Group #21433993 — subgroups

Subgroups — C₂×#16837718 #21433993

Order 7552.

512 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837718	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14273658	C₂	1	2	yes	maximal, complemented
4	C₂×#14273648	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#121×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₅₉	—	2	1	no	—
8	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₅₉	—	2	2	no	—
10	C₂×C₄⋊Dic₅₉	—	2	1	no	—
11	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅₉	C₂²	1	4	yes	complemented
13	#14273658	C₂²	1	8	yes	complemented
14	#14273648	C₂²	1	8	yes	complemented
15	C₂²⋊(C₅₉⋊C₈)	C₂²	1	4	yes	complemented
16	C₂²×D₄×C₅₉	—	2	1	no	—
17	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₅₉	C₂²	1	1	yes	characteristic
19	#121×C₅₉	—	2	2	no	—
20	#121×C₅₉	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
23	C₂⁴×C₅₉	—	2	1	no	—
24	C₂⁴×C₅₉	D₄	1	1	yes	characteristic
25	C₂×D₄×C₅₉	—	2	4	no	—
26	C₂×D₄×C₅₉	—	2	1	no	—
27	C₂×D₄×C₅₉	—	2	4	no	—
28	C₂×D₄×C₅₉	—	4	4	no	—
29	C₂×D₄×C₅₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₅₉	—	2	4	no	—
31	C₂²×C₄×C₅₉	—	2	1	no	—
32	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	D₄	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	2	2	no	—
37	C₂²×Dic₅₉	—	2	4	no	—
38	C₂²×Dic₅₉	—	4	4	no	—
39	C₄⋊C₄×C₅₉	—	2	1	no	—
40	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
41	C₂²⋊C₄×C₅₉	—	2	4	no	—
42	C₄⋊Dic₅₉	—	2	2	no	—
43	C₄⋊Dic₅₉	—	2	4	no	—
44	C₄⋊Dic₅₉	C₂³	1	4	yes	—
45	C₂×C₅₉⋊C₈	—	2	4	no	—
46	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
47	C₂³×C₅₉	—	2	2	no	—
48	C₂³×C₅₉	—	2	2	no	—
49	C₂³×C₅₉	—	2	2	no	—
50	C₂³×C₅₉	—	4	2	no	—
51	C₂³×C₅₉	—	4	2	no	—
52	C₂³×C₅₉	—	4	2	no	—
53	C₂³×C₅₉	SD₁₆	1	4	yes	—
54	C₂³×C₅₉	C₂×D₄	1	2	yes	—
55	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
56	D₄×C₅₉	—	2	4	no	—
57	D₄×C₅₉	—	4	8	no	—
58	D₄×C₅₉	—	4	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	4	no	—
61	C₂×C₄×C₅₉	—	2	4	no	—
62	C₂×C₄×C₅₉	—	2	2	no	—
63	C₂×C₄×C₅₉	—	4	2	no	—
64	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#41	1	4	yes	—
67	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₅₉	—	2	4	no	—
69	C₂×Dic₅₉	—	4	8	no	—
70	C₂×Dic₅₉	—	4	4	no	—
71	C₅₉⋊C₈	—	2	4	no	—
72	C₂²×C₅₉	—	2	2	no	—
73	C₂²×C₅₉	—	2	2	no	—
74	C₂²×C₅₉	—	2	8	no	—
75	C₂²×C₅₉	—	4	2	no	—
76	C₂²×C₅₉	—	4	2	no	—
77	C₂²×C₅₉	—	4	2	no	—
78	C₂²×C₅₉	—	8	4	no	—
79	C₂²×C₅₉	#123	1	2	yes	—
80	C₂²×C₅₉	C₂×SD₁₆	1	4	yes	—
81	C₂²×C₅₉	C₂×SD₁₆	1	2	yes	—
82	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₅₉	—	2	2	no	—
85	C₄×C₅₉	—	4	2	no	—
86	C₄×C₅₉	#123	1	2	yes	—
87	C₄×C₅₉	C₂×#41	1	2	yes	—
88	Dic₅₉	—	4	4	no	—
89	C₂×#481	—	59	1	no	maximal, Sylow
90	C₂×C₅₉	—	2	4	no	—
91	C₂×C₅₉	—	8	2	no	—
92	C₂×C₅₉	#481	1	4	yes	complemented
