Group #19132967 — subgroups

Subgroups — C₂×#14582626 #19132967

Order 5504.

344 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 125 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×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582626	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₇₂	C₄	1	1	yes	characteristic
6	C₂²×C₄×D₄₃	—	2	1	no	—
7	C₂²×C₄₃⋊C₈	—	2	1	no	—
8	C₄×D₁₇₂	—	2	1	no	—
9	C₄×D₁₇₂	—	2	1	no	—
10	C₄×D₁₇₂	C₄	1	2	yes	—
11	C₄×D₁₇₂	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
13	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
14	C₄×C₄₃⋊C₈	—	2	1	no	—
15	C₄×C₄₃⋊C₈	C₂²	1	2	yes	complemented
16	C₂²×C₈×C₄₃	—	2	1	no	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₄₃	—	2	1	no	—
19	C₄⋊C₈×C₄₃	C₂²	1	2	yes	complemented
20	C₂³×D₄₃	—	2	1	no	—
21	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×Dic₄₃	—	2	1	no	—
25	C₂×D₁₇₂	—	2	1	no	—
26	C₂×D₁₇₂	—	2	2	no	—
27	C₂×D₁₇₂	C₈	1	4	yes	complemented
28	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₄₃	—	2	2	no	—
30	C₂×C₄×D₄₃	—	2	2	no	—
31	C₂×C₄×D₄₃	—	4	2	no	—
32	C₂×C₄×D₄₃	—	4	2	no	—
33	C₂×C₈×C₄₃	—	2	2	no	—
34	C₂×C₈×C₄₃	—	4	2	no	—
35	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₃	D₄	1	1	yes	characteristic
37	C₄²×C₄₃	D₄	1	1	yes	characteristic
38	C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	(C₂×D₄₃)⋊C₄	—	2	2	no	—
40	(C₂×D₄₃)⋊C₄	—	2	2	no	—
41	C₄⋊Dic₄₃	—	2	1	no	—
42	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄₃⋊C₈	—	2	4	no	—
44	C₂×C₄₃⋊C₈	—	2	2	no	—
45	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×D₄₃	—	2	4	no	—
47	C₂²×D₄₃	—	2	2	no	—
48	C₂²×D₄₃	—	4	2	no	—
49	C₂²×D₄₃	—	4	2	no	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
56	C₂×C₄×C₄₃	D₈	1	2	yes	—
57	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
58	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₄₃	—	2	2	no	—
62	C₂×Dic₄₃	—	4	2	no	—
63	D₁₇₂	—	2	4	no	—
64	D₁₇₂	—	2	2	no	—
65	D₁₇₂	C₂×C₈	1	4	yes	complemented
66	C₄×D₄₃	—	4	4	no	—
67	C₄×D₄₃	—	4	2	no	—
68	C₄×D₄₃	—	4	2	no	—
69	C₈×C₄₃	—	4	2	no	—
70	C₄₃⋊C₈	—	2	4	no	—
71	C₂²×C₄₃	C₂²⋊C₈	1	2	yes	—
72	C₂²×C₄₃	#102	1	2	yes	—
73	C₂²×C₄₃	#104	1	2	yes	—
74	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×D₄₃	—	2	4	no	—
76	C₂×D₄₃	—	4	8	no	—
77	C₂×D₄₃	—	4	2	no	—
78	C₂×D₄₃	—	4	2	no	—
79	C₄×C₄₃	—	2	1	no	—
80	C₄×C₄₃	—	2	1	no	—
81	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
82	C₄×C₄₃	#102	1	2	yes	—
83	C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
84	C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
85	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
86	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
87	Dic₄₃	—	4	2	no	—
88	C₂×D₄⋊C₈	—	43	1	no	maximal, Sylow
89	C₂×C₄₃	D₄⋊C₈	1	4	yes	complemented
90	C₂×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
92	C₂×C₄₃	C₂×#104	1	1	yes	characteristic
