Group #19142653 — subgroups

Subgroups — #19142653

Order 5504.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13063325	C₂	1	1	yes	maximal, characteristic, complemented
2	#14582838	C₂	1	2	yes	maximal, complemented
3	#14582835	C₂	1	2	yes	maximal
4	#14582829	C₂	1	2	yes	maximal, complemented
5	#14582824	C₂	1	2	yes	maximal
6	#14582819	C₂	1	2	yes	maximal, complemented
7	#14582811	C₂	1	2	yes	maximal
8	#14582767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#126×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₈₆	C₂²	1	2	yes	—
11	#13063325	—	2	4	no	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
16	C₄₃⋊#95	C₂²	1	2	yes	—
17	C₄₃⋊#95	C₂²	1	4	yes	—
18	C₄₃⋊#95	C₂²	1	4	yes	—
19	C₄₃⋊#95	C₂²	1	2	yes	—
20	C₄₃⋊#95	C₂²	1	2	yes	—
21	#126×C₄₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
29	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₄₃	C₂³	1	4	yes	—
37	C₂²×Dic₄₃	C₂³	1	4	yes	—
38	C₂×Dic₈₆	—	2	4	no	—
39	C₂×Dic₈₆	—	4	4	no	—
40	C₄⋊C₄×C₄₃	—	2	2	no	—
41	C₄⋊C₄×C₄₃	—	2	2	no	—
42	C₄⋊C₄×C₄₃	—	2	2	no	—
43	C₂²⋊C₄×C₄₃	—	2	2	no	—
44	C₂²⋊C₄×C₄₃	—	2	2	no	—
45	C₂²⋊C₄×C₄₃	—	2	2	no	—
46	C₄²×C₄₃	D₄	1	4	yes	—
47	C₂²⋊Dic₄₃	—	4	4	no	—
48	C₄⋊Dic₄₃	—	2	4	no	—
49	C₄⋊Dic₄₃	—	2	4	no	—
50	Dic₄₃⋊C₄	—	4	4	no	—
51	C₂³×C₄₃	—	2	1	no	—
52	C₂³×C₄₃	—	2	2	no	—
53	C₂³×C₄₃	—	2	2	no	—
54	C₂³×C₄₃	—	2	2	no	—
55	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	—	2	2	no	—
58	C₂×C₄×C₄₃	—	2	2	no	—
59	C₂×C₄×C₄₃	—	2	1	no	—
60	C₂×C₄×C₄₃	—	2	1	no	—
61	C₂×C₄×C₄₃	—	2	1	no	—
62	C₂×C₄×C₄₃	—	2	2	no	—
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
68	C₂×Dic₄₃	—	2	4	no	—
69	C₂×Dic₄₃	—	2	4	no	—
70	C₂×Dic₄₃	—	2	4	no	—
71	C₂×Dic₄₃	—	2	4	no	—
72	C₂×Dic₄₃	—	2	4	no	—
73	C₂×Dic₄₃	—	2	4	no	—
74	Dic₈₆	—	4	8	no	—
75	C₂²×C₄₃	—	4	1	no	—
76	C₂²×C₄₃	—	4	1	no	—
77	C₂²×C₄₃	—	4	1	no	—
78	C₂²×C₄₃	—	4	4	no	—
79	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
81	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
82	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	—	4	1	no	—
87	C₄×C₄₃	—	4	1	no	—
88	C₄×C₄₃	—	4	1	no	—
89	Dic₄₃	—	4	4	no	—
90	Dic₄₃	—	4	4	no	—
91	#2821	—	43	1	no	maximal, Sylow
92	C₂×C₄₃	—	8	1	no	—
93	C₂×C₄₃	#535	1	1	yes	characteristic
94	C₂×C₄₃	#536	1	1	yes	characteristic
95	C₂×C₄₃	#536	1	1	yes	characteristic
96	C₂×C₄₃	#563	1	4	yes	—
97	C₂×C₄⋊Q₈	—	43	1	no	—
98	C₂×#126	D₄₃	1	1	yes	characteristic
99	#400	—	43	2	no	—
100	#399	—	43	2	no	—
101	#398	—	43	2	no	—
102	#397	—	43	2	no	—
103	#395	—	43	2	no	—
104	#393	—	43	2	no	—
105	#383	—	43	1	no	—
106	C₄₃	#2821	1	1	yes	characteristic, complemented, Sylow
107	C₂²×Q₈	—	43	2	no	—
108	C₄⋊Q₈	—	86	4	no	—
109	#126	—	2	4	no	—
110	C₂×C₄⋊C₄	—	43	2	no	—
111	C₂×C₄⋊C₄	—	43	2	no	—
112	C₂×C₄⋊C₄	—	43	4	no	—
113	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
116	C₂×C₂²⋊C₄	—	43	4	no	—
117	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
120	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
121	#95	—	43	2	no	—
122	#95	—	43	2	no	—
123	#95	—	43	2	no	—
124	#95	—	43	4	no	—
125	#95	—	43	4	no	—
126	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
127	C₂×Q₈	—	86	4	no	—
128	C₂×Q₈	—	172	4	no	—
129	C₂²×C₄	—	43	4	no	—
130	C₂²×C₄	—	43	4	no	—
131	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
132	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	86	4	no	—
141	C₄⋊C₄	—	86	4	no	—
142	C₄⋊C₄	—	172	4	no	—
143	C₂²⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	—	172	4	no	—
147	C₄²	C₂²⋊D₄₃	1	4	yes	—
148	C₂³	—	2	1	no	—
149	C₂³	—	2	2	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	C₂³×D₄₃	1	1	yes	characteristic
153	Q₈	—	172	8	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	86	4	no	—
164	C₂×C₄	—	86	4	no	—
165	C₂×C₄	—	86	4	no	—
166	C₂×C₄	—	86	4	no	—
167	C₂×C₄	—	86	4	no	—
168	C₂×C₄	—	86	4	no	—
169	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
171	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
172	C₂²	—	4	4	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	—	4	1	no	—
176	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
177	C₂²	#13063371	1	2	yes	—
178	C₂²	#13063372	1	2	yes	—
179	C₂²	#13063372	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	172	4	no	—
187	C₄	—	172	4	no	—
188	C₂	—	8	1	no	—
189	C₂	#14583463	1	1	yes	characteristic
190	C₂	#14583470	1	1	yes	characteristic
191	C₂	#14583470	1	1	yes	characteristic
192	C₂	#14583651	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic, complemented