93	C₂×C₅₉	C₂×#123	1	1	yes	characteristic
94	C₂×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
95	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×#121	D₅₉	1	1	yes	characteristic
97	C₂²×C₄⋊C₄	—	59	1	no	—
98	#481	—	59	8	no	—
99	C₂×#106	—	59	2	no	—
100	C₂×#102	—	59	2	no	—
101	C₂×C₂²⋊C₈	—	59	1	no	—
102	C₅₉	C₂×#481	1	1	yes	characteristic, complemented, Sylow
103	C₂²×D₄	—	2	1	no	—
104	C₂²×D₄	C₂×D₅₉	1	1	yes	characteristic
105	C₂³×C₄	—	118	1	no	—
106	C₂³×C₄	C₂×D₅₉	1	1	yes	characteristic
107	C₂²×C₈	—	59	2	no	—
108	#121	—	2	2	no	—
109	#121	C₂×D₅₉	1	4	yes	—
110	C₂×C₄⋊C₄	—	59	2	no	—
111	C₂×C₄⋊C₄	—	59	4	no	—
112	C₂×C₄⋊C₄	—	118	2	no	—
113	C₂×C₄⋊C₄	—	118	1	no	—
114	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
115	C₂×C₂²⋊C₄	—	2	1	no	—
116	#106	—	59	8	no	—
117	#102	—	59	8	no	—
118	C₂²⋊C₈	—	59	4	no	—
119	C₂⁴	—	2	1	no	—
120	C₂⁴	C₂²⋊D₅₉	1	1	yes	characteristic
121	C₂×D₄	—	2	4	no	—
122	C₂×D₄	—	2	4	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	4	4	no	—
125	C₂×D₄	C₂²×D₅₉	1	2	yes	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	118	4	no	—
129	C₂²×C₄	—	118	2	no	—
130	C₂²×C₄	—	236	4	no	—
131	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
132	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
134	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
135	C₂×C₈	—	59	4	no	—
136	C₂×C₈	—	118	4	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	59	4	no	—
139	C₄⋊C₄	—	118	4	no	—
140	C₄⋊C₄	—	118	2	no	—
141	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
142	C₂²⋊C₄	—	2	4	no	—
143	C₂³	—	2	2	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	4	2	no	—
147	C₂³	—	4	2	no	—
148	C₂³	—	4	2	no	—
149	C₂³	#11745670	1	4	yes	—
150	C₂³	C₂×C₂²⋊D₅₉	1	2	yes	—
151	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
152	D₄	—	2	4	no	—
153	D₄	—	4	8	no	—
154	D₄	—	4	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	118	4	no	—
161	C₂×C₄	—	236	8	no	—
162	C₂×C₄	—	236	4	no	—
163	C₂×C₄	#11745687	1	4	yes	—
164	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
165	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
166	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
167	C₈	—	118	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	8	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	8	4	no	—
175	C₂²	#14273708	1	2	yes	—
176	C₂²	C₂×#11745670	1	4	yes	—
177	C₂²	C₂×#11745670	1	2	yes	—
178	C₂²	#14273752	1	2	yes	—
179	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	236	4	no	—
183	C₄	#14273708	1	2	yes	—
184	C₄	C₂×#11745687	1	2	yes	—
185	C₂	—	2	4	no	—
186	C₂	—	8	2	no	—
187	C₂	#16837718	1	4	yes	direct product factor
188	C₂	C₂×#14273708	1	1	yes	characteristic
189	C₂	C₂²×#11745670	1	1	yes	characteristic
190	C₂	C₂×#14273752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic, complemented

