Group #19136115 — subgroups

Subgroups — #19136115

Order 5504.

274 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₂×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063257	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063247	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063247	C₂	1	1	yes	maximal, characteristic
5	#14582801	C₂	1	1	yes	maximal, characteristic
6	#14582765	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
13	#13063259	—	2	2	no	—
14	#13063257	—	2	2	no	—
15	#13063247	—	2	2	no	—
16	#13063247	—	2	2	no	—
17	C₄₃⋊#95	—	2	2	no	—
18	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₂³×D₄₃	—	2	1	no	—
22	C₂²×C₄×C₄₃	D₄	1	2	yes	—
23	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₃	—	2	1	no	—
25	C₂²×Dic₄₃	—	2	1	no	—
26	C₂²×Dic₄₃	—	2	1	no	—
27	C₂×D₁₇₂	—	2	1	no	—
28	C₂×D₁₇₂	—	2	1	no	—
29	C₂×D₁₇₂	—	2	1	no	—
30	C₂×D₁₇₂	—	2	1	no	—
31	C₂×D₁₇₂	—	2	1	no	—
32	C₂×D₁₇₂	Q₈	1	2	yes	—
33	C₂×C₈×C₄₃	—	2	1	no	—
34	C₂×C₈×C₄₃	—	2	1	no	—
35	C₂×C₈×C₄₃	—	2	1	no	—
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	D₄	1	2	yes	—
38	C₂×C₈×C₄₃	D₄	1	2	yes	—
39	C₄²×C₄₃	—	2	1	no	—
40	C₄²×C₄₃	—	2	1	no	—
41	(C₂×D₄₃)⋊C₄	—	4	4	no	—
42	C₄⋊Dic₄₃	—	2	1	no	—
43	C₄⋊Dic₄₃	—	2	1	no	—
44	C₄⋊Dic₄₃	—	2	1	no	—
45	C₄⋊Dic₄₃	—	2	1	no	—
46	C₄⋊Dic₄₃	—	2	1	no	—
47	C₄⋊Dic₄₃	—	2	1	no	—
48	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₄₃	—	2	2	no	—
50	C₂²×D₄₃	—	4	1	no	—
51	C₂²×D₄₃	—	4	1	no	—
52	C₂²×D₄₃	—	4	2	no	—
53	C₂²×D₄₃	—	4	2	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂×Dic₄₃	—	4	1	no	—
64	C₂×Dic₄₃	—	4	1	no	—
65	C₂×Dic₄₃	—	4	1	no	—
66	C₂×Dic₄₃	—	4	2	no	—
67	C₂×Dic₄₃	—	4	2	no	—
68	C₂×Dic₄₃	—	4	2	no	—
69	D₁₇₂	—	2	2	no	—
70	D₁₇₂	—	2	2	no	—
71	D₁₇₂	—	4	1	no	—
72	D₁₇₂	—	4	1	no	—
73	C₈×C₄₃	—	2	2	no	—
74	C₈×C₄₃	—	2	2	no	—
75	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₄₃	#123	1	1	yes	characteristic
77	C₂²×C₄₃	#127	1	1	yes	characteristic
78	C₂²×C₄₃	#135	1	2	yes	—
79	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×D₄₃	—	4	2	no	—
81	C₂×D₄₃	—	4	4	no	—
82	C₂×D₄₃	—	8	2	no	—
83	C₂×D₄₃	—	8	1	no	—
84	C₂×D₄₃	—	8	1	no	—
85	C₄×C₄₃	—	4	2	no	—
86	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₄₃	#123	1	1	yes	characteristic
89	C₄×C₄₃	#123	1	1	yes	characteristic
90	Dic₄₃	—	8	1	no	—
91	Dic₄₃	—	8	1	no	—
92	Dic₄₃	—	8	1	no	—
93	#2092	—	43	1	no	maximal, Sylow
94	C₂×C₄₃	#399	1	1	yes	characteristic
95	C₂×C₄₃	#478	1	2	yes	—
96	C₂×C₄₃	#482	1	2	yes	—
97	C₂×C₄₃	#495	1	1	yes	characteristic
98	C₂×C₄₃	#496	1	1	yes	characteristic
99	D₄₃	—	8	2	no	—
100	C₂×#107	—	43	1	no	—
101	C₂×#106	—	43	1	no	—