#14582838

C₂

yes

maximal, complemented

#14582835

C₂

yes

maximal

#14582829

C₂

yes

maximal, complemented

#14582824

C₂

yes

maximal

#14582819

C₂

yes

maximal, complemented

#14582811

C₂

yes

maximal

#14582767

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

#13063325

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

#126×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₂²×C₄₃

2₋¹⁺⁴

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

#2821

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄⋊Q₈

—

C₂×#126

D₄₃

yes

characteristic

#400

—

100

#399

—

101

#398

—

102

#397

—

103

#395

—

104

#393

—

105

#383

—

106

C₄₃

#2821

yes

characteristic, complemented, Sylow

107

C₂²×Q₈

—

108

C₄⋊Q₈

—

109

#126

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

114

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

115

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

116

C₂×C₂²⋊C₄

—

117

C₂×C₂²⋊C₄

C₂×D₄₃

yes

characteristic

118

C₂×C₂²⋊C₄

C₂×D₄₃

yes

characteristic

119

C₂×C₂²⋊C₄

C₂×D₄₃

yes

characteristic

120

C₂×C₄²

C₂×D₄₃

yes

characteristic

121

#95

—

122

#95

—

123

#95

—

124

#95

—

125

#95

—

126

C₂⁴

C₂²×D₄₃

yes

characteristic

127

C₂×Q₈

—

128

C₂×Q₈

—

172

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂²×D₄₃

yes

characteristic

132

C₂²×C₄

C₂²×D₄₃

yes

characteristic

133

C₂²×C₄

C₂²×D₄₃

yes

characteristic

134

C₂²×C₄

C₂²×D₄₃

yes

characteristic

135

C₂²×C₄

C₂²×D₄₃

yes

characteristic

136

C₂²×C₄

C₂²×D₄₃

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

172

—

143

C₂²⋊C₄

—

144

C₂²⋊C₄

—

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

172

—

147

C₄²

C₂²⋊D₄₃

yes

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

C₂³×D₄₃

yes

characteristic

153

Q₈

—

172

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

170

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

171

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

177

C₂²

#13063371

yes

—

178

C₂²

#13063372

yes

—

179

C₂²

#13063372

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

172

—

187

C₄

—

172

—

188

C₂

—

189

C₂

#14583463

yes

characteristic

190

C₂

#14583470

yes

characteristic

191

C₂

#14583470

yes

characteristic

192

C₂

#14583651

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×#126×C₄₃	C₂
hypercenter	C₂×#126	D₄₃
hypocenter	C₄₃	#2821
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₂×#126×C₄₃

C₂

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. C₂×#13063325	C₂	semidirect
2. #14582838	C₂	semidirect
4. #14582829	C₂	semidirect
6. #14582819	C₂	semidirect
8. #14582767	C₂	semidirect
106. C₄₃	#2821	semidirect

H ≅ G/N

type

1. C₂×#13063325

C₂

semidirect

2. #14582838

C₂

semidirect

4. #14582829

C₂

semidirect

6. #14582819

C₂

semidirect

8. #14582767

C₂

semidirect

106. C₄₃

#2821

semidirect

Subgroups — #19142653

Conjugacy classes of subgroups

Complements