Group #19134866 — subgroups

Subgroups — #19134866

Order 5504.

298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 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
2	C₂×#13063257	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063250	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063250	C₂	1	1	yes	maximal, characteristic
5	#14582797	C₂	1	1	yes	maximal, characteristic
6	#14582662	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₄×Dic₈₆	—	2	2	no	—
10	C₄×Dic₈₆	—	2	2	no	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₃⋊C₄	—	2	1	no	—
15	C₂×Dic₄₃⋊C₄	—	2	1	no	—
16	C₂×C₄×Dic₄₃	—	2	1	no	—
17	#13063257	—	2	2	no	—
18	#13063250	—	2	2	no	—
19	#13063250	C₄	1	4	yes	complemented
20	C₄₃⋊#95	—	2	1	no	—
21	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₃	—	2	1	no	—
28	C₂²×Dic₄₃	—	2	1	no	—
29	C₂²×Dic₄₃	—	2	1	no	—
30	C₂²×Dic₄₃	—	2	1	no	—
31	C₂×Dic₈₆	—	2	1	no	—
32	C₂×Dic₈₆	—	2	1	no	—
33	C₂×Dic₈₆	—	2	1	no	—
34	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
35	C₂×Dic₈₆	D₄	1	2	yes	—
36	C₂×Dic₈₆	Q₈	1	2	yes	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	—	2	1	no	—
39	C₂×C₈×C₄₃	—	2	1	no	—
40	C₂×C₈×C₄₃	—	2	1	no	—
41	C₂×C₈×C₄₃	—	2	1	no	—
42	C₂×C₈×C₄₃	C₂×C₄	1	2	yes	—
43	C₄²×C₄₃	—	2	1	no	—
44	C₄²×C₄₃	—	2	1	no	—
45	C₄⋊Dic₄₃	—	2	1	no	—
46	C₄⋊Dic₄₃	—	2	1	no	—
47	C₄⋊Dic₄₃	—	2	1	no	—
48	C₄⋊Dic₄₃	—	2	1	no	—
49	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₄₃	D₄	1	2	yes	—
51	Dic₄₃⋊C₄	—	2	4	no	—
52	Dic₄₃⋊C₄	—	4	2	no	—
53	C₄×Dic₄₃	—	2	4	no	—
54	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	—	2	1	no	—
56	C₂×C₄×C₄₃	—	2	1	no	—
57	C₂×C₄×C₄₃	—	2	1	no	—
58	C₂×C₄×C₄₃	—	2	1	no	—
59	C₂×C₄×C₄₃	—	2	1	no	—
60	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
61	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
67	C₂×Dic₄₃	—	2	2	no	—
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	—	4	1	no	—
73	C₂×Dic₄₃	—	4	1	no	—
74	C₂×Dic₄₃	—	4	1	no	—
75	C₂×Dic₄₃	—	4	1	no	—
76	C₂×Dic₄₃	—	4	1	no	—
77	C₂×Dic₄₃	—	4	1	no	—
78	C₂×Dic₄₃	—	4	1	no	—
79	Dic₈₆	—	2	2	no	—
80	Dic₈₆	—	2	2	no	—
81	Dic₈₆	—	2	2	no	—
82	Dic₈₆	C₄⋊C₄	1	4	yes	complemented
83	C₈×C₄₃	—	2	2	no	—
84	C₈×C₄₃	—	4	1	no	—
85	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₄₃	#120	1	1	yes	characteristic
87	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂²×C₄₃	#135	1	1	yes	characteristic
90	C₂²×C₄₃	#137	1	1	yes	characteristic
91	C₂²×C₄₃	#137	1	1	yes	characteristic
92	C₄×C₄₃	—	2	2	no	—
93	C₄×C₄₃	—	4	1	no	—
94	C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
96	C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₄×C₄₃	#123	1	1	yes	characteristic
98	Dic₄₃	—	2	4	no	—
99	Dic₄₃	—	4	2	no	—
100	Dic₄₃	—	4	2	no	—
101	Dic₄₃	—	8	1	no	—
102	#1865	—	43	1	no	maximal, Sylow
103	C₂×C₄₃	#385	1	1	yes	characteristic
104	C₂×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
105	C₂×C₄₃	#441	1	1	yes	characteristic
106	C₂×C₄₃	#448	1	1	yes	characteristic
107	C₂×C₄₃	#452	1	1	yes	characteristic
108	C₂×C₄₃	#475	1	1	yes	characteristic
109	C₂×C₄₃	#479	1	1	yes	characteristic