#16837718

C₂

yes

maximal, direct product factor

C₂×#14273658

C₂

yes

maximal, complemented

C₂×#14273648

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×#121×C₅₉

C₂

yes

maximal, characteristic

C₂³×Dic₅₉

—

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

complemented

#14273658

C₂²

yes

complemented

#14273648

C₂²

yes

complemented

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

complemented

C₂²×D₄×C₅₉

—

C₂²×D₄×C₅₉

C₂²

yes

characteristic

C₂³×C₄×C₅₉

C₂²

yes

characteristic

#121×C₅₉

—

#121×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₉

—

C₂⁴×C₅₉

—

C₂⁴×C₅₉

D₄

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

SD₁₆

yes

—

C₂³×C₅₉

C₂×D₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#123

yes

—

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#123

yes

—

C₄×C₅₉

C₂×#41

yes

—

Dic₅₉

—

C₂×#481

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

C₂²×SD₁₆

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

C₂²×C₄⋊C₄

—

#481

—

C₂×#106

—

100

C₂×#102

—

101

C₂×C₂²⋊C₈

—

102

C₅₉

C₂×#481

yes

characteristic, complemented, Sylow

103

C₂²×D₄

—

104

C₂²×D₄

C₂×D₅₉

yes

characteristic

105

C₂³×C₄

—

118

—

106

C₂³×C₄

C₂×D₅₉

yes

characteristic

107

C₂²×C₈

—

108

#121

—

109

#121

C₂×D₅₉

yes

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

118

—

113

C₂×C₄⋊C₄

—

118

—

114

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

115

C₂×C₂²⋊C₄

—

116

#106

—

117

#102

—

118

C₂²⋊C₈

—

119

C₂⁴

—

120

C₂⁴

C₂²⋊D₅₉

yes

characteristic

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

C₂²×D₅₉

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

118

—

129

C₂²×C₄

—

118

—

130

C₂²×C₄

—

236

—

131

C₂²×C₄

C₂²⋊D₅₉

yes

—

132

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

133

C₂²×C₄

C₂²×D₅₉

yes

—

134

C₂²×C₄

C₂²×D₅₉

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

118

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

118

—

140

C₄⋊C₄

—

118

—

141

C₄⋊C₄

C₂²×D₅₉

yes

—

142

C₂²⋊C₄

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

#11745670

yes

—

150

C₂³

C₂×C₂²⋊D₅₉

yes

—

151

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

152

D₄

—

153

D₄

—

154

D₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

118

—

161

C₂×C₄

—

236

—

162

C₂×C₄

—

236

—

163

C₂×C₄

#11745687

yes

—

164

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

165

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

166

C₂×C₄

C₂³×D₅₉

yes

characteristic

167

C₈

—

118

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#14273708

yes

—

176

C₂²

C₂×#11745670

yes

—

177

C₂²

C₂×#11745670

yes

—

178

C₂²

#14273752

yes

—

179

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

236

—

183

C₄

#14273708

yes

—

184

C₄

C₂×#11745687

yes

—

185

C₂

—

186

C₂

—

187

C₂

#16837718

yes

direct product factor

188

C₂

C₂×#14273708

yes

characteristic

189

C₂

C₂²×#11745670

yes

characteristic

190

C₂

C₂×#14273752

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×#121×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#121	D₅₉
hypocenter	C₅₉	C₂×#481
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×#121×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

N	H ≅ G/N	type
1. C₂²×C₄⋊Dic₅₉	C₂	semidirect
2. #16837718	C₂	direct (N × H)
3. C₂×#14273658	C₂	semidirect
4. C₂×#14273648	C₂	semidirect
5. C₂×C₂²⋊(C₅₉⋊C₈)	C₂	semidirect
12. C₂×C₄⋊Dic₅₉	C₂²	semidirect
13. #14273658	C₂²	semidirect
14. #14273648	C₂²	semidirect
15. C₂²⋊(C₅₉⋊C₈)	C₂²	semidirect
92. C₂×C₅₉	#481	semidirect
102. C₅₉	C₂×#481	semidirect
187. C₂	#16837718	direct (N × H)

H ≅ G/N

type

1. C₂²×C₄⋊Dic₅₉

C₂

semidirect

2. #16837718

C₂

direct (N × H)

3. C₂×#14273658

C₂

semidirect

4. C₂×#14273648

C₂

semidirect

5. C₂×C₂²⋊(C₅₉⋊C₈)

C₂

semidirect

12. C₂×C₄⋊Dic₅₉

C₂²

semidirect

13. #14273658

C₂²

semidirect

14. #14273648

C₂²

semidirect

15. C₂²⋊(C₅₉⋊C₈)

C₂²

semidirect

92. C₂×C₅₉

#481

semidirect

102. C₅₉

C₂×#481

semidirect

187. C₂

#16837718

direct (N × H)

Subgroups — C₂×#16837718 #21433993

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837718 #21433993

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837718 #21433993