Group #19145602 — subgroups

Subgroups — #19145602

Order 5504.

390 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 121 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	#14583449	C₂	1	1	yes	maximal, characteristic
2	#14583291	C₂	1	2	yes	maximal, complemented
3	#14583261	C₂	1	2	yes	maximal, complemented
4	#14583109	C₂	1	2	yes	maximal, complemented
5	#14583099	C₂	1	2	yes	maximal, complemented
6	#14583059	C₂	1	4	yes	maximal, complemented
7	#14582956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10621898	C₂²	1	2	yes	—
10	C₂×#10621885	C₂²	1	4	yes	—
11	C₂×#10621884	C₂²	1	4	yes	—
12	#13063325	C₂²	1	1	yes	characteristic
13	#13063322	C₂²	1	1	yes	characteristic
14	#13063313	C₂²	1	2	yes	—
15	#13063299	—	2	1	no	—
16	#13063299	—	2	1	no	—
17	C₄×D₁₇₂	—	2	1	no	—
18	C₄×D₁₇₂	—	2	1	no	—
19	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
20	#13063248	C₂²	1	4	yes	—
21	#13063245	C₂²	1	4	yes	—
22	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
23	#13063236	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
26	#125×C₄₃	—	2	1	no	—
27	C₂²⋊Q₈×C₄₃	—	2	1	no	—
28	C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
29	C₄×Q₈×C₄₃	—	2	1	no	—
30	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
31	#117×C₄₃	—	2	1	no	—
32	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×Q₈×C₄₃	—	2	1	no	—
34	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₄₃	—	2	2	no	—
39	#10621898	—	4	4	no	—
40	C₂×D₁₇₂	—	2	1	no	—
41	C₂×D₁₇₂	—	2	1	no	—
42	C₂×D₁₇₂	C₂³	1	2	yes	—
43	C₂×C₄×D₄₃	—	2	2	no	—
44	C₂×Dic₈₆	C₂³	1	2	yes	—
45	C₄⋊C₄×C₄₃	—	2	1	no	—
46	C₄⋊C₄×C₄₃	—	2	1	no	—
47	C₄⋊C₄×C₄₃	—	2	1	no	—
48	C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₄⋊C₄×C₄₃	—	2	2	no	—
50	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₄₃	—	2	2	no	—
52	C₄²×C₄₃	—	2	2	no	—
53	C₄²×C₄₃	D₄	1	1	yes	characteristic
54	C₄²×C₄₃	D₄	1	1	yes	characteristic
55	C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	#10621885	—	4	4	no	—
58	#10621884	—	4	4	no	—
59	(C₂×D₄₃)⋊C₄	—	2	2	no	—
60	(C₂×D₄₃)⋊C₄	—	2	2	no	—
61	C₄⋊Dic₄₃	—	2	1	no	—
62	C₄⋊Dic₄₃	—	2	1	no	—
63	C₂×C₄₃⋊C₈	C₂³	1	4	yes	—
64	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₄₃	—	2	2	no	—
66	Q₈×C₄₃	—	2	2	no	—
67	Q₈×C₄₃	—	2	2	no	—
68	Q₈×C₄₃	—	4	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	—	2	2	no	—
73	C₂×C₄×C₄₃	—	4	1	no	—
74	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₄₃	—	4	4	no	—
81	C₂×Dic₄₃	—	2	2	no	—
82	D₁₇₂	—	2	4	no	—
83	D₁₇₂	—	4	2	no	—
84	D₁₇₂	—	4	2	no	—
85	C₄×D₄₃	—	4	4	no	—
86	Dic₈₆	—	2	4	no	—
87	C₄₃⋊C₈	—	2	4	no	—
88	C₂²×C₄₃	—	2	1	no	—
89	C₂²×C₄₃	—	2	1	no	—
90	C₂²×C₄₃	—	2	1	no	—
91	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×D₄₃	—	4	2	no	—
93	C₂×D₄₃	—	4	2	no	—
94	C₂×D₄₃	—	4	2	no	—
95	C₄×C₄₃	—	2	1	no	—
96	C₄×C₄₃	—	2	1	no	—
97	C₄×C₄₃	—	2	1	no	—
98	C₄×C₄₃	—	2	1	no	—
99	C₄×C₄₃	—	4	2	no	—
100	C₄×C₄₃	—	4	2	no	—
