Group #24999716 — subgroups

Subgroups — #24999716

Order 11008.

402 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19142982	C₂	1	1	yes	maximal, characteristic, complemented
2	#19135423	C₂	1	1	yes	maximal, characteristic
3	#19133156	C₂	1	2	yes	maximal
4	#19132454	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583449	C₂²	1	1	yes	characteristic
7	#14582923	C₂²	1	1	yes	characteristic
8	C₂×#13063253	—	2	2	no	—
9	#14582797	—	2	2	no	—
10	#14582783	—	2	4	no	—
11	#14582765	C₂²	1	2	yes	—
12	#14582634	—	2	4	no	—
13	C₄₃×#431	C₂²	1	2	yes	—
14	C₄³×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#10621898	—	2	1	no	—
16	C₂×#10621877	—	2	1	no	—
17	#13063325	Q₈	1	1	yes	characteristic
18	#13063322	Q₈	1	1	yes	characteristic
19	#13063313	—	2	1	no	—
20	#13063299	—	2	2	no	—
21	C₄×D₁₇₂	—	2	2	no	—
22	C₂×C₄⋊Dic₄₃	—	2	2	no	—
23	C₂×Dic₄₃⋊C₄	—	2	4	no	—
24	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
25	#13063253	—	4	4	no	—
26	C₄₃⋊#95	—	2	4	no	—
27	C₄₃⋊#95	—	2	2	no	—
28	C₄₃⋊#95	D₄	1	4	yes	—
29	C₂×C₄²×C₄₃	—	2	2	no	—
30	C₂×C₄²×C₄₃	D₄	1	2	yes	—
31	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₃	—	2	2	no	—
33	C₂²⋊C₈×C₄₃	—	2	2	no	—
34	C₄×C₈×C₄₃	—	2	2	no	—
35	C₂²×C₄×C₄₃	—	2	2	no	—
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
38	C₂×C₂²⋊D₄₃	—	4	1	no	—
39	C₂²×Dic₄₃	—	2	4	no	—
40	#10621898	—	4	2	no	—
41	#10621898	—	4	2	no	—
42	C₂×D₁₇₂	—	2	1	no	—
43	C₂×D₁₇₂	—	2	2	no	—
44	C₂×C₄×D₄₃	—	4	1	no	—
45	C₂×Dic₈₆	—	2	1	no	—
46	C₂×C₈×C₄₃	—	4	2	no	—
47	C₄²×C₄₃	—	2	2	no	—
48	C₄²×C₄₃	—	2	2	no	—
49	C₄²×C₄₃	—	4	4	no	—
50	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₄₃	#41	1	2	yes	—
52	C₄²×C₄₃	#41	1	1	yes	characteristic
53	(C₂×D₄₃)⋊C₄	—	4	2	no	—
54	C₄⋊Dic₄₃	—	2	2	no	—
55	C₄⋊Dic₄₃	—	4	2	no	—
56	C₄⋊Dic₄₃	—	4	2	no	—
57	Dic₄₃⋊C₄	—	4	8	no	—
58	#10621877	—	4	2	no	—
59	C₂×C₄₃⋊C₈	—	4	1	no	—
60	C₂³×C₄₃	#120	1	1	yes	characteristic
61	C₂²×D₄₃	—	4	1	no	—
62	C₂×C₄×C₄₃	—	2	4	no	—
63	C₂×C₄×C₄₃	—	4	2	no	—
64	C₂×C₄×C₄₃	—	4	2	no	—
65	C₂×C₄×C₄₃	—	4	2	no	—
66	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#124	1	2	yes	—
69	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
70	C₂×C₄×C₄₃	#135	1	2	yes	—
71	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²⋊D₄₃	—	8	2	no	—
73	C₂×Dic₄₃	—	4	8	no	—
74	C₂×Dic₄₃	—	4	1	no	—
75	C₂×Dic₄₃	—	4	4	no	—
76	D₁₇₂	—	4	2	no	—
77	D₁₇₂	—	8	2	no	—
78	C₄×D₄₃	—	8	1	no	—
79	C₄×D₄₃	—	8	1	no	—
80	Dic₈₆	—	4	2	no	—
81	C₈×C₄₃	—	8	2	no	—
82	C₄₃⋊C₈	—	8	1	no	—
83	#10368	—	43	1	no	maximal, Sylow
84	C₂²×C₄₃	—	2	2	no	—
85	C₂²×C₄₃	#393	1	1	yes	characteristic
86	C₂²×C₄₃	#449	1	2	yes	—
87	C₂×D₄₃	—	8	1	no	—
88	C₂×D₄₃	—	8	2	no	—
89	C₄×C₄₃	—	2	2	no	—
90	C₄×C₄₃	—	2	2	no	—
91	C₄×C₄₃	—	4	4	no	—
92	C₄×C₄₃	#478	1	2	yes	—
93	C₄×C₄₃	#488	1	1	yes	characteristic
94	C₄×C₄₃	#490	1	1	yes	characteristic
