Group #21435551 — subgroups

Subgroups — #21435551

Order 7552.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16838019	C₂	1	2	yes	maximal, complemented
2	#16837696	C₂	1	2	yes	maximal, complemented
3	#16837680	C₂	1	2	yes	maximal, complemented
4	#16837646	C₂	1	2	yes	maximal, complemented
5	#16837643	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837623	C₂	1	2	yes	maximal, complemented
7	#16837615	C₂	1	2	yes	maximal, complemented
8	#16837428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#466	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
11	C₂×#11745670	C₂²	1	2	yes	—
12	C₂×#11745664	C₂²	1	1	yes	characteristic
13	#14273726	C₂²	1	2	yes	—
14	#14273720	—	2	2	no	—
15	#14273714	—	2	2	no	—
16	#14273714	C₂²	1	2	yes	—
17	#14273708	—	2	2	no	—
18	#14273706	C₂²	1	2	yes	—
19	#14273704	—	2	2	no	—
20	#14273692	C₂²	1	2	yes	—
21	D₄×Dic₅₉	C₂²	1	2	yes	—
22	#14273686	C₂²	1	1	yes	characteristic
23	#14273661	C₂²	1	1	yes	characteristic
24	#14273660	C₂²	1	2	yes	—
25	Dic₅₉⋊C₈	C₂²	1	2	yes	—
26	Dic₅₉⋊C₈	C₂²	1	2	yes	—
27	#14273652	C₂²	1	2	yes	—
28	#14273650	C₂²	1	2	yes	—
29	#14273648	C₂²	1	2	yes	—
30	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
32	#121×C₅₉	C₂²	1	2	yes	—
33	#107×C₅₉	C₂²	1	1	yes	characteristic
34	#102×C₅₉	C₂²	1	2	yes	—
35	C₂×D₄×C₅₉	—	2	2	no	—
36	C₂×D₄×C₅₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	2	2	no	—
39	C₂×Dic₁₁₈	—	2	2	no	—
40	C₂×Dic₁₁₈	C₂³	1	2	yes	—
41	D₈×C₅₉	—	4	1	no	—
42	C₂×C₈×C₅₉	—	4	1	no	—
43	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₅₉	—	2	2	no	—
47	C₂²⋊Dic₅₉	—	2	2	no	—
48	C₂²⋊Dic₅₉	—	2	2	no	—
49	C₂²⋊Dic₅₉	—	2	2	no	—
50	C₂²⋊Dic₅₉	—	2	2	no	—
51	C₂²⋊Dic₅₉	D₄	1	2	yes	—
52	#11745670	—	4	2	no	—
53	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
54	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
55	C₄⋊Dic₅₉	C₂³	1	2	yes	—
56	Dic₅₉⋊C₄	—	2	2	no	—
57	Dic₅₉⋊C₄	—	2	2	no	—
58	Dic₅₉⋊C₄	—	2	2	no	—
59	C₄×Dic₅₉	—	2	2	no	—
60	C₄×Dic₅₉	C₂³	1	2	yes	—
61	#11745664	—	4	1	no	—
62	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
63	Dic₂₃₆	—	4	1	no	—
64	C₂³×C₅₉	—	2	2	no	—
65	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
66	D₄×C₅₉	—	2	2	no	—
67	D₄×C₅₉	—	2	2	no	—
68	D₄×C₅₉	—	8	2	no	—
69	C₂×C₄×C₅₉	—	2	1	no	—
70	C₂×C₄×C₅₉	—	2	1	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₅₉	—	2	2	no	—
75	C₂×Dic₅₉	—	2	2	no	—
76	C₂×Dic₅₉	—	8	2	no	—
77	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
78	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
79	Dic₁₁₈	—	2	2	no	—
80	Dic₁₁₈	—	2	2	no	—
81	Dic₁₁₈	—	8	2	no	—
82	C₈×C₅₉	—	2	1	no	—
83	C₈×C₅₉	—	2	1	no	—
84	C₅₉⋊C₈	—	2	2	no	—
85	C₂²×C₅₉	—	2	1	no	—
86	C₂²×C₅₉	—	2	1	no	—
87	C₂²×C₅₉	—	2	1	no	—
88	C₂²×C₅₉	—	4	2	no	—
89	C₂²×C₅₉	—	4	2	no	—
90	C₂²×C₅₉	—	4	2	no	—
91	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₅₉	—	2	1	no	—
93	C₄×C₅₉	—	4	2	no	—
94	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₅₉	—	2	2	no	—
97	Dic₅₉	—	2	2	no	—
98	Dic₅₉	—	4	2	no	—
99	Dic₅₉	—	4	2	no	—
100	#3262	—	59	1	no	maximal, Sylow
101	C₂×C₅₉	—	4	1	no	—
102	C₂×C₅₉	—	8	2	no	—
103	C₂×C₅₉	#537	1	1	yes	characteristic
104	C₂×C₅₉	#576	1	1	yes	characteristic
105	C₂×C₅₉	#578	1	1	yes	characteristic
106	#542	—	59	2	no	—
107	#477	—	59	2	no	—
108	#474	—	59	2	no	—
109	#468	—	59	2	no	—
110	#467	—	59	1	no	—
111	#466	D₅₉	1	1	yes	characteristic
112	#464	—	59	2	no	—
113	#463	—	59	2	no	—
114	#424	—	59	1	no	—
115	C₅₉	#3262	1	1	yes	characteristic, complemented, Sylow
116	C₂×Q₁₆	—	59	1	no	—
117	C₂×SD₁₆	—	59	2	no	—
118	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
119	C₂×M₁₆	—	59	1	no	—
120	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
121	C₄⋊Q₈	—	59	2	no	—
122	#126	—	118	2	no	—
123	#124	—	59	2	no	—
124	#124	—	118	2	no	—
125	#123	—	118	2	no	—
126	C₂²⋊Q₈	—	59	2	no	—
127	C₂²⋊Q₈	—	118	2	no	—
128	#121	C₂×D₅₉	1	2	yes	—
129	C₄×Q₈	—	59	2	no	—
130	C₄×D₄	—	59	2	no	—
131	#117	—	59	1	no	—
132	#107	—	59	1	no	—
133	#107	—	59	2	no	—
134	#107	C₂×D₅₉	1	1	yes	characteristic
135	C₄⋊C₈	—	59	2	no	—
136	C₄⋊C₈	—	59	2	no	—
137	#103	—	59	2	no	—
138	#103	—	59	2	no	—
139	#102	—	59	2	no	—
140	#102	C₂×D₅₉	1	2	yes	—
141	C₂×Q₈	—	59	2	no	—
142	C₂×Q₈	—	118	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂²×D₅₉	1	2	yes	—
145	C₂²×C₄	—	118	2	no	—
146	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
147	Q₁₆	—	236	1	no	—
148	SD₁₆	—	236	2	no	—
149	D₈	—	4	1	no	—
150	M₁₆	—	236	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	59	2	no	—
153	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
154	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
155	C₄⋊C₄	—	59	1	no	—
156	C₄⋊C₄	—	59	1	no	—
157	C₄⋊C₄	—	59	2	no	—
158	C₄⋊C₄	—	118	2	no	—
159	C₄⋊C₄	—	118	2	no	—
160	C₄⋊C₄	—	118	2	no	—
161	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	59	2	no	—
164	C₂²⋊C₄	—	118	2	no	—
165	C₂²⋊C₄	—	118	2	no	—
166	C₂²⋊C₄	—	118	2	no	—
167	C₂²⋊C₄	—	118	2	no	—
168	C₄²	—	59	2	no	—
169	C₄²	—	118	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	D₄×D₅₉	1	1	yes	characteristic
172	Q₈	—	118	2	no	—
173	Q₈	—	118	2	no	—
174	Q₈	—	472	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	8	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	59	2	no	—
182	C₂×C₄	—	59	2	no	—
183	C₂×C₄	—	118	2	no	—
184	C₂×C₄	—	118	2	no	—
185	C₂×C₄	—	472	2	no	—
186	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
187	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	118	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	4	2	no	—
200	C₄	—	118	2	no	—
201	C₄	—	118	2	no	—
202	C₄	—	236	2	no	—
203	C₄	—	236	2	no	—
204	C₄	#14273772	1	1	yes	characteristic
205	C₄	#14273774	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	8	2	no	—
208	C₂	#16837979	1	1	yes	characteristic
209	C₂	#16838203	1	1	yes	characteristic
210	C₂	#16838216	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838019