102	C₂×#102	—	43	1	no	—
103	C₂×#102	—	43	1	no	—
104	#390	—	43	1	no	—
105	#382	—	43	1	no	—
106	#336	D₄₃	1	1	yes	characteristic, complemented
107	C₄₃	#2092	1	1	yes	characteristic, complemented, Sylow
108	C₂²×D₄	—	43	1	no	—
109	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	43	1	no	—
112	C₂×C₄⋊C₄	—	43	1	no	—
113	C₂×C₄⋊C₄	—	43	1	no	—
114	C₂×C₂²⋊C₄	—	86	2	no	—
115	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
116	#107	—	86	2	no	—
117	#106	—	86	2	no	—
118	#102	—	86	2	no	—
119	#102	—	86	2	no	—
120	#95	—	86	2	no	—
121	C₂⁴	—	86	1	no	—
122	C₂×D₄	—	43	2	no	—
123	C₂×D₄	—	86	1	no	—
124	C₂×D₄	—	86	1	no	—
125	C₂×D₄	—	86	1	no	—
126	C₂×D₄	—	86	1	no	—
127	C₂×D₄	—	86	1	no	—
128	C₂²×C₄	—	86	1	no	—
129	C₂²×C₄	—	86	1	no	—
130	C₂²×C₄	—	86	1	no	—
131	C₂²×C₄	D₁₇₂	1	2	yes	—
132	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	D₁₇₂	1	2	yes	—
138	C₂×C₈	D₁₇₂	1	2	yes	—
139	C₄⋊C₄	—	86	1	no	—
140	C₄⋊C₄	—	86	1	no	—
141	C₄⋊C₄	—	86	1	no	—
142	C₄⋊C₄	—	86	1	no	—
143	C₄⋊C₄	—	86	1	no	—
144	C₄⋊C₄	—	86	1	no	—
145	C₂²⋊C₄	—	172	4	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₂³	—	86	2	no	—
149	C₂³	—	172	1	no	—
150	C₂³	—	172	1	no	—
151	C₂³	—	172	2	no	—
152	C₂³	—	172	2	no	—
153	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
154	D₄	—	86	2	no	—
155	D₄	—	86	2	no	—
156	D₄	—	172	1	no	—
157	D₄	—	172	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	172	1	no	—
162	C₂×C₄	—	172	1	no	—
163	C₂×C₄	—	172	1	no	—
164	C₂×C₄	—	172	2	no	—
165	C₂×C₄	—	172	2	no	—
166	C₂×C₄	—	172	2	no	—
167	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
168	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
169	C₂×C₄	#10621900	1	1	yes	characteristic
170	C₂×C₄	#10621900	1	1	yes	characteristic
171	C₂×C₄	#10621900	1	1	yes	characteristic
172	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₂²	—	344	2	no	—
176	C₂²	—	344	1	no	—
177	C₂²	—	344	1	no	—
178	C₂²	—	172	2	no	—
179	C₂²	—	172	4	no	—
180	C₂²	#13063305	1	1	yes	characteristic
181	C₂²	#13063309	1	1	yes	characteristic
182	C₂²	#13063322	1	1	yes	characteristic
183	C₂²	#13063347	1	2	yes	—
184	C₂²	#13063353	1	2	yes	—
185	C₄	—	4	2	no	—
186	C₄	—	344	1	no	—
187	C₄	—	344	1	no	—
188	C₄	—	344	1	no	—
189	C₄	#13063305	1	1	yes	characteristic
190	C₄	#13063305	1	1	yes	characteristic
191	C₄	#13063309	1	1	yes	characteristic
192	C₄	#13063309	1	1	yes	characteristic
193	C₂	—	344	2	no	—
194	C₂	#14582836	1	1	yes	characteristic
195	C₂	#14583200	1	2	yes	—
196	C₂	#14583224	1	2	yes	—
197	C₂	#14583283	1	1	yes	characteristic
198	C₂	#14583288	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063259

