Group #21425621 — subgroups

Subgroups — #21425621

Order 7552.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14273660	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273649	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273647	C₂	1	1	yes	maximal, characteristic
4	#16837303	C₂	1	1	yes	maximal, characteristic
5	#16837271	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837156	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#107×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₃₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₉⋊C₄	—	2	2	no	—
12	C₂×(C₂×D₅₉)⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
14	#14273660	—	2	2	no	—
15	#14273649	—	2	2	no	—
16	#14273647	—	2	2	no	—
17	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
20	#107×C₅₉	—	2	2	no	—
21	C₂³×D₅₉	—	2	1	no	—
22	C₂²×C₄×C₅₉	—	2	1	no	—
23	C₂²×C₄×C₅₉	—	2	1	no	—
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₉	—	2	1	no	—
26	C₂²×Dic₅₉	D₄	1	2	yes	—
27	C₂×D₂₃₆	—	2	1	no	—
28	C₂×D₂₃₆	—	2	1	no	—
29	C₂×D₂₃₆	—	2	1	no	—
30	C₂×D₂₃₆	—	2	1	no	—
31	C₂×D₂₃₆	—	2	1	no	—
32	C₂×D₂₃₆	Q₈	1	2	yes	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₂×C₈×C₅₉	D₄	1	2	yes	—
36	C₄⋊C₄×C₅₉	—	2	1	no	—
37	C₄⋊C₄×C₅₉	—	2	1	no	—
38	C₄⋊C₄×C₅₉	—	2	1	no	—
39	C₄⋊C₄×C₅₉	—	2	1	no	—
40	(C₂×D₅₉)⋊C₄	—	4	4	no	—
41	C₄⋊Dic₅₉	—	2	1	no	—
42	C₄⋊Dic₅₉	—	2	1	no	—
43	Dic₅₉⋊C₄	—	4	4	no	—
44	C₄×Dic₅₉	—	2	1	no	—
45	C₄×Dic₅₉	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
49	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×D₅₉	—	2	2	no	—
51	C₂²×D₅₉	—	4	1	no	—
52	C₂²×D₅₉	—	4	1	no	—
53	C₂²×D₅₉	—	4	2	no	—
54	C₂²×D₅₉	—	4	2	no	—
55	C₂×C₄×C₅₉	—	4	1	no	—
56	C₂×C₄×C₅₉	—	4	1	no	—
57	C₂×C₄×C₅₉	—	4	2	no	—
58	C₂×C₄×C₅₉	—	4	2	no	—
59	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
65	C₂×Dic₅₉	—	2	2	no	—
66	C₂×Dic₅₉	—	2	2	no	—
67	C₂×Dic₅₉	—	4	1	no	—
68	C₂×Dic₅₉	—	4	2	no	—
69	C₂×Dic₅₉	D₈	1	4	yes	—
70	D₂₃₆	—	2	2	no	—
71	D₂₃₆	—	2	2	no	—
72	D₂₃₆	—	4	1	no	—
73	D₂₃₆	—	4	1	no	—
74	C₈×C₅₉	—	2	2	no	—
75	C₅₉⋊C₈	—	2	2	no	—
76	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂²×C₅₉	#123	1	1	yes	characteristic
78	C₂²×C₅₉	#127	1	1	yes	characteristic
79	C₂²×C₅₉	C₂×D₈	1	2	yes	—
80	C₂²×C₅₉	#137	1	2	yes	—
81	C₂×D₅₉	—	4	2	no	—
82	C₂×D₅₉	—	4	4	no	—
83	C₂×D₅₉	—	8	2	no	—
84	C₂×D₅₉	—	8	1	no	—
85	C₂×D₅₉	—	8	1	no	—
86	C₄×C₅₉	—	8	1	no	—
87	C₄×C₅₉	—	8	1	no	—
88	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
89	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
90	C₄×C₅₉	#123	1	1	yes	characteristic
91	C₄×C₅₉	#123	1	1	yes	characteristic
92	Dic₅₉	—	2	4	no	—
93	Dic₅₉	—	8	1	no	—
94	#2093	—	59	1	no	maximal, Sylow
95	C₂×C₅₉	#399	1	1	yes	characteristic
96	C₂×C₅₉	#474	1	2	yes	—
