Group #19140084 — subgroups

Subgroups — #19140084

Order 5504.

520 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 153 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×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14582824	C₂	1	2	yes	maximal, complemented
3	#14582811	C₂	1	2	yes	maximal, complemented
4	#14582806	C₂	1	2	yes	maximal, complemented
5	#14582802	C₂	1	2	yes	maximal
6	#14582793	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#124×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₈₆	—	2	4	no	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
16	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
18	C₄₃⋊#95	C₂²	1	4	yes	—
19	C₄₃⋊#95	C₂²	1	4	yes	—
20	C₄₃⋊#95	C₂²	1	2	yes	—
21	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
23	#124×C₄₃	—	2	4	no	—
24	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
27	C₂⁴×C₄₃	C₂³	1	2	yes	—
28	C₂×Q₈×C₄₃	—	2	2	no	—
29	C₂×Q₈×C₄₃	—	2	4	no	—
30	C₂×D₄×C₄₃	—	2	2	no	—
31	C₂×D₄×C₄₃	—	2	4	no	—
32	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
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	2	yes	—
38	C₂²×Dic₄₃	C₂³	1	2	yes	—
39	C₂×Dic₈₆	—	2	2	no	—
40	C₂×Dic₈₆	—	2	4	no	—
41	C₂²⋊C₄×C₄₃	—	2	4	no	—
42	C₂²⋊C₄×C₄₃	—	2	4	no	—
43	C₄²×C₄₃	D₄	1	4	yes	—
44	C₂²⋊Dic₄₃	—	2	8	no	—
45	C₂²⋊Dic₄₃	—	4	4	no	—
46	C₄⋊Dic₄₃	—	2	2	no	—
47	C₄⋊Dic₄₃	—	2	4	no	—
48	Dic₄₃⋊C₄	—	2	4	no	—
49	C₄×Dic₄₃	—	2	4	no	—
50	C₄×Dic₄₃	—	4	2	no	—
51	C₂³×C₄₃	—	2	2	no	—
52	C₂³×C₄₃	—	2	4	no	—
53	C₂³×C₄₃	—	2	4	no	—
54	C₂³×C₄₃	—	2	2	no	—
55	C₂³×C₄₃	—	2	2	no	—
56	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
57	Q₈×C₄₃	—	4	4	no	—
58	D₄×C₄₃	—	4	4	no	—
59	C₂×C₄×C₄₃	—	2	2	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	4	no	—
64	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₃	#41	1	4	yes	—
68	C₂×Dic₄₃	—	2	2	no	—
69	C₂×Dic₄₃	—	2	2	no	—
70	C₂×Dic₄₃	—	2	2	no	—
71	C₂×Dic₄₃	—	2	4	no	—
72	C₂×Dic₄₃	—	2	4	no	—
73	C₂×Dic₄₃	—	2	4	no	—
74	C₂×Dic₄₃	—	2	2	no	—
75	C₂×Dic₄₃	—	2	2	no	—
76	C₂×Dic₄₃	#41	1	4	yes	—
77	Dic₈₆	—	2	8	no	—
78	C₂²×C₄₃	—	4	2	no	—
79	C₂²×C₄₃	—	4	2	no	—
80	C₂²×C₄₃	—	4	2	no	—
81	C₂²×C₄₃	—	4	4	no	—
82	C₂²×C₄₃	—	4	4	no	—
83	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₄₃	C₂×#41	1	2	yes	—
85	C₂²×C₄₃	C₂×#41	1	2	yes	—
86	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
88	C₄×C₄₃	—	2	2	no	—
89	C₄×C₄₃	—	2	2	no	—
90	C₄×C₄₃	—	4	2	no	—
91	C₄×C₄₃	#124	1	4	yes	—
92	Dic₄₃	—	2	4	no	—
93	Dic₄₃	—	4	2	no	—
94	Dic₄₃	—	4	4	no	—
95	#2592	—	43	1	no	maximal, Sylow
96	C₂×C₄₃	—	8	2	no	—
97	C₂×C₄₃	#525	1	1	yes	characteristic
98	C₂×C₄₃	C₂×#124	1	1	yes	characteristic
99	C₂×C₄₃	#537	1	1	yes	characteristic
100	C₂×C₄₃	#556	1	2	yes	—
101	C₂×C₄₃	#559	1	2	yes	—
102	C₂×#124	D₄₃	1	1	yes	characteristic
103	C₂×C₄×Q₈	—	43	1	no	—
104	#397	—	43	2	no	—
105	#393	—	43	2	no	—
106	#391	—	43	2	no	—
107	#390	—	43	2	no	—
108	#388	—	43	4	no	—
109	C₄×C₄⋊C₄	—	43	1	no	—
110	C₄₃	#2592	1	1	yes	characteristic, complemented, Sylow
111	C₂²×Q₈	—	43	1	no	—
112	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
113	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
114	#124	—	2	4	no	—
115	C₄×Q₈	—	86	4	no	—
116	C₂×C₄⋊C₄	—	43	2	no	—
117	C₂×C₄⋊C₄	—	43	1	no	—
118	C₂×C₄⋊C₄	—	43	2	no	—
119	C₂×C₂²⋊C₄	—	43	4	no	—
120	C₂×C₂²⋊C₄	—	43	4	no	—
121	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
122	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
123	C₂×C₄²	—	43	2	no	—
124	C₂×C₄²	—	43	2	no	—
125	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
126	#95	—	43	2	no	—
127	#95	—	43	4	no	—
128	#95	—	43	4	no	—
129	C₂⁴	C₂²×D₄₃	1	2	yes	—
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	86	4	no	—
133	C₂×Q₈	—	86	2	no	—
134	C₂×D₄	—	2	4	no	—
135	C₂×D₄	—	2	2	no	—
136	C₂²×C₄	—	43	2	no	—
137	C₂²×C₄	—	43	4	no	—
138	C₂²×C₄	—	43	2	no	—
139	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
140	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
143	C₄⋊C₄	—	86	4	no	—
144	C₄⋊C₄	—	86	4	no	—
145	C₄⋊C₄	—	86	2	no	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	86	8	no	—
149	C₂²⋊C₄	—	172	4	no	—
150	C₄²	—	86	4	no	—
151	C₄²	—	172	2	no	—
152	C₄²	C₂²⋊D₄₃	1	4	yes	—
153	C₂³	—	2	2	no	—
154	C₂³	—	2	4	no	—
155	C₂³	—	2	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	C₂³×D₄₃	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	Q₈	—	86	8	no	—
161	D₄	—	4	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	43	4	no	—
168	C₂×C₄	—	86	2	no	—
169	C₂×C₄	—	86	2	no	—
170	C₂×C₄	—	86	2	no	—
171	C₂×C₄	—	86	4	no	—
172	C₂×C₄	—	86	4	no	—
173	C₂×C₄	—	86	4	no	—
174	C₂×C₄	—	86	2	no	—
175	C₂×C₄	—	86	2	no	—
176	C₂×C₄	#10621900	1	4	yes	—
177	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
186	C₂²	#41×D₄₃	1	2	yes	—
187	C₂²	C₂×#10621900	1	2	yes	—
188	C₂²	#13063371	1	1	yes	characteristic
189	C₂²	#13063372	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	—	86	4	no	—
194	C₄	—	172	4	no	—
195	C₄	—	172	2	no	—
196	C₄	#13063312	1	4	yes	—
197	C₂	—	8	2	no	—
198	C₂	#14583406	1	1	yes	characteristic
199	C₂	C₂×#13063312	1	1	yes	characteristic
200	C₂	#14583477	1	1	yes	characteristic
201	C₂	#14583618	1	2	yes	—
202	C₂	#14583631	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic, complemented

