Group #19139805 — subgroups

Subgroups — #19139805

Order 5504.

500 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 185 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×#13063324	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#14582820	C₂	1	4	yes	maximal
4	#14582806	C₂	1	1	yes	maximal, characteristic
5	#14582796	C₂	1	4	yes	maximal
6	#14582769	C₂	1	1	yes	maximal, characteristic
7	#14582768	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₄₃⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#391	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₈₆	C₂²	1	2	yes	—
11	#13063324	—	2	4	no	—
12	Q₈×Dic₄₃	—	2	4	no	—
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₃	C₂²	1	4	yes	—
17	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
19	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
23	C₄₃⋊#95	C₂²	1	4	yes	—
24	C₄₃⋊#95	C₂²	1	2	yes	—
25	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
28	#95×C₄₃	C₂²	1	4	yes	—
29	C₂×Q₈×C₄₃	—	2	2	no	—
30	C₂×Q₈×C₄₃	—	2	2	no	—
31	C₂×Q₈×C₄₃	Q₈	1	4	yes	complemented
32	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₄₃	C₂³	1	2	yes	—
37	C₂²×Dic₄₃	C₂³	1	2	yes	—
38	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
39	C₂²×Dic₄₃	C₂³	1	2	yes	—
40	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
41	C₂×Dic₈₆	—	2	4	no	—
42	C₂×Dic₈₆	—	4	4	no	—
43	C₄⋊C₄×C₄₃	—	2	2	no	—
44	C₄²×C₄₃	—	2	2	no	—
45	C₄⋊Dic₄₃	—	2	2	no	—
46	C₄⋊Dic₄₃	—	2	2	no	—
47	C₄⋊Dic₄₃	—	2	4	no	—
48	C₄⋊Dic₄₃	—	4	4	no	—
49	Dic₄₃⋊C₄	—	2	4	no	—
50	Dic₄₃⋊C₄	—	4	4	no	—
51	Dic₄₃⋊C₄	Q₈	1	8	yes	complemented
52	C₄×Dic₄₃	—	2	2	no	—
53	C₄×Dic₄₃	—	2	4	no	—
54	C₄×Dic₄₃	D₄	1	4	yes	—
55	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
56	Q₈×C₄₃	—	2	8	no	—
57	C₂×C₄×C₄₃	—	2	2	no	—
58	C₂×C₄×C₄₃	—	2	2	no	—
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	4	no	—
63	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
67	C₂×C₄×C₄₃	#41	1	2	yes	—
68	C₂×Dic₄₃	—	2	2	no	—
69	C₂×Dic₄₃	—	2	2	no	—
70	C₂×Dic₄₃	—	2	2	no	—
71	C₂×Dic₄₃	—	2	2	no	—
72	C₂×Dic₄₃	—	2	2	no	—
73	C₂×Dic₄₃	—	2	4	no	—
74	C₂×Dic₄₃	—	2	4	no	—
75	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
76	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
77	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
78	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
79	Dic₈₆	—	4	8	no	—
80	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₄₃	C₂²×Q₈	1	2	yes	—
82	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
84	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
85	C₄×C₄₃	—	2	4	no	—
86	C₄×C₄₃	—	2	4	no	—
87	C₄×C₄₃	—	4	2	no	—
88	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
89	Dic₄₃	—	2	2	no	—
90	Dic₄₃	—	2	4	no	—
91	Dic₄₃	—	4	2	no	—
92	Dic₄₃	—	4	2	no	—
93	Dic₄₃	C₄⋊Q₈	1	4	yes	—
94	#2567	—	43	1	no	maximal, Sylow
95	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄₃	#535	1	1	yes	characteristic
98	C₂×C₄₃	#554	1	2	yes	—
99	C₂×C₄₃	Q₈²	1	2	yes	—
100	C₂×C₄⋊Q₈	—	43	1	no	—
101	C₂×C₄×Q₈	—	43	1	no	—
102	#395	—	43	4	no	—
103	#391	—	43	1	no	—
104	#391	D₄₃	1	1	yes	characteristic
105	#389	—	43	4	no	—
106	#384	—	43	1	no	—
107	#384	—	43	1	no	—
108	C₄×C₄⋊C₄	—	43	1	no	—
109	C₄₃	#2567	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	43	2	no	—
111	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
112	C₄⋊Q₈	—	86	4	no	—
113	C₄×Q₈	—	86	4	no	—
114	C₂×C₄⋊C₄	—	43	2	no	—
115	C₂×C₄⋊C₄	—	43	1	no	—
116	C₂×C₄⋊C₄	—	43	2	no	—
117	C₂×C₄⋊C₄	—	43	2	no	—
118	C₂×C₄⋊C₄	—	43	1	no	—
119	C₂×C₄⋊C₄	—	43	4	no	—
120	C₂×C₄⋊C₄	—	43	4	no	—
121	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
122	C₂×C₄²	—	43	1	no	—
123	C₂×C₄²	—	43	2	no	—
124	C₂×C₄²	—	43	1	no	—
125	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
126	#95	—	43	2	no	—
127	#95	—	43	4	no	—
128	#95	C₂×D₄₃	1	4	yes	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	86	4	no	—
132	C₂×Q₈	—	172	4	no	—
133	C₂×Q₈	Dic₈₆	1	4	yes	complemented
134	C₂²×C₄	—	43	1	no	—
135	C₂²×C₄	—	43	2	no	—
136	C₂²×C₄	—	43	1	no	—
137	C₂²×C₄	—	43	2	no	—
138	C₂²×C₄	—	43	2	no	—
139	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
140	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
141	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
142	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	43	8	no	—
145	C₄⋊C₄	—	86	4	no	—
146	C₄⋊C₄	—	86	4	no	—
147	C₄⋊C₄	—	86	2	no	—
148	C₄⋊C₄	—	86	2	no	—
149	C₄⋊C₄	—	172	4	no	—
150	C₄⋊C₄	—	172	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	43	4	no	—
153	C₄²	—	86	4	no	—
154	C₄²	—	86	2	no	—
155	C₂³	C₂³×D₄₃	1	1	yes	characteristic
156	Q₈	—	2	8	no	—
157	Q₈	—	172	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	43	2	no	—
165	C₂×C₄	—	43	4	no	—
166	C₂×C₄	—	43	2	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	2	no	—
172	C₂×C₄	—	86	2	no	—
173	C₂×C₄	—	86	4	no	—
174	C₂×C₄	—	86	4	no	—
175	C₂×C₄	C₂×Dic₈₆	1	4	yes	—
176	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
177	C₂×C₄	D₄×D₄₃	1	2	yes	—
178	C₂×C₄	#10621900	1	2	yes	—
179	C₂×C₄	Q₈×D₄₃	1	4	yes	—
180	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
181	C₂²	C₂×Q₈×D₄₃	1	2	yes	—
182	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
183	C₂²	C₂×#10621900	1	1	yes	characteristic
184	C₂²	#13063371	1	2	yes	—
185	C₄	—	2	4	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	2	no	—
188	C₄	—	43	4	no	—
189	C₄	—	86	4	no	—
190	C₄	—	86	2	no	—
191	C₄	—	172	2	no	—
192	C₄	—	172	2	no	—
193	C₄	#13063302	1	4	yes	—
194	C₂	C₂×#13063302	1	1	yes	characteristic
195	C₂	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
196	C₂	#14583467	1	1	yes	characteristic
197	C₂	#14583607	1	2	yes	—
198	C₂	Q₈×Dic₈₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063324

