Group #21435537 — subgroups

Subgroups — #21435537

Order 7552.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16838007	C₂	1	1	yes	maximal, characteristic
2	#16837709	C₂	1	2	yes	maximal, complemented
3	#16837699	C₂	1	2	yes	maximal, complemented
4	#16837650	C₂	1	2	yes	maximal, complemented
5	#16837633	C₂	1	2	yes	maximal, complemented
6	#16837604	C₂	1	2	yes	maximal, complemented
7	#16837602	C₂	1	2	yes	maximal, complemented
8	#16837429	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#541	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11745671	C₂²	1	2	yes	—
11	C₂×#11745671	C₂²	1	2	yes	—
12	C₂×#11745664	C₂²	1	1	yes	characteristic
13	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
14	#14273724	C₂²	1	1	yes	characteristic
15	#14273719	C₂²	1	1	yes	characteristic
16	#14273712	—	2	2	no	—
17	#14273701	C₂²	1	2	yes	—
18	#14273700	—	2	2	no	—
19	C₄×D₂₃₆	C₂²	1	2	yes	—
20	#14273660	C₂²	1	2	yes	—
21	#14273658	C₂²	1	2	yes	—
22	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
23	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
24	#14273651	C₂²	1	2	yes	—
25	#14273651	C₂²	1	2	yes	—
26	#14273647	C₂²	1	2	yes	—
27	#14273647	C₂²	1	2	yes	—
28	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
29	#126×C₅₉	—	2	2	no	—
30	#125×C₅₉	—	2	2	no	—
31	#125×C₅₉	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₅₉	C₂²	1	2	yes	—
33	C₄×Q₈×C₅₉	C₂²	1	2	yes	—
34	#117×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₅₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊D₅₉	—	2	2	no	—
38	C₂×D₂₃₆	—	2	2	no	—
39	C₂×D₂₃₆	C₂³	1	2	yes	—
40	C₂×C₄×D₅₉	—	2	2	no	—
41	C₄⋊C₄×C₅₉	—	2	2	no	—
42	C₄⋊C₄×C₅₉	—	2	2	no	—
43	C₄⋊C₄×C₅₉	—	2	2	no	—
44	C₄⋊C₄×C₅₉	—	2	2	no	—
45	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
46	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₅₉	—	2	2	no	—
48	C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
49	C₄²×C₅₉	—	2	2	no	—
50	C₄²×C₅₉	C₂³	1	1	yes	characteristic
51	C₄²×C₅₉	C₂³	1	1	yes	characteristic
52	#11745671	—	4	2	no	—
53	#11745671	—	4	2	no	—
54	(C₂×D₅₉)⋊C₄	—	2	2	no	—
55	(C₂×D₅₉)⋊C₄	—	2	2	no	—
56	C₄⋊Dic₅₉	C₂³	1	2	yes	—
57	Dic₅₉⋊C₄	—	2	2	no	—
58	#11745664	—	4	1	no	—
59	C₂×C₅₉⋊C₈	—	4	1	no	—
60	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
61	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
62	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×D₅₉	—	2	2	no	—
64	Q₈×C₅₉	—	2	2	no	—
65	Q₈×C₅₉	—	2	2	no	—
66	C₂×C₄×C₅₉	—	2	1	no	—
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₅₉	—	8	2	no	—
75	C₂×Dic₅₉	—	2	2	no	—
76	D₂₃₆	—	2	2	no	—
77	D₂₃₆	—	2	2	no	—
78	D₂₃₆	—	8	2	no	—
79	C₄×D₅₉	—	8	2	no	—
80	C₅₉⋊C₈	—	2	2	no	—
81	C₅₉⋊C₈	—	2	2	no	—
82	C₂²×C₅₉	—	2	1	no	—
83	C₂²×C₅₉	—	2	2	no	—
84	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₅₉	—	4	2	no	—
86	C₂×D₅₉	—	4	2	no	—
87	C₂×D₅₉	—	4	2	no	—