110	C₂×C₄×Q₈	—	43	1	no	—
111	C₂×#106	—	43	1	no	—
112	C₂×#103	—	43	1	no	—
113	C₂×#103	—	43	1	no	—
114	#389	—	43	1	no	—
115	#340	—	43	1	no	—
116	#336	D₄₃	1	1	yes	characteristic
117	C₄₃	#1865	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	43	1	no	—
119	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
121	C₄×Q₈	—	86	2	no	—
122	C₄×Q₈	—	86	2	no	—
123	C₂×C₄⋊C₄	—	43	1	no	—
124	C₂×C₄⋊C₄	—	43	1	no	—
125	C₂×C₄⋊C₄	—	43	1	no	—
126	C₂×C₄⋊C₄	—	86	1	no	—
127	C₂×C₄⋊C₄	—	86	1	no	—
128	C₂×C₄²	—	86	1	no	—
129	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
130	#106	—	86	2	no	—
131	#103	—	43	4	no	—
132	#103	—	86	2	no	—
133	#95	—	86	1	no	—
134	C₂×Q₈	—	43	2	no	—
135	C₂×Q₈	—	43	2	no	—
136	C₂×Q₈	—	43	2	no	—
137	C₂×Q₈	—	86	1	no	—
138	C₂×Q₈	—	86	1	no	—
139	C₂×Q₈	—	86	1	no	—
140	C₂²×C₄	—	86	1	no	—
141	C₂²×C₄	—	86	1	no	—
142	C₂²×C₄	—	86	1	no	—
143	C₂²×C₄	—	86	1	no	—
144	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
145	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	C₄×D₄₃	1	2	yes	—
153	C₄⋊C₄	—	43	2	no	—
154	C₄⋊C₄	—	43	2	no	—
155	C₄⋊C₄	—	86	4	no	—
156	C₄⋊C₄	—	86	1	no	—
157	C₄⋊C₄	—	86	1	no	—
158	C₄⋊C₄	—	86	1	no	—
159	C₄⋊C₄	—	86	1	no	—
160	C₄⋊C₄	—	172	2	no	—
161	C₄²	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	86	4	no	—
164	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
165	Q₈	—	43	4	no	—
166	Q₈	—	86	2	no	—
167	Q₈	—	86	2	no	—
168	Q₈	—	86	2	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	86	2	no	—
175	C₂×C₄	—	86	2	no	—
176	C₂×C₄	—	86	2	no	—
177	C₂×C₄	—	86	2	no	—
178	C₂×C₄	—	86	2	no	—
179	C₂×C₄	—	172	1	no	—
180	C₂×C₄	—	172	1	no	—
181	C₂×C₄	—	172	1	no	—
182	C₂×C₄	—	172	1	no	—
183	C₂×C₄	—	172	1	no	—
184	C₂×C₄	—	172	1	no	—
185	C₂×C₄	—	172	1	no	—
186	C₂×C₄	#10621873	1	2	yes	—
187	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
188	C₂×C₄	#10621898	1	1	yes	characteristic
189	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
190	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
191	C₂×C₄	#10621900	1	1	yes	characteristic
192	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	4	1	no	—
195	C₂²	C₄×D₁₇₂	1	1	yes	characteristic
196	C₂²	#13063295	1	1	yes	characteristic
197	C₂²	#13063305	1	1	yes	characteristic
198	C₂²	C₂×#10621873	1	1	yes	characteristic
199	C₂²	#13063347	1	1	yes	characteristic
200	C₂²	#13063358	1	1	yes	characteristic
201	C₂²	#13063358	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	4	1	no	—
204	C₄	—	86	4	no	—
205	C₄	—	344	1	no	—
206	C₄	—	172	2	no	—
207	C₄	—	172	2	no	—
208	C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
209	C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
210	C₄	#13063303	1	1	yes	characteristic
211	C₄	#13063307	1	1	yes	characteristic
212	C₂	#14582773	1	1	yes	characteristic
213	C₂	C₄×#10621873	1	1	yes	characteristic
214	C₂	#14583007	1	1	yes	characteristic
215	C₂	#14583040	1	1	yes	characteristic
216	C₂	#14583063	1	1	yes	characteristic
217	C₂	#14583185	1	1	yes	characteristic
218	C₂	#14583207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic

C₂×#13063257

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal, characteristic

#14582797

C₂

yes

maximal, characteristic

#14582662

C₂

yes

maximal, characteristic

C₄₃×#336

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

#13063257

—

#13063250

—

#13063250

C₄

yes

complemented

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×Dic₈₆

D₄

yes

—

C₂×Dic₈₆

Q₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

D₄

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

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×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

C₄⋊C₄

yes

complemented

C₈×C₄₃

—

C₈×C₄₃

—

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₂²×C₄₃

#120

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₃

C₄×D₄

yes

characteristic

C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₄×C₄₃

#123

yes

characteristic

Dic₄₃

—

Dic₄₃

—

100

Dic₄₃

—

101

Dic₄₃

—

102

#1865

—

maximal, Sylow

103

C₂×C₄₃

#385

yes

characteristic

104

C₂×C₄₃

C₄×SD₁₆

yes

characteristic

105

C₂×C₄₃

#441

yes

characteristic

106

C₂×C₄₃

#448

yes

characteristic

107

C₂×C₄₃

#452

yes

characteristic

108

C₂×C₄₃

#475

yes

characteristic

109

C₂×C₄₃

#479

yes

characteristic

110

C₂×C₄×Q₈

—

111

C₂×#106

—

112

C₂×#103

—

113

C₂×#103

—

114

#389

—

115

#340

—

116

#336

D₄₃

yes

characteristic

117

C₄₃

#1865

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

119

C₂²×C₈

C₂×D₄₃

yes

characteristic

120

C₂²×C₈

C₂×D₄₃

yes

characteristic

121

C₄×Q₈

—

122

C₄×Q₈

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

—

129

C₂×C₄²

C₂×D₄₃

yes

characteristic

130

#106

—

131

#103

—

132

#103

—

133

#95

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

D₁₇₂

yes

characteristic

145

C₂²×C₄

D₁₇₂

yes

characteristic

146

C₂²×C₄

C₂²×D₄₃

yes

characteristic

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₄×D₄₃

yes

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

172

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₂³

C₂×D₁₇₂

yes

characteristic

165

Q₈

—

166

Q₈

—

167

Q₈

—

168

Q₈

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

172

—

180

C₂×C₄

—

172

—

181

C₂×C₄

—

172

—

182

C₂×C₄

—

172

—

183

C₂×C₄

—

172

—

184

C₂×C₄

—

172

—

185

C₂×C₄

—

172

—

186

C₂×C₄

#10621873

yes

—

187

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

188

C₂×C₄

#10621898

yes

characteristic

189

C₂×C₄

D₄×D₄₃

yes

characteristic

190

C₂×C₄

D₄×D₄₃

yes

characteristic

191

C₂×C₄

#10621900

yes

characteristic

192

C₂×C₄

Q₈×D₄₃

yes

characteristic

193

C₈

—

194

C₈

—

195

C₂²

C₄×D₁₇₂

yes

characteristic

196

C₂²

#13063295

yes

characteristic

197

C₂²

#13063305

yes

characteristic

198

C₂²

C₂×#10621873

yes

characteristic

199

C₂²

#13063347

yes

characteristic

200

C₂²

#13063358

yes

characteristic

201

C₂²

#13063358

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

344

—

206

C₄

—

172

—

207

C₄

—

172

—

208

C₄

C₄⋊C₄×D₄₃

yes

characteristic

209

C₄

(C₂²⋊D₄₃)⋊C₄

yes

characteristic

210

C₄

#13063303

yes

characteristic

211

C₄

#13063307

yes

characteristic

212

C₂

#14582773

yes

characteristic

213

C₂

C₄×#10621873

yes

characteristic

214

C₂

#14583007

yes

characteristic

215

C₂

#14583040

yes

characteristic

216

C₂

#14583063

yes

characteristic

217

C₂

#14583185

yes

characteristic

218

C₂

#14583207

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#336	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#336	D₄₃
hypocenter	C₄₃	#1865
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#336

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
19. #13063250	C₄	semidirect
82. Dic₈₆	C₄⋊C₄	semidirect
117. C₄₃	#1865	semidirect

H ≅ G/N

type

19. #13063250

C₄

semidirect

82. Dic₈₆

C₄⋊C₄

semidirect

117. C₄₃

#1865

semidirect

Subgroups — #19134866

Conjugacy classes of subgroups

Complements