Group #21433866 — subgroups

Subgroups — C₂×#16837696 #21433866

Order 7552.

416 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14273726	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273692	C₂	1	1	yes	maximal, characteristic
3	#16837696	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14273660	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273652	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273650	C₂	1	1	yes	maximal, characteristic
8	C₂×#107×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₁₈	—	2	1	no	—
10	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	#14273726	—	2	2	no	—
13	#14273726	C₂²	1	4	yes	—
14	#14273692	—	2	2	no	—
15	#14273692	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₉⋊C₄	—	2	1	no	—
18	C₂×Dic₅₉⋊C₄	—	2	1	no	—
19	C₂×C₄×Dic₅₉	—	2	1	no	—
20	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
21	#14273660	C₂²	1	4	yes	—
22	Dic₅₉⋊C₈	C₂²	1	4	yes	—
23	#14273652	C₂²	1	4	yes	—
24	#14273650	C₂²	1	4	yes	—
25	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
28	#107×C₅₉	C₂²	1	4	yes	—
29	C₂²×C₄×C₅₉	—	2	1	no	—
30	C₂²×C₄×C₅₉	—	2	1	no	—
31	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₉	—	2	1	no	—
33	C₂²×Dic₅₉	—	2	1	no	—
34	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
35	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
36	C₂×Dic₁₁₈	—	2	1	no	—
37	C₂×Dic₁₁₈	—	2	2	no	—
38	C₂×Dic₁₁₈	—	2	4	no	—
39	C₂×Dic₁₁₈	—	4	4	no	—
40	C₂×Dic₁₁₈	Q₈	1	4	yes	complemented
41	C₂×Dic₁₁₈	C₂³	1	2	yes	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	C₂³	1	2	yes	—
44	C₄⋊C₄×C₅₉	—	2	1	no	—
45	C₄⋊C₄×C₅₉	—	2	1	no	—
46	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
47	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
48	C₄⋊Dic₅₉	—	2	1	no	—
49	C₄⋊Dic₅₉	C₂³	1	2	yes	—
50	Dic₅₉⋊C₄	—	2	4	no	—
51	Dic₅₉⋊C₄	—	2	4	no	—
52	C₄×Dic₅₉	—	2	4	no	—
53	C₄×Dic₅₉	D₄	1	2	yes	—
54	C₄×Dic₅₉	C₂³	1	2	yes	—
55	C₂×C₅₉⋊C₈	—	2	2	no	—
56	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
57	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	4	2	no	—
61	C₂×C₄×C₅₉	—	4	2	no	—
62	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₅₉	#41	1	2	yes	—
65	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₅₉	—	2	4	no	—
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	—	2	2	no	—
69	C₂×Dic₅₉	—	2	2	no	—
70	C₂×Dic₅₉	—	4	2	no	—
71	C₂×Dic₅₉	Q₁₆	1	4	yes	—
72	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
73	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
74	Dic₁₁₈	—	2	4	no	—
75	Dic₁₁₈	—	2	2	no	—
76	Dic₁₁₈	—	4	8	no	—
77	Dic₁₁₈	C₂×Q₈	1	4	yes	complemented
78	C₈×C₅₉	—	2	2	no	—
79	C₅₉⋊C₈	—	2	2	no	—
80	C₂²×C₅₉	C₂²⋊Q₈	1	2	yes	—
81	C₂²×C₅₉	C₂×Q₁₆	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	—	4	2	no	—
85	C₄×C₅₉	—	4	2	no	—
86	C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
87	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
88	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
