Group #19135859 — subgroups

Subgroups — #19135859

Order 5504.

276 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×#10621873	C₂	1	1	yes	maximal, characteristic
2	#14582925	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582909	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582905	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063243	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063241	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×#10621873	—	2	2	no	—
11	C₂×#10621873	—	2	2	no	—
12	C₂×#10621873	—	2	2	no	—
13	C₂×#10621877	C₂²	1	1	yes	characteristic
14	#13063284	C₂²	1	1	yes	characteristic
15	Dic₄₃⋊C₈	—	2	2	no	—
16	#13063250	—	2	2	no	—
17	#13063247	—	2	2	no	—
18	#13063243	—	2	2	no	—
19	#13063241	—	2	2	no	—
20	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂³×D₄₃	—	2	1	no	—
24	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₃	—	2	1	no	—
26	C₂×D₁₇₂	—	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	D₄	1	2	yes	—
32	C₂×Dic₈₆	—	2	1	no	—
33	C₂×Dic₈₆	—	2	1	no	—
34	C₂×Dic₈₆	—	2	1	no	—
35	C₂×Dic₈₆	—	2	1	no	—
36	C₂×Dic₈₆	—	2	1	no	—
37	C₂×Dic₈₆	D₄	1	2	yes	—
38	M₁₆×C₄₃	—	2	1	no	—
39	M₁₆×C₄₃	—	2	1	no	—
40	M₁₆×C₄₃	—	2	1	no	—
41	M₁₆×C₄₃	—	2	1	no	—
42	C₂×C₈×C₄₃	—	2	1	no	—
43	C₂×C₈×C₄₃	—	2	1	no	—
44	C₂×C₈×C₄₃	—	2	1	no	—
45	C₂×C₈×C₄₃	—	2	1	no	—
46	C₂×C₈×C₄₃	D₄	1	2	yes	—
47	C₂²⋊Dic₄₃	—	2	1	no	—
48	C₄⋊Dic₄₃	D₄	1	2	yes	—
49	C₄×Dic₄₃	—	2	1	no	—
50	#10621877	—	2	1	no	—
51	#10621877	—	2	1	no	—
52	C₂×C₄₃⋊C₈	—	2	1	no	—
53	#10621873	—	2	8	no	—
54	C₂³×C₄₃	#41	1	1	yes	characteristic
55	C₂²×D₄₃	—	2	2	no	—
56	C₂²×D₄₃	—	4	1	no	—
57	C₂²×D₄₃	—	4	1	no	—
58	C₂²×D₄₃	—	8	2	no	—
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
65	C₂×Dic₄₃	—	2	2	no	—
66	C₂×Dic₄₃	—	2	2	no	—
67	C₂×Dic₄₃	—	4	1	no	—
68	C₂×Dic₄₃	—	4	1	no	—
69	D₁₇₂	—	2	2	no	—
70	D₁₇₂	—	2	2	no	—
71	D₁₇₂	—	4	1	no	—
72	D₁₇₂	—	4	1	no	—
73	Dic₈₆	—	2	2	no	—
74	Dic₈₆	—	2	2	no	—
75	Dic₈₆	—	4	1	no	—
76	Dic₈₆	—	4	1	no	—
77	C₈×C₄₃	—	2	2	no	—
78	C₈×C₄₃	—	4	1	no	—
79	C₈×C₄₃	—	4	1	no	—
80	C₄₃⋊C₈	—	4	1	no	—
81	C₂²×C₄₃	—	2	2	no	—
82	C₂²×C₄₃	#121	1	1	yes	characteristic
83	C₂²×C₄₃	#123	1	1	yes	characteristic
84	C₂²×C₄₃	#123	1	1	yes	characteristic
85	C₂×D₄₃	—	4	2	no	—
86	C₂×D₄₃	—	8	2	no	—
87	C₂×D₄₃	—	8	2	no	—
88	C₂×D₄₃	—	8	2	no	—
89	C₄×C₄₃	#120	1	1	yes	characteristic
90	C₄×C₄₃	#121	1	1	yes	characteristic
91	C₄×C₄₃	#121	1	1	yes	characteristic
92	C₄×C₄₃	#124	1	1	yes	characteristic
93	Dic₄₃	—	4	2	no	—
94	Dic₄₃	—	4	2	no	—
95	#2046	—	43	1	no	maximal, Sylow
96	C₂×C₄₃	—	2	1	no	—
97	C₂×C₄₃	—	2	1	no	—
98	C₂×C₄₃	#394	1	1	yes	characteristic
99	C₂×C₄₃	#471	1	2	yes	—
100	D₄₃	—	8	2	no	—
101	C₂²×SD₁₆	—	43	1	no	—
102	#424	—	43	1	no	—
103	#419	—	43	1	no	—