95	Dic₄₃	—	8	4	no	—
96	Dic₄₃	—	8	1	no	—
97	#2864	—	43	1	no	—
98	#1962	—	43	1	no	—
99	#1517	—	43	2	no	—
100	#1286	—	43	2	no	—
101	#1276	D₄₃	1	1	yes	characteristic, complemented
102	C₂×C₄₃	—	2	2	no	—
103	C₂×C₄₃	#2025	1	1	yes	characteristic
104	C₂×C₄₃	#2031	1	2	yes	—
105	D₄₃	—	16	1	no	—
106	#532	—	43	1	no	—
107	#431	C₂×D₄₃	1	2	yes	—
108	#423	—	43	1	no	—
109	C₂×#104	—	86	2	no	—
110	#389	—	86	2	no	—
111	#387	—	86	4	no	—
112	#382	—	43	2	no	—
113	C₄³	C₂×D₄₃	1	1	yes	characteristic
114	#327	—	86	4	no	—
115	C₄₃	#10368	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	—	86	1	no	—
117	C₂×M₁₆	—	86	1	no	—
118	C₄⋊Q₈	—	43	1	no	—
119	#127	—	43	1	no	—
120	#124	—	86	1	no	—
121	#121	—	86	2	no	—
122	C₄×D₄	—	86	2	no	—
123	C₂×C₄⋊C₄	—	86	4	no	—
124	C₂×C₄⋊C₄	—	86	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	D₁₇₂	1	2	yes	—
127	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	86	2	no	—
130	#104	—	172	4	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	#95	—	43	4	no	—
134	#95	—	86	4	no	—
135	#95	—	86	2	no	—
136	#41	—	172	2	no	—
137	#41	—	172	2	no	—
138	C₂×Q₈	—	86	1	no	—
139	C₂×D₄	—	86	2	no	—
140	C₂×D₄	—	86	1	no	—
141	C₂×D₄	—	172	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	86	4	no	—
144	C₂²×C₄	—	172	1	no	—
145	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
146	C₂²×C₄	D₄×D₄₃	1	2	yes	—
147	M₁₆	—	172	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	172	1	no	—
150	C₄⋊C₄	—	86	2	no	—
151	C₄⋊C₄	—	172	8	no	—
152	C₄⋊C₄	—	172	2	no	—
153	C₄⋊C₄	—	172	2	no	—
154	C₂²⋊C₄	—	172	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	4	no	—
158	C₄²	#10621898	1	2	yes	—
159	C₄²	#10621900	1	1	yes	characteristic
160	C₄²	Q₈×D₄₃	1	1	yes	characteristic
161	C₂³	—	172	1	no	—
162	C₂³	#13063295	1	1	yes	characteristic
163	Q₈	—	172	2	no	—
164	D₄	—	344	2	no	—
165	D₄	—	344	2	no	—
166	D₄	—	172	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	344	1	no	—
172	C₂×C₄	—	344	1	no	—
173	C₂×C₄	—	172	8	no	—
174	C₂×C₄	—	172	4	no	—
175	C₂×C₄	—	172	1	no	—
176	C₂×C₄	#13063303	1	2	yes	—
177	C₂×C₄	#13063305	1	1	yes	characteristic
178	C₂×C₄	#13063311	1	2	yes	—
179	C₂×C₄	#13063313	1	1	yes	characteristic
180	C₂×C₄	#13063347	1	2	yes	—
181	C₂×C₄	#13063353	1	2	yes	—
182	C₈	—	8	2	no	—
183	C₈	—	344	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	344	2	no	—
186	C₂²	—	344	1	no	—
187	C₂²	#14582812	1	1	yes	characteristic
188	C₂²	#14583046	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	344	4	no	—
193	C₄	—	344	1	no	—
194	C₄	#14583200	1	2	yes	—
195	C₄	#14583254	1	1	yes	characteristic
196	C₄	#14583264	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	688	1	no	—
199	C₂	#19135742	1	1	yes	characteristic
200	C₂	#19135768	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19142982