89	Dic₅₉	—	2	4	no	—
90	Dic₅₉	—	2	2	no	—
91	Dic₅₉	—	4	2	no	—
92	Dic₅₉	C₂×Q₁₆	1	4	yes	—
93	C₂×#477	—	59	1	no	maximal, Sylow
94	C₂×C₅₉	#477	1	4	yes	complemented
95	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
97	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄⋊Q₈	—	59	1	no	—
99	C₂×C₄×Q₈	—	59	1	no	—
100	#477	—	59	8	no	—
101	C₂×#107	—	59	1	no	—
102	C₂×#107	D₅₉	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	59	1	no	—
104	C₂×#103	—	59	1	no	—
105	C₂×#103	—	59	1	no	—
106	C₅₉	C₂×#477	1	1	yes	characteristic, complemented, Sylow
107	C₂²×Q₈	—	59	1	no	—
108	C₂²×Q₈	—	118	1	no	—
109	C₂²×C₈	—	59	1	no	—
110	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₄⋊Q₈	—	59	4	no	—
112	C₄⋊Q₈	—	118	2	no	—
113	C₄×Q₈	—	59	4	no	—
114	C₄×Q₈	—	118	2	no	—
115	C₂×C₄⋊C₄	—	59	1	no	—
116	C₂×C₄⋊C₄	—	118	1	no	—
117	C₂×C₄⋊C₄	—	118	1	no	—
118	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
120	C₂×C₄²	—	59	1	no	—
121	C₂×C₄²	—	118	1	no	—
122	#107	—	59	4	no	—
123	#107	C₂×D₅₉	1	4	yes	—
124	C₄⋊C₈	—	59	4	no	—
125	#103	—	59	4	no	—
126	#103	—	59	4	no	—
127	C₂×Q₈	—	59	4	no	—
128	C₂×Q₈	—	59	2	no	—
129	C₂×Q₈	—	118	4	no	—
130	C₂×Q₈	—	118	2	no	—
131	C₂×Q₈	—	118	1	no	—
132	C₂×Q₈	—	236	4	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	59	1	no	—
136	C₂²×C₄	—	59	1	no	—
137	C₂²×C₄	—	118	1	no	—
138	C₂²×C₄	—	118	1	no	—
139	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	59	2	no	—
142	C₂×C₈	—	118	2	no	—
143	C₂×C₈	C₂²×D₅₉	1	2	yes	—
144	C₄⋊C₄	—	2	1	no	—
145	C₄⋊C₄	—	2	1	no	—
146	C₄⋊C₄	—	59	2	no	—
147	C₄⋊C₄	—	118	4	no	—
148	C₄⋊C₄	—	118	4	no	—
149	C₄⋊C₄	—	118	1	no	—
150	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
151	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
152	C₄²	—	59	2	no	—
153	C₄²	—	59	2	no	—
154	C₄²	—	118	4	no	—
155	C₂³	D₄×D₅₉	1	1	yes	characteristic
156	Q₈	—	59	4	no	—
157	Q₈	—	118	4	no	—
158	Q₈	—	118	2	no	—
159	Q₈	—	236	8	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	59	4	no	—
165	C₂×C₄	—	59	2	no	—
166	C₂×C₄	—	59	2	no	—
167	C₂×C₄	—	118	4	no	—
168	C₂×C₄	—	118	2	no	—
169	C₂×C₄	—	118	2	no	—
170	C₂×C₄	—	118	2	no	—
171	C₂×C₄	—	236	2	no	—
172	C₂×C₄	#11745685	1	2	yes	—
173	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
174	C₂×C₄	Q₈×D₅₉	1	2	yes	—
175	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	118	2	no	—
178	C₂²	#14273705	1	2	yes	—
179	C₂²	Q₁₆×D₅₉	1	2	yes	—
180	C₂²	#14273751	1	2	yes	—
181	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	59	4	no	—
185	C₄	—	118	4	no	—
186	C₄	—	118	2	no	—
187	C₄	—	236	2	no	—
188	C₄	#14273705	1	2	yes	—
189	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
190	C₄	C₂×#11745685	1	1	yes	characteristic
191	C₂	#16837696	1	4	yes	direct product factor
192	C₂	C₂×#14273705	1	1	yes	characteristic
193	C₂	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
194	C₂	C₂×#14273751	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273726