#14582824

C₂

yes

maximal, complemented

#14582811

C₂

yes

maximal, complemented

#14582806

C₂

yes

maximal, complemented

#14582802

C₂

yes

maximal

#14582793

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₂²×D₄×C₄₃

C₂²

yes

characteristic

#124×C₄₃

—

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

C₂³

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

—

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

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂⁴

yes

characteristic

Q₈×C₄₃

—

D₄×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×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

#41

yes

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

—

C₂²×C₄₃

C₂×#41

yes

—

C₂²×C₄₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₄₃

2₋¹⁺⁴

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#124

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#2592

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

100

C₂×C₄₃

#556

yes

—

101

C₂×C₄₃

#559

yes

—

102

C₂×#124

D₄₃

yes

characteristic

103

C₂×C₄×Q₈

—

104

#397

—

105

#393

—

106

#391

—

107

#390

—

108

#388

—

109

C₄×C₄⋊C₄

—

110

C₄₃

#2592

yes

characteristic, complemented, Sylow

111

C₂²×Q₈

—

112

C₂²×Q₈

C₂×D₄₃

yes

characteristic

113

C₂²×D₄

C₂×D₄₃

yes

characteristic

114

#124

—

115

C₄×Q₈

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₂²⋊C₄

—

121

C₂×C₂²⋊C₄

C₂×D₄₃

yes

—

122

C₂×C₂²⋊C₄

C₂×D₄₃

yes

—

123

C₂×C₄²

—

124

C₂×C₄²

—

125

C₂×C₄²

C₂×D₄₃

yes

characteristic

126

#95

—

127

#95

—

128

#95

—

129

C₂⁴

C₂²×D₄₃

yes

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²×D₄₃

yes

—

140

C₂²×C₄

C₂²×D₄₃

yes

characteristic

141

C₂²×C₄

C₂²×D₄₃

yes

characteristic

142

C₂²×C₄

C₂²×D₄₃

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

172

—

150

C₄²

—

151

C₄²

—

172

—

152

C₄²

C₂²⋊D₄₃

yes

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

C₂³×D₄₃

yes

characteristic

159

Q₈

—

160

Q₈

—

161

D₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

#10621900

yes

—

177

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

178

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

179

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

186

C₂²

#41×D₄₃

yes

—

187

C₂²

C₂×#10621900

yes

—

188

C₂²

#13063371

yes

characteristic

189

C₂²

#13063372

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

172

—

195

C₄

—

172

—

196

C₄

#13063312

yes

—

197

C₂

—

198

C₂

#14583406

yes

characteristic

199

C₂

C₂×#13063312

yes

characteristic

200

C₂

#14583477

yes

characteristic

201

C₂

#14583618

yes

—

202

C₂

#14583631

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×#124×C₄₃	C₂
hypercenter	C₂×#124	D₄₃
hypocenter	C₄₃	#2592
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₂×#124×C₄₃

C₂

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×C₄×Dic₈₆	C₂	semidirect
2. #14582824	C₂	semidirect
3. #14582811	C₂	semidirect
4. #14582806	C₂	semidirect
6. #14582793	C₂	semidirect
7. C₄×C₄⋊Dic₄₃	C₂	semidirect
110. C₄₃	#2592	semidirect

H ≅ G/N

type

1. C₂×C₄×Dic₈₆

C₂

semidirect

2. #14582824

C₂

semidirect

3. #14582811

C₂

semidirect

4. #14582806

C₂

semidirect

6. #14582793

C₂

semidirect

7. C₄×C₄⋊Dic₄₃

C₂

semidirect

110. C₄₃

#2592

semidirect

Subgroups — #19140084

Conjugacy classes of subgroups

Complements