104	#418	—	43	1	no	—
105	C₂×#101	—	43	1	no	—
106	C₂×#100	—	43	1	no	—
107	#341	D₄₃	1	1	yes	characteristic, complemented
108	C₄₃	#2046	1	1	yes	characteristic, complemented, Sylow
109	C₂²×Q₈	—	43	1	no	—
110	C₂²×D₄	—	43	1	no	—
111	C₂×SD₁₆	—	86	2	no	—
112	C₂×SD₁₆	—	86	2	no	—
113	C₂×SD₁₆	—	86	2	no	—
114	C₂×M₁₆	—	43	1	no	—
115	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
116	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
117	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
118	#117	—	43	1	no	—
119	C₄⋊C₈	—	86	2	no	—
120	#103	—	86	2	no	—
121	#102	—	86	2	no	—
122	#101	—	86	2	no	—
123	#100	—	86	2	no	—
124	C₂⁴	—	86	1	no	—
125	C₂×Q₈	—	43	2	no	—
126	C₂×Q₈	—	86	1	no	—
127	C₂×Q₈	—	86	1	no	—
128	C₂×Q₈	—	86	1	no	—
129	C₂×Q₈	—	86	1	no	—
130	C₂×Q₈	—	86	1	no	—
131	C₂×D₄	—	43	2	no	—
132	C₂×D₄	—	86	1	no	—
133	C₂×D₄	—	86	1	no	—
134	C₂×D₄	—	86	1	no	—
135	C₂×D₄	—	86	1	no	—
136	C₂×D₄	—	86	1	no	—
137	C₂²×C₄	—	86	1	no	—
138	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
139	SD₁₆	—	86	8	no	—
140	M₁₆	—	2	1	no	—
141	M₁₆	—	2	1	no	—
142	M₁₆	—	2	1	no	—
143	M₁₆	—	2	1	no	—
144	M₁₆	—	86	1	no	—
145	M₁₆	—	86	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	86	1	no	—
151	C₂×C₈	D₁₇₂	1	2	yes	—
152	C₄⋊C₄	—	43	2	no	—
153	C₂²⋊C₄	—	86	1	no	—
154	C₄²	—	86	1	no	—
155	C₂³	—	86	2	no	—
156	C₂³	—	344	2	no	—
157	C₂³	—	172	1	no	—
158	C₂³	—	172	1	no	—
159	C₂³	#10621902	1	1	yes	characteristic
160	Q₈	—	86	2	no	—
161	Q₈	—	86	2	no	—
162	Q₈	—	172	1	no	—
163	Q₈	—	172	1	no	—
164	D₄	—	86	2	no	—
165	D₄	—	86	2	no	—
166	D₄	—	172	1	no	—
167	D₄	—	172	1	no	—
168	C₂×C₄	—	86	2	no	—
169	C₂×C₄	—	86	2	no	—
170	C₂×C₄	—	172	1	no	—
171	C₂×C₄	—	172	1	no	—
172	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
173	C₂×C₄	#10621898	1	1	yes	characteristic
174	C₂×C₄	#10621898	1	1	yes	characteristic
175	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
176	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
177	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	4	1	no	—
180	C₈	—	4	1	no	—
181	C₈	—	172	1	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	344	2	no	—
184	C₂²	—	344	2	no	—
185	C₂²	—	344	2	no	—
186	C₂²	—	172	2	no	—
187	C₂²	#13063300	1	1	yes	characteristic
188	C₂²	#13063310	1	1	yes	characteristic
189	C₂²	#13063310	1	1	yes	characteristic
190	C₄	—	172	2	no	—
191	C₄	—	172	2	no	—
192	C₄	#13063295	1	1	yes	characteristic
193	C₄	#13063298	1	1	yes	characteristic
194	C₄	#13063298	1	1	yes	characteristic
195	C₄	#13063313	1	1	yes	characteristic
196	C₂	—	2	1	no	—
197	C₂	—	2	1	no	—
198	C₂	—	344	2	no	—
199	C₂	#14582817	1	1	yes	characteristic
200	C₂	#14583163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10621873