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₄₃

C₂

yes

maximal, characteristic

#14582820

C₂

yes

maximal

#14582806

C₂

yes

maximal, characteristic

#14582796

C₂

yes

maximal

#14582769

C₂

yes

maximal, characteristic

#14582768

C₂

yes

maximal, characteristic

C₄×Dic₄₃⋊C₄

C₂

yes

maximal, characteristic

C₄₃×#391

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

#13063324

—

Q₈×Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

C₂²

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

Q₈

yes

complemented

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

Q₈

yes

complemented

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

D₄

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

Q₈×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×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×Q₈

yes

—

C₂×Dic₄₃

C₂×Q₈

yes

—

Dic₈₆

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

—

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

C₄⋊Q₈

yes

—

#2567

—

maximal, Sylow

C₂×C₄₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₃

C₂×C₄⋊Q₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

Q₈²

yes

—

100

C₂×C₄⋊Q₈

—

101

C₂×C₄×Q₈

—

102

#395

—

103

#391

—

104

#391

D₄₃

yes

characteristic

105

#389

—

106

#384

—

107

#384

—

108

C₄×C₄⋊C₄

—

109

C₄₃

#2567

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₄₃

yes

characteristic

112

C₄⋊Q₈

—

113

C₄×Q₈

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

122

C₂×C₄²

—

123

C₂×C₄²

—

124

C₂×C₄²

—

125

C₂×C₄²

C₂×D₄₃

yes

characteristic

126

#95

—

127

#95

—

128

#95

C₂×D₄₃

yes

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

172

—

133

C₂×Q₈

Dic₈₆

yes

complemented

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂²×D₄₃

yes

—

140

C₂²×C₄

C₂²×D₄₃

yes

—

141

C₂²×C₄

C₂²×D₄₃

yes

—

142

C₂²×C₄

C₂²×D₄₃

yes

characteristic

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

172

—

150

C₄⋊C₄

—

172

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₂³

C₂³×D₄₃

yes

characteristic

156

Q₈

—

157

Q₈

—

172

—

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

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₂×Dic₈₆

yes

—

176

C₂×C₄

C₂×Dic₈₆

yes

—

177

C₂×C₄

D₄×D₄₃

yes

—

178

C₂×C₄

#10621900

yes

—

179

C₂×C₄

Q₈×D₄₃

yes

—

180

C₂²

C₂×D₄×D₄₃

yes

characteristic

181

C₂²

C₂×Q₈×D₄₃

yes

—

182

C₂²

C₂²×Dic₈₆

yes

characteristic

183

C₂²

C₂×#10621900

yes

characteristic

184

C₂²

#13063371

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

172

—

192

C₄

—

172

—

193

C₄

#13063302

yes

—

194

C₂

C₂×#13063302

yes

characteristic

195

C₂

C₄⋊Q₈×D₄₃

yes

characteristic

196

C₂

#14583467

yes

characteristic

197

C₂

#14583607

yes

—

198

C₂

Q₈×Dic₈₆

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄₃×#391	C₂
hypercenter	#391	D₄₃
hypocenter	C₄₃	#2567
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₄₃×#391

C₂

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
31. C₂×Q₈×C₄₃	Q₈	semidirect
51. Dic₄₃⋊C₄	Q₈	semidirect
109. C₄₃	#2567	semidirect
133. C₂×Q₈	Dic₈₆	semidirect

H ≅ G/N

type

31. C₂×Q₈×C₄₃

Q₈

semidirect

51. Dic₄₃⋊C₄

Q₈

semidirect

109. C₄₃

#2567

semidirect

133. C₂×Q₈

Dic₈₆

semidirect

Subgroups — #19139805

Conjugacy classes of subgroups

Complements