C₂

yes

maximal, characteristic, complemented

C₂×#13063257

C₂

yes

maximal, characteristic, complemented

C₂×#13063247

C₂

yes

maximal, characteristic

C₂×#13063247

C₂

yes

maximal, characteristic

#14582801

C₂

yes

maximal, characteristic

#14582765

C₂

yes

maximal, characteristic, complemented

C₄₃×#336

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

—

—

—

—

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

Q₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

D₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₈×C₄₃

—

C₂²×C₄₃

C₂²⋊Q₈

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#482

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

100

C₂×#107

—

101

C₂×#106

—

102

C₂×#102

—

103

C₂×#102

—

104

#390

—

105

#382

—

106

#336

D₄₃

yes

characteristic, complemented

107

C₄₃

#2092

yes

characteristic, complemented, Sylow

108

C₂²×D₄

—

109

C₂²×C₈

C₂×D₄₃

yes

characteristic

110

C₂²×C₈

C₂×D₄₃

yes

characteristic

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₂²⋊C₄

—

115

C₂×C₄²

C₂×D₄₃

yes

characteristic

116

#107

—

117

#106

—

118

#102

—

119

#102

—

120

#95

—

121

C₂⁴

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

D₁₇₂

yes

—

132

C₂²×C₄

C₂²×D₄₃

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

D₁₇₂

yes

—

138

C₂×C₈

D₁₇₂

yes

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₂²⋊C₄

—

172

—

146

C₄²

—

147

C₄²

—

148

C₂³

—

149

C₂³

—

172

—

150

C₂³

—

172

—

151

C₂³

—

172

—

152

C₂³

—

172

—

153

C₂³

C₂×D₁₇₂

yes

characteristic

154

D₄

—

155

D₄

—

156

D₄

—

172

—

157

D₄

—

172

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

172

—

162

C₂×C₄

—

172

—

163

C₂×C₄

—

172

—

164

C₂×C₄

—

172

—

165

C₂×C₄

—

172

—

166

C₂×C₄

—

172

—

167

C₂×C₄

C₂×D₁₇₂

yes

characteristic

168

C₂×C₄

C₂×D₁₇₂

yes

characteristic

169

C₂×C₄

#10621900

yes

characteristic

170

C₂×C₄

#10621900

yes

characteristic

171

C₂×C₄

#10621900

yes

characteristic

172

C₂×C₄

Q₈×D₄₃

yes

characteristic

173

C₈

—

174

C₈

—

175

C₂²

—

344

—

176

C₂²

—

344

—

177

C₂²

—

344

—

178

C₂²

—

172

—

179

C₂²

—

172

—

180

C₂²

#13063305

yes

characteristic

181

C₂²

#13063309

yes

characteristic

182

C₂²

#13063322

yes

characteristic

183

C₂²

#13063347

yes

—

184

C₂²

#13063353

yes

—

185

C₄

—

186

C₄

—

344

—

187

C₄

—

344

—

188

C₄

—

344

—

189

C₄

#13063305

yes

characteristic

190

C₄

#13063305

yes

characteristic

191

C₄

#13063309

yes

characteristic

192

C₄

#13063309

yes

characteristic

193

C₂

—

344

—

194

C₂

#14582836

yes

characteristic

195

C₂

#14583200

yes

—

196

C₂

#14583224

yes

—

197

C₂

#14583283

yes

characteristic

198

C₂

#14583288

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#336	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#336	D₄₃
hypocenter	C₄₃	#2092
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#336

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#13063259	C₂	semidirect
2. C₂×#13063257	C₂	semidirect
6. #14582765	C₂	semidirect
7. C₄₃×#336	C₂	semidirect
106. #336	D₄₃	semidirect
107. C₄₃	#2092	semidirect

H ≅ G/N

type

1. C₂×#13063259

C₂

semidirect

2. C₂×#13063257

C₂

semidirect

6. #14582765

C₂

semidirect

7. C₄₃×#336

C₂

semidirect

106. #336

D₄₃

semidirect

107. C₄₃

#2092

semidirect

Subgroups — #19136115

Conjugacy classes of subgroups

Complements