97	C₂×C₅₉	#480	1	2	yes	—
98	C₂×C₅₉	#493	1	1	yes	characteristic
99	C₂×C₅₉	#497	1	1	yes	characteristic
100	D₅₉	—	8	2	no	—
101	C₂×#107	—	59	1	no	—
102	C₂×#107	D₅₉	1	1	yes	characteristic, complemented
103	C₂×#102	—	59	1	no	—
104	C₂×#102	—	59	1	no	—
105	#390	—	59	1	no	—
106	#384	—	59	1	no	—
107	#336	—	59	1	no	—
108	C₅₉	#2093	1	1	yes	characteristic, complemented, Sylow
109	C₂²×D₄	—	59	1	no	—
110	C₂²×C₈	—	59	1	no	—
111	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
112	C₂×C₄⋊C₄	—	59	1	no	—
113	C₂×C₄⋊C₄	—	118	2	no	—
114	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
116	C₂×C₂²⋊C₄	—	118	2	no	—
117	C₂×C₄²	—	59	1	no	—
118	#107	—	2	2	no	—
119	#107	—	118	2	no	—
120	#102	—	118	2	no	—
121	#102	—	118	2	no	—
122	C₂⁴	—	118	1	no	—
123	C₂×D₄	—	59	2	no	—
124	C₂×D₄	—	118	1	no	—
125	C₂×D₄	—	118	1	no	—
126	C₂×D₄	—	118	1	no	—
127	C₂×D₄	—	118	1	no	—
128	C₂×D₄	—	118	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	59	2	no	—
132	C₂²×C₄	—	118	1	no	—
133	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	59	2	no	—
137	C₂×C₈	—	118	1	no	—
138	C₂×C₈	—	118	1	no	—
139	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	118	1	no	—
145	C₄⋊C₄	—	118	1	no	—
146	C₄⋊C₄	—	236	4	no	—
147	C₂²⋊C₄	—	236	4	no	—
148	C₄²	—	118	1	no	—
149	C₄²	—	118	1	no	—
150	C₂³	—	118	2	no	—
151	C₂³	—	236	1	no	—
152	C₂³	—	236	1	no	—
153	C₂³	—	236	2	no	—
154	C₂³	—	236	2	no	—
155	C₂³	D₄×D₅₉	1	1	yes	characteristic
156	D₄	—	118	2	no	—
157	D₄	—	118	2	no	—
158	D₄	—	236	1	no	—
159	D₄	—	236	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	59	4	no	—
165	C₂×C₄	—	118	2	no	—
166	C₂×C₄	—	118	2	no	—
167	C₂×C₄	—	236	1	no	—
168	C₂×C₄	—	236	2	no	—
169	C₂×C₄	#11745685	1	1	yes	characteristic
170	C₂×C₄	#11745685	1	1	yes	characteristic
171	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
172	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
173	C₂×C₄	#11745689	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	118	2	no	—
177	C₂²	—	472	2	no	—
178	C₂²	—	472	1	no	—
179	C₂²	—	472	1	no	—
180	C₂²	—	236	4	no	—
181	C₂²	—	236	2	no	—
182	C₂²	#14273705	1	1	yes	characteristic
183	C₂²	#14273712	1	1	yes	characteristic
184	C₂²	#14273723	1	1	yes	characteristic
185	C₂²	D₈×D₅₉	1	2	yes	—
186	C₂²	#14273759	1	2	yes	—
187	C₄	—	8	1	no	—
188	C₄	—	8	1	no	—
189	C₄	—	118	4	no	—
190	C₄	—	472	1	no	—
191	C₄	#14273703	1	1	yes	characteristic
192	C₄	#14273705	1	1	yes	characteristic
193	C₄	#14273710	1	1	yes	characteristic
194	C₄	#14273712	1	1	yes	characteristic
195	C₂	—	472	2	no	—
196	C₂	#16837337	1	1	yes	characteristic
197	C₂	#16837681	1	2	yes	—
198	C₂	#16837717	1	2	yes	—
199	C₂	#16837777	1	1	yes	characteristic
200	C₂	#16837795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273660