C₂

yes

maximal, characteristic

#14582925

C₂

yes

maximal, characteristic, complemented

#14582909

C₂

yes

maximal, characteristic, complemented

#14582905

C₂

yes

maximal, characteristic

C₂×#13063243

C₂

yes

maximal, characteristic, complemented

C₂×#13063241

C₂

yes

maximal, characteristic

C₄₃×#341

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂²×D₁₇₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

#13063284

C₂²

yes

characteristic

Dic₄₃⋊C₈

—

—

—

—

—

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

#10621873

—

C₂³×C₄₃

#41

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

#2046

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#394

yes

characteristic

C₂×C₄₃

#471

yes

—

100

D₄₃

—

101

C₂²×SD₁₆

—

102

#424

—

103

#419

—

104

#418

—

105

C₂×#101

—

106

C₂×#100

—

107

#341

D₄₃

yes

characteristic, complemented

108

C₄₃

#2046

yes

characteristic, complemented, Sylow

109

C₂²×Q₈

—

110

C₂²×D₄

—

111

C₂×SD₁₆

—

112

C₂×SD₁₆

—

113

C₂×SD₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

C₂×D₄₃

yes

characteristic

116

C₂×M₁₆

C₂×D₄₃

yes

characteristic

117

C₂²×C₈

C₂×D₄₃

yes

characteristic

118

#117

—

119

C₄⋊C₈

—

120

#103

—

121

#102

—

122

#101

—

123

#100

—

124

C₂⁴

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²×D₄₃

yes

characteristic

139

SD₁₆

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

D₁₇₂

yes

—

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₄²

—

155

C₂³

—

156

C₂³

—

344

—

157

C₂³

—

172

—

158

C₂³

—

172

—

159

C₂³

#10621902

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

172

—

163

Q₈

—

172

—

164

D₄

—

165

D₄

—

166

D₄

—

172

—

167

D₄

—

172

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

172

—

171

C₂×C₄

—

172

—

172

C₂×C₄

C₂×D₁₇₂

yes

characteristic

173

C₂×C₄

#10621898

yes

characteristic

174

C₂×C₄

#10621898

yes

characteristic

175

C₂×C₄

D₄×D₄₃

yes

characteristic

176

C₂×C₄

D₄×D₄₃

yes

characteristic

177

C₂×C₄

D₄×D₄₃

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

172

—

182

C₂²

—

183

C₂²

—

344

—

184

C₂²

—

344

—

185

C₂²

—

344

—

186

C₂²

—

172

—

187

C₂²

#13063300

yes

characteristic

188

C₂²

#13063310

yes

characteristic

189

C₂²

#13063310

yes

characteristic

190

C₄

—

172

—

191

C₄

—

172

—

192

C₄

#13063295

yes

characteristic

193

C₄

#13063298

yes

characteristic

194

C₄

#13063298

yes

characteristic

195

C₄

#13063313

yes

characteristic

196

C₂

—

197

C₂

—

198

C₂

—

344

—

199

C₂

#14582817

yes

characteristic

200

C₂

#14583163

yes

—

	subgroup	G/H
center	C₂²	#13063300
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#341	C₂
socle	C₂²×C₄₃	#121
hypercenter	#341	D₄₃
hypocenter	C₄₃	#2046
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#13063300

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#341

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. #14582925	C₂	semidirect
3. #14582909	C₂	semidirect
5. C₂×#13063243	C₂	semidirect
7. C₄₃×#341	C₂	semidirect
107. #341	D₄₃	semidirect
108. C₄₃	#2046	semidirect

H ≅ G/N

type

2. #14582925

C₂

semidirect

3. #14582909

C₂

semidirect

5. C₂×#13063243

C₂

semidirect

7. C₄₃×#341

C₂

semidirect

107. #341

D₄₃

semidirect

108. C₄₃

#2046

semidirect

Subgroups — #19135859

Conjugacy classes of subgroups

Complements