C₂

yes

maximal, characteristic

C₂×#14273692

C₂

yes

maximal, characteristic

#16837696

C₂

yes

maximal, direct product factor

C₂×#14273660

C₂

yes

maximal, characteristic

C₂×Dic₅₉⋊C₈

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal, characteristic

C₂×#14273650

C₂

yes

maximal, characteristic

C₂×#107×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

—

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273726

—

#14273726

C₂²

yes

—

#14273692

—

#14273692

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273660

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

#14273652

C₂²

yes

—

#14273650

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

#107×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₈

yes

complemented

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

Q₁₆

yes

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×D₄

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

C₂×Q₈

yes

complemented

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂²×C₅₉

C₂×Q₁₆

yes

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

—

C₄×C₅₉

C₂²×Q₈

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

C₂×Q₁₆

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₅₉

C₂²×Q₁₆

yes

characteristic

C₂×C₅₉

C₂×AGL(1,ℤ/8)

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

100

#477

—

101

C₂×#107

—

102

C₂×#107

D₅₉

yes

characteristic

103

C₂×C₄⋊C₈

—

104

C₂×#103

—

105

C₂×#103

—

106

C₅₉

C₂×#477

yes

characteristic, complemented, Sylow

107

C₂²×Q₈

—

108

C₂²×Q₈

—

118

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂×D₅₉

yes

characteristic

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

118

—

113

C₄×Q₈

—

114

C₄×Q₈

—

118

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

118

—

117

C₂×C₄⋊C₄

—

118

—

118

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

120

C₂×C₄²

—

121

C₂×C₄²

—

118

—

122

#107

—

123

#107

C₂×D₅₉

yes

—

124

C₄⋊C₈

—

125

#103

—

126

#103

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

118

—

130

C₂×Q₈

—

118

—

131

C₂×Q₈

—

118

—

132

C₂×Q₈

—

236

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

118

—

138

C₂²×C₄

—

118

—

139

C₂²×C₄

C₂²×D₅₉

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

118

—

143

C₂×C₈

C₂²×D₅₉

yes

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

118

—

148

C₄⋊C₄

—

118

—

149

C₄⋊C₄

—

118

—

150

C₄⋊C₄

C₂²×D₅₉

yes

—

151

C₄⋊C₄

C₂²×D₅₉

yes

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

118

—

155

C₂³

D₄×D₅₉

yes

characteristic

156

Q₈

—

157

Q₈

—

118

—

158

Q₈

—

118

—

159

Q₈

—

236

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

118

—

168

C₂×C₄

—

118

—

169

C₂×C₄

—

118

—

170

C₂×C₄

—

118

—

171

C₂×C₄

—

236

—

172

C₂×C₄

#11745685

yes

—

173

C₂×C₄

D₄×D₅₉

yes

characteristic

174

C₂×C₄

Q₈×D₅₉

yes

—

175

C₂×C₄

C₂³×D₅₉

yes

characteristic

176

C₈

—

177

C₈

—

118

—

178

C₂²

#14273705

yes

—

179

C₂²

Q₁₆×D₅₉

yes

—

180

C₂²

#14273751

yes

—

181

C₂²

C₂×D₄×D₅₉

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

118

—

186

C₄

—

118

—

187

C₄

—

236

—

188

C₄

#14273705

yes

—

189

C₄

C₂×Q₈×D₅₉

yes

characteristic

190

C₄

C₂×#11745685

yes

characteristic

191

C₂

#16837696

yes

direct product factor

192

C₂

C₂×#14273705

yes

characteristic

193

C₂

C₂×Q₁₆×D₅₉

yes

characteristic

194

C₂

C₂×#14273751

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×#107×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#107	D₅₉
hypocenter	C₅₉	C₂×#477
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₂×#107×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #16837696	C₂	direct (N × H)
40. C₂×Dic₁₁₈	Q₈	semidirect
77. Dic₁₁₈	C₂×Q₈	semidirect
94. C₂×C₅₉	#477	semidirect
106. C₅₉	C₂×#477	semidirect
191. C₂	#16837696	direct (N × H)

H ≅ G/N

type

3. #16837696

C₂

direct (N × H)

40. C₂×Dic₁₁₈

Q₈

semidirect

77. Dic₁₁₈

C₂×Q₈

semidirect

94. C₂×C₅₉

#477

semidirect

106. C₅₉

C₂×#477

semidirect

191. C₂

#16837696

direct (N × H)

Subgroups — C₂×#16837696 #21433866

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16837696 #21433866

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837696 #21433866