101	C₄×C₄₃	#137	1	2	yes	—
102	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
103	Dic₄₃	—	4	2	no	—
104	#3216	—	43	1	no	maximal, Sylow
105	C₂×C₄₃	—	4	1	no	—
106	C₂×C₄₃	#535	1	1	yes	characteristic
107	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
108	C₂×C₄₃	#575	1	1	yes	characteristic
109	D₄₃	—	8	2	no	—
110	#533	D₄₃	1	1	yes	characteristic, complemented
111	#532	—	43	1	no	—
112	#497	—	43	2	no	—
113	#489	—	43	2	no	—
114	#462	—	43	2	no	—
115	#460	—	43	2	no	—
116	#452	—	43	4	no	—
117	#432	—	43	1	no	—
118	C₄₃	#3216	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	—	43	2	no	—
120	C₂×Q₁₆	—	43	4	no	—
121	C₂×SD₁₆	—	43	4	no	—
122	C₄⋊Q₈	—	43	1	no	—
123	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
125	#127	—	43	1	no	—
126	#125	—	2	1	no	—
127	#124	—	43	2	no	—
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂²⋊Q₈	C₂×D₄₃	1	2	yes	—
130	#121	—	86	1	no	—
131	#121	—	86	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	C₂×D₄₃	1	2	yes	—
134	C₄×D₄	—	86	1	no	—
135	C₄×D₄	—	86	1	no	—
136	#117	—	2	1	no	—
137	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
138	C₄⋊C₈	—	43	2	no	—
139	#103	—	43	4	no	—
140	#102	—	43	4	no	—
141	C₂²⋊C₈	—	43	2	no	—
142	#97	—	43	2	no	—
143	#41	—	172	4	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	43	2	no	—
146	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
147	C₂×D₄	—	43	2	no	—
148	C₂×D₄	—	86	2	no	—
149	C₂×D₄	—	86	1	no	—
150	C₂×D₄	—	86	1	no	—
151	C₂²×C₄	—	86	2	no	—
152	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
155	Q₁₆	—	172	4	no	—
156	SD₁₆	—	172	4	no	—
157	C₂×C₈	—	43	4	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	86	1	no	—
164	C₄⋊C₄	—	86	1	no	—
165	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	86	2	no	—
168	C₂²⋊C₄	—	86	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
171	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
172	C₄²	C₂²×D₄₃	1	1	yes	characteristic
173	C₄²	C₂²×D₄₃	1	1	yes	characteristic
174	C₂³	—	86	2	no	—
175	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	4	2	no	—
179	Q₈	—	86	4	no	—
180	D₄	—	86	4	no	—
181	D₄	—	172	2	no	—
182	D₄	—	172	4	no	—
183	D₄	—	172	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	86	2	no	—
190	C₂×C₄	—	172	4	no	—
191	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
192	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
196	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
197	C₈	—	86	4	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	172	2	no	—
202	C₂²	—	172	2	no	—
203	C₂²	—	172	2	no	—
204	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	172	2	no	—
212	C₄	#13063356	1	2	yes	—
213	C₄	#13063371	1	2	yes	—
214	C₂	—	4	1	no	—
215	C₂	—	344	2	no	—
216	C₂	#14583463	1	1	yes	characteristic
217	C₂	C₂×#13063356	1	1	yes	characteristic
218	C₂	#14583697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