C₂

yes

maximal, characteristic, complemented

#19135423

C₂

yes

maximal, characteristic

#19133156

C₂

yes

maximal

#19132454

C₂

yes

maximal, complemented

C₄₃×#1276

C₂

yes

maximal, characteristic, complemented

#14583449

C₂²

yes

characteristic

#14582923

C₂²

yes

—

—

—

C₂²

yes

—

#14582634

—

C₄₃×#431

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

—

—

Q₈

yes

characteristic

#13063322

Q₈

yes

characteristic

#13063313

—

#13063299

—

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

—

C₂×Dic₄₃⋊C₄

—

C₄⋊(C₄₃⋊C₈)

—

#13063253

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₄₃⋊#95

D₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₂²⋊D₄₃

—

C₂²×Dic₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄²×C₄₃

#41

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#124

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10368

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#393

yes

characteristic

C₂²×C₄₃

#449

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#478

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

#2864

—

#1962

—

#1517

—

100

#1286

—

101

#1276

D₄₃

yes

characteristic, complemented

102

C₂×C₄₃

—

103

C₂×C₄₃

#2025

yes

characteristic

104

C₂×C₄₃

#2031

yes

—

105

D₄₃

—

106

#532

—

107

#431

C₂×D₄₃

yes

—

108

#423

—

109

C₂×#104

—

110

#389

—

111

#387

—

112

#382

—

113

C₄³

C₂×D₄₃

yes

characteristic

114

#327

—

115

C₄₃

#10368

yes

characteristic, complemented, Sylow

116

C₂×#41

—

117

C₂×M₁₆

—

118

C₄⋊Q₈

—

119

#127

—

120

#124

—

121

#121

—

122

C₄×D₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

—

126

C₂×C₄²

D₁₇₂

yes

—

127

C₂×C₄²

C₂²×D₄₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

#104

—

172

—

131

C₂²⋊C₈

—

132

C₄×C₈

—

133

#95

—

134

#95

—

135

#95

—

136

#41

—

172

—

137

#41

—

172

—

138

C₂×Q₈

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

172

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

172

—

145

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

146

C₂²×C₄

D₄×D₄₃

yes

—

147

M₁₆

—

172

—

148

C₂×C₈

—

149

C₂×C₈

—

172

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

172

—

152

C₄⋊C₄

—

172

—

153

C₄⋊C₄

—

172

—

154

C₂²⋊C₄

—

172

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#10621898

yes

—

159

C₄²

#10621900

yes

characteristic

160

C₄²

Q₈×D₄₃

yes

characteristic

161

C₂³

—

172

—

162

C₂³

#13063295

yes

characteristic

163

Q₈

—

172

—

164

D₄

—

344

—

165

D₄

—

344

—

166

D₄

—

172

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

344

—

172

C₂×C₄

—

344

—

173

C₂×C₄

—

172

—

174

C₂×C₄

—

172

—

175

C₂×C₄

—

172

—

176

C₂×C₄

#13063303

yes

—

177

C₂×C₄

#13063305

yes

characteristic

178

C₂×C₄

#13063311

yes

—

179

C₂×C₄

#13063313

yes

characteristic

180

C₂×C₄

#13063347

yes

—

181

C₂×C₄

#13063353

yes

—

182

C₈

—

183

C₈

—

344

—

184

C₂²

—

185

C₂²

—

344

—

186

C₂²

—

344

—

187

C₂²

#14582812

yes

characteristic

188

C₂²

#14583046

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

344

—

193

C₄

—

344

—

194

C₄

#14583200

yes

—

195

C₄

#14583254

yes

characteristic

196

C₄

#14583264

yes

characteristic

197

C₂

—

198

C₂

—

688

—

199

C₂

#19135742

yes

characteristic

200

C₂

#19135768

yes

—

	subgroup	G/H
center	C₂²	#14582812
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1276	C₂
socle	C₂²×C₄₃	#393
prime radical	#14583449	C₂²
hypercenter	#1276	D₄₃
hypocenter	C₄₃	#10368
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#14582812

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1276

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
1. #19142982	C₂	semidirect
4. #19132454	C₂	semidirect
5. C₄₃×#1276	C₂	semidirect
101. #1276	D₄₃	semidirect
115. C₄₃	#10368	semidirect

H ≅ G/N

type

1. #19142982

C₂

semidirect

4. #19132454

C₂

semidirect

5. C₄₃×#1276

C₂

semidirect

101. #1276

D₄₃

semidirect

115. C₄₃

#10368

semidirect

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

subgroup	index
#19142982	2
#19135423	2
#19133156	2
#19132454	2
C₄₃×#1276	2
#10368	43

subgroup

index

C₄₃×#1276

Subgroups — #24999716

Conjugacy classes of subgroups

Complements

Maximal subgroups