×C₈	—	172	1	no	—
147	C₂×C₈	—	172	1	no	—
148	C₄⋊C₄	—	86	2	no	—
149	C₄⋊C₄	—	86	2	no	—
150	C₄⋊C₄	—	86	2	no	—
151	C₄⋊C₄	—	86	2	no	—
152	C₄⋊C₄	—	172	2	no	—
153	C₄⋊C₄	—	172	2	no	—
154	C₂²⋊C₄	—	172	2	no	—
155	C₄²	—	172	2	no	—
156	C₄²	D₄×D₄₃	1	2	yes	—
157	C₄²	D₄×D₄₃	1	1	yes	characteristic
158	C₄²	#10621902	1	1	yes	characteristic
159	C₂³	#13063313	1	1	yes	characteristic
160	Q₈	—	344	2	no	—
161	Q₈	—	172	2	no	—
162	Q₈	—	172	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	172	2	no	—
167	C₂×C₄	—	172	2	no	—
168	C₂×C₄	#13063295	1	1	yes	characteristic
169	C₂×C₄	#13063298	1	2	yes	—
170	C₂×C₄	#13063300	1	1	yes	characteristic
171	C₂×C₄	#13063310	1	2	yes	—
172	C₈	—	4	2	no	—
173	C₈	—	8	2	no	—
174	C₈	—	344	1	no	—
175	C₈	—	172	2	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	2	no	—
178	C₂²	#14582817	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	344	2	no	—
183	C₄	—	344	2	no	—
184	C₄	#14583074	1	1	yes	characteristic
185	C₄	#14583079	1	1	yes	characteristic
186	C₂	—	4	1	no	—
187	C₂	#19135825	1	1	yes	characteristic
188	C₂	#19135878	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145989

C₂

yes

maximal, characteristic

#19134735

C₂

yes

maximal, characteristic

#19133326

C₂

yes

maximal

#19133211

C₂

yes

maximal

C₄₃×#1307

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#13063236

C₂²

yes

—

—

—

—

C₄₃×#432

C₂²

yes

—

C₄₃×#423

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×#10621877

—

C₂²×C₄₃⋊C₈

—

#13063325

D₄

yes

—

#13063325

D₄

yes

—

#13063317

—

#13063304

—

#13063304

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

#13063284

—

#13063257

—

C₄⋊(C₄₃⋊C₈)

—

#13063250

—

#13063236

—

#13063236

D₄

yes

—

C₂×M₁₆×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

C₂³×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

#123

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10529

—

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

—

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

#2040

yes

characteristic

C₂×C₄₃

#2049

yes

—

100

#533

—

101

#475

—

102

#470

—

103

#462

—

104

#432

C₂×D₄₃

yes

—

105

#423

—

106

#423

C₂×D₄₃

yes

characteristic

107

C₂×#97

—

108

#341

—

109

#333

—

110

#332

—

111

#330

—

112

C₄₃

#10529

yes

characteristic, complemented, Sylow

113

C₂×Q₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

D₁₇₂

yes

—

116

C₂²×C₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

#125

—

120

C₂²⋊Q₈

—

121

C₂²⋊Q₈

—

122

C₄×Q₈

—

123

C₄×Q₈

—

124

#117

—

125

C₂×C₄²

C₂²×D₄₃

yes

characteristic

126

#106

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

#103

—

131

C₂²⋊C₈

—

132

#97

—

133

#97

—

134

#97

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

138

C₂²×C₄

#10621898

yes

—

139

Q₁₆

—

344

—

140

M₁₆

—

141

M₁₆

—

172

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

172

—

146

C₂×C₈

—

172

—

147

C₂×C₈

—

172

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

172

—

153

C₄⋊C₄

—

172

—

154

C₂²⋊C₄

—

172

—

155

C₄²

—

172

—

156

C₄²

D₄×D₄₃

yes

—

157

C₄²

D₄×D₄₃

yes

characteristic

158

C₄²

#10621902

yes

characteristic

159

C₂³

#13063313

yes

characteristic

160

Q₈

—

344

—

161

Q₈

—

172

—

162

Q₈

—

172

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

172

—

167

C₂×C₄

—

172

—

168

C₂×C₄

#13063295

yes

characteristic

169

C₂×C₄

#13063298

yes

—

170

C₂×C₄

#13063300

yes

characteristic

171

C₂×C₄

#13063310

yes

—

172

C₈

—

173

C₈

—

174

C₈

—

344

—

175

C₈

—

172

—

176

C₂²

—

177

C₂²

—

178

C₂²

#14582817

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

344

—

183

C₄

—

344

—

184

C₄

#14583074

yes

characteristic

185

C₄

#14583079

yes

characteristic

186

C₂

—

187

C₂

#19135825

yes

characteristic

188

C₂

#19135878

yes

—

	subgroup	G/H
center	C₂²	#14582817
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1307	C₂
socle	C₂²×C₄₃	#394
prime radical	C₂³×C₄₃	#124
hypercenter	#1307	D₄₃
hypocenter	C₄₃	#10529
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#14582817

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1307

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
112. C₄₃	#10529	semidirect

H ≅ G/N

type

112. C₄₃

#10529

semidirect

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

subgroup	index
#19145989	2
#19134735	2
#19133326	2
#19133211	2
C₄₃×#1307	2
#10529	43

subgroup

index

C₄₃×#1307

Subgroups — #25000765

Conjugacy classes of subgroups

Complements

Maximal subgroups