88	C₄×C₅₉	—	2	1	no	—
89	C₄×C₅₉	—	2	2	no	—
90	C₄×C₅₉	—	2	2	no	—
91	C₄×C₅₉	—	4	2	no	—
92	C₄×C₅₉	—	4	2	no	—
93	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
94	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
95	Dic₅₉	—	4	2	no	—
96	#3260	—	59	1	no	maximal, Sylow
97	C₂×C₅₉	—	4	1	no	—
98	C₂×C₅₉	#537	1	1	yes	characteristic
99	C₂×C₅₉	#577	1	2	yes	—
100	D₅₉	—	8	2	no	—
101	#541	D₅₉	1	1	yes	characteristic, complemented
102	#540	—	59	1	no	—
103	#479	—	59	2	no	—
104	#478	—	59	2	no	—
105	#468	—	59	2	no	—
106	#465	—	59	2	no	—
107	#461	—	59	2	no	—
108	#460	—	59	2	no	—
109	#424	—	59	1	no	—
110	C₅₉	#3260	1	1	yes	characteristic, complemented, Sylow
111	C₂×SD₁₆	—	59	2	no	—
112	C₂×SD₁₆	—	59	2	no	—
113	C₂×M₁₆	—	59	1	no	—
114	C₂²×C₈	—	59	1	no	—
115	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
116	#127	—	59	1	no	—
117	#126	—	2	2	no	—
118	#125	—	2	2	no	—
119	#125	—	59	1	no	—
120	#125	C₂×D₅₉	1	1	yes	characteristic
121	#123	—	118	2	no	—
122	C₂²⋊Q₈	C₂×D₅₉	1	2	yes	—
123	#121	—	59	2	no	—
124	#121	—	118	2	no	—
125	C₄×Q₈	C₂×D₅₉	1	2	yes	—
126	C₄×D₄	—	59	2	no	—
127	#117	C₂×D₅₉	1	1	yes	characteristic
128	#107	—	59	2	no	—
129	#106	—	59	2	no	—
130	C₄⋊C₈	—	59	2	no	—
131	C₄⋊C₈	—	59	2	no	—
132	#103	—	59	2	no	—
133	#103	—	59	2	no	—
134	#102	—	59	2	no	—
135	#102	—	59	2	no	—
136	C₂×Q₈	C₂²×D₅₉	1	2	yes	—
137	C₂×D₄	—	59	2	no	—
138	C₂×D₄	—	118	2	no	—
139	C₂×D₄	—	118	2	no	—
140	C₂²×C₄	—	118	2	no	—
141	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
142	SD₁₆	—	236	2	no	—
143	SD₁₆	—	236	2	no	—
144	M₁₆	—	236	1	no	—
145	C₂×C₈	—	59	2	no	—
146	C₂×C₈	—	59	2	no	—
147	C₂×C₈	—	236	1	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	59	2	no	—
153	C₄⋊C₄	—	118	2	no	—
154	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
155	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	118	2	no	—
158	C₂²⋊C₄	—	118	2	no	—
159	C₂²⋊C₄	C₂²⋊D₅₉	1	2	yes	—
160	C₄²	—	2	2	no	—
161	C₄²	C₂²×D₅₉	1	1	yes	characteristic
162	C₄²	C₂²×D₅₉	1	1	yes	characteristic
163	C₂³	—	118	2	no	—
164	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	Q₈	—	2	2	no	—
167	D₄	—	118	2	no	—
168	D₄	—	118	2	no	—
169	D₄	—	472	2	no	—
170	D₄	—	472	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	118	2	no	—
176	C₂×C₄	—	472	2	no	—
177	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
178	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
181	C₈	—	118	2	no	—
182	C₈	—	118	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	236	2	no	—
186	C₂²	—	236	2	no	—
187	C₂²	—	236	2	no	—
188	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	236	2	no	—
195	C₄	#14273773	1	1	yes	characteristic
196	C₄	#14273774	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	472	2	no	—
199	C₂	#16837980	1	1	yes	characteristic
200	C₂	#16838210	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16838007