#14583291

C₂

yes

maximal, complemented

#14583261

C₂

yes

maximal, complemented

#14583109

C₂

yes

maximal, complemented

#14583099

C₂

yes

maximal, complemented

#14583059

C₂

yes

maximal, complemented

#14582956

C₂

yes

maximal, characteristic, complemented

C₄₃×#533

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₂²

yes

—

C₂×#10621885

C₂²

yes

—

C₂×#10621884

C₂²

yes

—

#13063325

C₂²

yes

characteristic

#13063322

C₂²

yes

characteristic

#13063313

C₂²

yes

—

#13063299

—

#13063299

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

#13063248

C₂²

yes

—

#13063245

C₂²

yes

—

C₂²⋊(C₄₃⋊C₈)

C₂²

yes

—

#13063236

C₂²

yes

—

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

#125×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

C₂²

yes

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

C₂²

yes

—

#117×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

C₂³

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

#10621885

—

#10621884

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

C₂³

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

#137

yes

—

102

C₄×C₄₃

H(5,𝔽₂)

yes

—

103

Dic₄₃

—

104

#3216

—

maximal, Sylow

105

C₂×C₄₃

—

106

C₂×C₄₃

#535

yes

characteristic

107

C₂×C₄₃

C₂×#137

yes

characteristic

108

C₂×C₄₃

#575

yes

characteristic

109

D₄₃

—

110

#533

D₄₃

yes

characteristic, complemented

111

#532

—

112

#497

—

113

#489

—

114

#462

—

115

#460

—

116

#452

—

117

#432

—

118

C₄₃

#3216

yes

characteristic, complemented, Sylow

119

C₂×#41

—

120

C₂×Q₁₆

—

121

C₂×SD₁₆

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

124

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

125

#127

—

126

#125

—

127

#124

—

128

C₂²⋊Q₈

—

129

C₂²⋊Q₈

C₂×D₄₃

yes

—

130

#121

—

131

#121

—

132

C₄×Q₈

—

133

C₄×Q₈

C₂×D₄₃

yes

—

134

C₄×D₄

—

135

C₄×D₄

—

136

#117

—

137

C₂×C₄²

C₂×D₄₃

yes

characteristic

138

C₄⋊C₈

—

139

#103

—

140

#102

—

141

C₂²⋊C₈

—

142

#97

—

143

#41

—

172

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

C₂²×D₄₃

yes

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

153

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

154

C₂²×C₄

C₂²×D₄₃

yes

characteristic

155

Q₁₆

—

172

—

156

SD₁₆

—

172

—

157

C₂×C₈

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

C₂²×D₄₃

yes

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

—

170

C₄²

C₂²⋊D₄₃

yes

characteristic

171

C₄²

C₂²⋊D₄₃

yes

characteristic

172

C₄²

C₂²×D₄₃

yes

characteristic

173

C₄²

C₂²×D₄₃

yes

characteristic

174

C₂³

—

175

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

176

Q₈

—

177

Q₈

—

178

Q₈

—

179

Q₈

—

180

D₄

—

181

D₄

—

172

—

182

D₄

—

172

—

183

D₄

—

172

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

172

—

191

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

192

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

193

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

194

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

195

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

196

C₂×C₄

C₂³×D₄₃

yes

characteristic

197

C₈

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

172

—

202

C₂²

—

172

—

203

C₂²

—

172

—

204

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

172

—

212

C₄

#13063356

yes

—

213

C₄

#13063371

yes

—

214

C₂

—

215

C₂

—

344

—

216

C₂

#14583463

yes

characteristic

217

C₂

C₂×#13063356

yes

characteristic

218

C₂

#14583697

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#533	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#533	D₄₃
hypocenter	C₄₃	#3216
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#533

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14583291	C₂	semidirect
3. #14583261	C₂	semidirect
4. #14583109	C₂	semidirect
5. #14583099	C₂	semidirect
6. #14583059	C₂	semidirect
7. #14582956	C₂	semidirect
8. C₄₃×#533	C₂	semidirect
110. #533	D₄₃	semidirect
118. C₄₃	#3216	semidirect

H ≅ G/N

type

2. #14583291

C₂

semidirect

3. #14583261

C₂

semidirect

4. #14583109

C₂

semidirect

5. #14583099

C₂

semidirect

6. #14583059

C₂

semidirect

7. #14582956

C₂

semidirect

8. C₄₃×#533

C₂

semidirect

110. #533

D₄₃

semidirect

118. C₄₃

#3216

semidirect

Subgroups — #19145602

Conjugacy classes of subgroups

Complements