93	D₄₃	—	4	4	no	—
94	C₂×C₄×D₄	—	43	1	no	—
95	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
96	C₂×C₄×C₈	—	43	1	no	—
97	D₄⋊C₈	—	43	4	no	—
98	C₄₃	C₂×D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
99	C₂²×D₄	—	43	1	no	—
100	C₂³×C₄	—	86	1	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	86	1	no	—
103	C₄×D₄	—	43	2	no	—
104	C₄×D₄	—	43	2	no	—
105	C₄×D₄	—	86	1	no	—
106	C₄×D₄	—	86	1	no	—
107	C₂×C₄⋊C₄	—	43	1	no	—
108	C₂×C₂²⋊C₄	—	86	1	no	—
109	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
110	C₄⋊C₈	—	2	1	no	—
111	C₄⋊C₈	C₂×D₄₃	1	2	yes	complemented
112	C₄×C₈	—	43	2	no	—
113	C₄×C₈	—	86	1	no	—
114	C₂⁴	—	86	1	no	—
115	C₂×D₄	—	43	4	no	—
116	C₂×D₄	—	43	2	no	—
117	C₂×D₄	—	86	2	no	—
118	C₂×D₄	—	86	1	no	—
119	C₂²×C₄	—	86	1	no	—
120	C₂²×C₄	—	86	2	no	—
121	C₂²×C₄	—	86	2	no	—
122	C₂²×C₄	—	172	2	no	—
123	C₂²×C₄	—	172	2	no	—
124	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
125	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	86	4	no	—
130	C₂×C₈	—	86	2	no	—
131	C₄⋊C₄	—	43	2	no	—
132	C₄⋊C₄	—	86	1	no	—
133	C₂²⋊C₄	—	86	2	no	—
134	C₂²⋊C₄	—	86	2	no	—
135	C₄²	C₄×D₄₃	1	1	yes	characteristic
136	C₄²	D₁₇₂	1	1	yes	characteristic
137	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
138	C₄²	C₂²×D₄₃	1	1	yes	characteristic
139	C₂³	—	86	4	no	—
140	C₂³	—	86	2	no	—
141	C₂³	—	172	2	no	—
142	C₂³	—	172	2	no	—
143	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
144	D₄	—	43	4	no	—
145	D₄	—	86	4	no	—
146	D₄	—	86	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	86	2	no	—
149	C₂×C₄	—	172	2	no	—
150	C₂×C₄	—	172	4	no	—
151	C₂×C₄	—	172	2	no	—
152	C₂×C₄	—	172	2	no	—
153	C₂×C₄	C₈×D₄₃	1	2	yes	—
154	C₂×C₄	#10621872	1	2	yes	—
155	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
156	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
157	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
158	C₂×C₄	#10621882	1	2	yes	—
159	C₂×C₄	#10621884	1	2	yes	—
160	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
161	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
163	C₈	—	4	2	no	—
164	C₈	—	86	4	no	—
165	C₂²	—	86	4	no	—
166	C₂²	—	172	8	no	—
167	C₂²	—	172	2	no	—
168	C₂²	—	172	2	no	—
169	C₂²	(C₂×D₄₃)⋊C₈	1	2	yes	—
170	C₂²	#13063245	1	2	yes	—
171	C₂²	#13063251	1	2	yes	—
172	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	172	2	no	—
176	C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
177	C₄	#13063245	1	2	yes	—
178	C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
179	C₄	C₂×#10621872	1	1	yes	characteristic
180	C₄	C₂×#10621882	1	1	yes	characteristic
181	C₄	C₂×#10621884	1	1	yes	characteristic
182	C₂	—	172	4	no	—
183	C₂	#14582626	1	4	yes	direct product factor
184	C₂	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
185	C₂	C₂×#13063245	1	1	yes	characteristic
186	C₂	C₂×#13063251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₁₇₂