C₂

yes

maximal, characteristic

#16837709

C₂

yes

maximal, complemented

#16837699

C₂

yes

maximal, complemented

#16837650

C₂

yes

maximal, complemented

#16837633

C₂

yes

maximal, complemented

#16837604

C₂

yes

maximal, complemented

#16837602

C₂

yes

maximal, complemented

#16837429

C₂

yes

maximal, characteristic, complemented

C₅₉×#541

C₂

yes

maximal, characteristic, complemented

C₂×#11745671

C₂²

yes

—

C₂×#11745671

C₂²

yes

—

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273724

C₂²

yes

characteristic

#14273719

C₂²

yes

characteristic

#14273712

—

#14273701

C₂²

yes

—

#14273700

—

C₄×D₂₃₆

C₂²

yes

—

#14273660

C₂²

yes

—

#14273658

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273651

C₂²

yes

—

#14273651

C₂²

yes

—

#14273647

C₂²

yes

—

#14273647

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#126×C₅₉

—

#125×C₅₉

—

#125×C₅₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₅₉

C₂²

yes

—

C₄×Q₈×C₅₉

C₂²

yes

—

#117×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂³

yes

—

C₂×C₄×D₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

#11745671

—

#11745671

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×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

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

#3260

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#537

yes

characteristic

C₂×C₅₉

#577

yes

—

100

D₅₉

—

101

#541

D₅₉

yes

characteristic, complemented

102

#540

—

103

#479

—

104

#478

—

105

#468

—

106

#465

—

107

#461

—

108

#460

—

109

#424

—

110

C₅₉

#3260

yes

characteristic, complemented, Sylow

111

C₂×SD₁₆

—

112

C₂×SD₁₆

—

113

C₂×M₁₆

—

114

C₂²×C₈

—

115

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

116

#127

—

117

#126

—

118

#125

—

119

#125

—

120

#125

C₂×D₅₉

yes

characteristic

121

#123

—

118

—

122

C₂²⋊Q₈

C₂×D₅₉

yes

—

123

#121

—

124

#121

—

118

—

125

C₄×Q₈

C₂×D₅₉

yes

—

126

C₄×D₄

—

127

#117

C₂×D₅₉

yes

characteristic

128

#107

—

129

#106

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

#103

—

134

#102

—

135

#102

—

136

C₂×Q₈

C₂²×D₅₉

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

118

—

139

C₂×D₄

—

118

—

140

C₂²×C₄

—

118

—

141

C₂²×C₄

C₂²×D₅₉

yes

characteristic

142

SD₁₆

—

236

—

143

SD₁₆

—

236

—

144

M₁₆

—

236

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

236

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

118

—

154

C₄⋊C₄

C₂²⋊D₅₉

yes

—

155

C₄⋊C₄

C₂²×D₅₉

yes

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

118

—

158

C₂²⋊C₄

—

118

—

159

C₂²⋊C₄

C₂²⋊D₅₉

yes

—

160

C₄²

—

161

C₄²

C₂²×D₅₉

yes

characteristic

162

C₄²

C₂²×D₅₉

yes

characteristic

163

C₂³

—

118

—

164

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

165

Q₈

—

166

Q₈

—

167

D₄

—

118

—

168

D₄

—

118

—

169

D₄

—

472

—

170

D₄

—

472

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

118

—

176

C₂×C₄

—

472

—

177

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

178

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

179

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

180

C₂×C₄

C₂³×D₅₉

yes

characteristic

181

C₈

—

118

—

182

C₈

—

118

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

236

—

186

C₂²

—

236

—

187

C₂²

—

236

—

188

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

236

—

195

C₄

#14273773

yes

characteristic

196

C₄

#14273774

yes

characteristic

197

C₂

—

198

C₂

—

472

—

199

C₂

#16837980

yes

characteristic

200

C₂

#16838210

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#541	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#541	D₅₉
hypocenter	C₅₉	#3260
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #541

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#541

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #541

N	H ≅ G/N	type
2. #16837709	C₂	semidirect
3. #16837699	C₂	semidirect
4. #16837650	C₂	semidirect
5. #16837633	C₂	semidirect
6. #16837604	C₂	semidirect
7. #16837602	C₂	semidirect
8. #16837429	C₂	semidirect
9. C₅₉×#541	C₂	semidirect
101. #541	D₅₉	semidirect
110. C₅₉	#3260	semidirect

H ≅ G/N

type

2. #16837709

C₂

semidirect

3. #16837699

C₂

semidirect

4. #16837650

C₂

semidirect

5. #16837633

C₂

semidirect

6. #16837604

C₂

semidirect

7. #16837602

C₂

semidirect

8. #16837429

C₂

semidirect

9. C₅₉×#541

C₂

semidirect

101. #541

D₅₉

semidirect

110. C₅₉

#3260

semidirect

Subgroups — #21435537

Conjugacy classes of subgroups

Complements