C₂

yes

maximal, characteristic, complemented

C₂×#14273649

C₂

yes

maximal, characteristic

C₂×#14273647

C₂

yes

maximal, characteristic

#16837303

C₂

yes

maximal, characteristic

#16837271

C₂

yes

maximal, characteristic, complemented

#16837156

C₂

yes

maximal, characteristic, complemented

C₂×#107×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×(C₂×D₅₉)⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

—

—

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

#107×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

Q₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²⋊Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×D₈

yes

—

C₂²×C₅₉

#137

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#480

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

100

D₅₉

—

101

C₂×#107

—

102

C₂×#107

D₅₉

yes

characteristic, complemented

103

C₂×#102

—

104

C₂×#102

—

105

#390

—

106

#384

—

107

#336

—

108

C₅₉

#2093

yes

characteristic, complemented, Sylow

109

C₂²×D₄

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂×D₅₉

yes

characteristic

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

118

—

114

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

115

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

116

C₂×C₂²⋊C₄

—

118

—

117

C₂×C₄²

—

118

#107

—

119

#107

—

118

—

120

#102

—

118

—

121

#102

—

118

—

122

C₂⁴

—

118

—

123

C₂×D₄

—

124

C₂×D₄

—

118

—

125

C₂×D₄

—

118

—

126

C₂×D₄

—

118

—

127

C₂×D₄

—

118

—

128

C₂×D₄

—

118

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

118

—

133

C₂²×C₄

C₂²×D₅₉

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

118

—

138

C₂×C₈

—

118

—

139

C₂×C₈

C₂²⋊D₅₉

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

118

—

145

C₄⋊C₄

—

118

—

146

C₄⋊C₄

—

236

—

147

C₂²⋊C₄

—

236

—

148

C₄²

—

118

—

149

C₄²

—

118

—

150

C₂³

—

118

—

151

C₂³

—

236

—

152

C₂³

—

236

—

153

C₂³

—

236

—

154

C₂³

—

236

—

155

C₂³

D₄×D₅₉

yes

characteristic

156

D₄

—

118

—

157

D₄

—

118

—

158

D₄

—

236

—

159

D₄

—

236

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

118

—

166

C₂×C₄

—

118

—

167

C₂×C₄

—

236

—

168

C₂×C₄

—

236

—

169

C₂×C₄

#11745685

yes

characteristic

170

C₂×C₄

#11745685

yes

characteristic

171

C₂×C₄

D₄×D₅₉

yes

characteristic

172

C₂×C₄

Q₈×D₅₉

yes

characteristic

173

C₂×C₄

#11745689

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

175

C₈

—

176

C₈

—

118

—

177

C₂²

—

472

—

178

C₂²

—

472

—

179

C₂²

—

472

—

180

C₂²

—

236

—

181

C₂²

—

236

—

182

C₂²

#14273705

yes

characteristic

183

C₂²

#14273712

yes

characteristic

184

C₂²

#14273723

yes

characteristic

185

C₂²

D₈×D₅₉

yes

—

186

C₂²

#14273759

yes

—

187

C₄

—

188

C₄

—

189

C₄

—

118

—

190

C₄

—

472

—

191

C₄

#14273703

yes

characteristic

192

C₄

#14273705

yes

characteristic

193

C₄

#14273710

yes

characteristic

194

C₄

#14273712

yes

characteristic

195

C₂

—

472

—

196

C₂

#16837337

yes

characteristic

197

C₂

#16837681

yes

—

198

C₂

#16837717

yes

—

199

C₂

#16837777

yes

characteristic

200

C₂

#16837795

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#107×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#107	D₅₉
hypocenter	C₅₉	#2093
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#107×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
1. C₂×#14273660	C₂	semidirect
5. #16837271	C₂	semidirect
6. #16837156	C₂	semidirect
7. C₂×#107×C₅₉	C₂	semidirect
102. C₂×#107	D₅₉	semidirect
108. C₅₉	#2093	semidirect

H ≅ G/N

type

1. C₂×#14273660

C₂

semidirect

5. #16837271

C₂

semidirect

6. #16837156

C₂

semidirect

7. C₂×#107×C₅₉

C₂

semidirect

102. C₂×#107

D₅₉

semidirect

108. C₅₉

#2093

semidirect

Subgroups — #21425621

Conjugacy classes of subgroups

Complements