C₂

yes

maximal, characteristic

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic, complemented

#14582626

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₄

yes

characteristic

C₂²×C₄×D₄₃

—

C₂²×C₄₃⋊C₈

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₄

yes

—

C₄×D₁₇₂

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₄

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

C₂²

yes

complemented

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

C₂²

yes

complemented

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₈

yes

complemented

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

D₈

yes

—

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×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

C₂×C₈

yes

complemented

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂²⋊C₈

yes

—

C₂²×C₄₃

#102

yes

—

C₂²×C₄₃

#104

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

#102

yes

—

C₄×C₄₃

C₂²×C₈

yes

characteristic

C₄×C₄₃

C₂×M₁₆

yes

characteristic

C₄×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

Dic₄₃

—

C₂×D₄⋊C₈

—

maximal, Sylow

C₂×C₄₃

D₄⋊C₈

yes

complemented

C₂×C₄₃

C₂×C₂²⋊C₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×C₄×D₄

—

C₂×C₄⋊C₈

D₄₃

yes

characteristic, complemented

C₂×C₄×C₈

—

D₄⋊C₈

—

C₄₃

C₂×D₄⋊C₈

yes

characteristic, complemented, Sylow

C₂²×D₄

—

100

C₂³×C₄

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₄×D₄

—

104

C₄×D₄

—

105

C₄×D₄

—

106

C₄×D₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₄²

C₂×D₄₃

yes

characteristic

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×D₄₃

yes

complemented

112

C₄×C₈

—

113

C₄×C₈

—

114

C₂⁴

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

172

—

123

C₂²×C₄

—

172

—

124

C₂²×C₄

C₄×D₄₃

yes

characteristic

125

C₂²×C₄

D₁₇₂

yes

characteristic

126

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

C₄×D₄₃

yes

characteristic

136

C₄²

D₁₇₂

yes

characteristic

137

C₄²

C₂²⋊D₄₃

yes

characteristic

138

C₄²

C₂²×D₄₃

yes

characteristic

139

C₂³

—

140

C₂³

—

141

C₂³

—

172

—

142

C₂³

—

172

—

143

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

144

D₄

—

145

D₄

—

146

D₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

172

—

150

C₂×C₄

—

172

—

151

C₂×C₄

—

172

—

152

C₂×C₄

—

172

—

153

C₂×C₄

C₈×D₄₃

yes

—

154

C₂×C₄

#10621872

yes

—

155

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

156

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

157

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

158

C₂×C₄

#10621882

yes

—

159

C₂×C₄

#10621884

yes

—

160

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

161

C₂×C₄

C₂×D₁₇₂

yes

characteristic

162

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

163

C₈

—

164

C₈

—

165

C₂²

—

166

C₂²

—

172

—

167

C₂²

—

172

—

168

C₂²

—

172

—

169

C₂²

(C₂×D₄₃)⋊C₈

yes

—

170

C₂²

#13063245

yes

—

171

C₂²

#13063251

yes

—

172

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

172

—

176

C₄

(C₂×D₄₃)⋊C₈

yes

—

177

C₄

#13063245

yes

—

178

C₄

C₂×C₈×D₄₃

yes

characteristic

179

C₄

C₂×#10621872

yes

characteristic

180

C₄

C₂×#10621882

yes

characteristic

181

C₄

C₂×#10621884

yes

characteristic

182

C₂

—

172

—

183

C₂

#14582626

yes

direct product factor

184

C₂

C₂×(C₂×D₄₃)⋊C₈

yes

characteristic

185

C₂

C₂×#13063245

yes

characteristic

186

C₂

C₂×#13063251

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂²×C₈
Frattini	C₄²	C₂²×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	C₂×D₄⋊C₈
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₄₃

commutator

C₄×C₄₃

C₂²×C₈

Frattini

C₄²

C₂²×D₄₃

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂²⋊C₄

hypercenter

C₂×C₄⋊C₈

D₄₃

hypocenter

C₄₃

C₂×D₄⋊C₈

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. C₂×C₄×C₄₃⋊C₈	C₂	semidirect
3. #14582626	C₂	direct (N × H)
4. C₂×C₄⋊C₈×C₄₃	C₂	semidirect
15. C₄×C₄₃⋊C₈	C₂²	semidirect
19. C₄⋊C₈×C₄₃	C₂²	semidirect
27. C₂×D₁₇₂	C₈	semidirect
65. D₁₇₂	C₂×C₈	semidirect
89. C₂×C₄₃	D₄⋊C₈	semidirect
95. C₂×C₄⋊C₈	D₄₃	semidirect
98. C₄₃	C₂×D₄⋊C₈	semidirect
111. C₄⋊C₈	C₂×D₄₃	semidirect
183. C₂	#14582626	direct (N × H)

H ≅ G/N

type

2. C₂×C₄×C₄₃⋊C₈

C₂

semidirect

3. #14582626

C₂

direct (N × H)

4. C₂×C₄⋊C₈×C₄₃

C₂

semidirect

15. C₄×C₄₃⋊C₈

C₂²

semidirect

19. C₄⋊C₈×C₄₃

C₂²

semidirect

27. C₂×D₁₇₂

C₈

semidirect

65. D₁₇₂

C₂×C₈

semidirect

89. C₂×C₄₃

D₄⋊C₈

semidirect

95. C₂×C₄⋊C₈

D₄₃

semidirect

98. C₄₃

C₂×D₄⋊C₈

semidirect

111. C₄⋊C₈

C₂×D₄₃

semidirect

183. C₂

#14582626

direct (N × H)

Subgroups — C₂×#14582626 #19132967

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14582626 #19132967

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14582626 #19132967