C₂

yes

maximal, complemented

#16837696

C₂

yes

maximal, complemented

#16837680

C₂

yes

maximal, complemented

#16837646

C₂

yes

maximal, complemented

#16837643

C₂

yes

maximal, characteristic, complemented

#16837623

C₂

yes

maximal, complemented

#16837615

C₂

yes

maximal, complemented

#16837428

C₂

yes

maximal, characteristic, complemented

C₅₉×#466

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745670

C₂²

yes

—

C₂×#11745664

C₂²

yes

characteristic

#14273726

C₂²

yes

—

#14273720

—

#14273714

—

#14273714

C₂²

yes

—

#14273708

—

#14273706

C₂²

yes

—

#14273704

—

#14273692

C₂²

yes

—

D₄×Dic₅₉

C₂²

yes

—

#14273686

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273660

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

Dic₅₉⋊C₈

C₂²

yes

—

#14273652

C₂²

yes

—

#14273650

C₂²

yes

—

#14273648

C₂²

yes

—

C₂×D₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#121×C₅₉

C₂²

yes

—

#107×C₅₉

C₂²

yes

characteristic

#102×C₅₉

C₂²

yes

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

D₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

D₄

yes

—

#11745670

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂³

yes

—

#11745664

—

C₂×C₅₉⋊C₈

C₂³

yes

—

Dic₂₃₆

—

C₂³×C₅₉

—

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×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×D₄

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

100

#3262

—

maximal, Sylow

101

C₂×C₅₉

—

102

C₂×C₅₉

—

103

C₂×C₅₉

#537

yes

characteristic

104

C₂×C₅₉

#576

yes

characteristic

105

C₂×C₅₉

#578

yes

characteristic

106

#542

—

107

#477

—

108

#474

—

109

#468

—

110

#467

—

111

#466

D₅₉

yes

characteristic

112

#464

—

113

#463

—

114

#424

—

115

C₅₉

#3262

yes

characteristic, complemented, Sylow

116

C₂×Q₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

C₂×D₅₉

yes

characteristic

119

C₂×M₁₆

—

120

C₂²×C₈

C₂×D₅₉

yes

characteristic

121

C₄⋊Q₈

—

122

#126

—

118

—

123

#124

—

124

#124

—

118

—

125

#123

—

118

—

126

C₂²⋊Q₈

—

127

C₂²⋊Q₈

—

118

—

128

#121

C₂×D₅₉

yes

—

129

C₄×Q₈

—

130

C₄×D₄

—

131

#117

—

132

#107

—

133

#107

—

134

#107

C₂×D₅₉

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#103

—

138

#103

—

139

#102

—

140

#102

C₂×D₅₉

yes

—

141

C₂×Q₈

—

142

C₂×Q₈

—

118

—

143

C₂×D₄

—

144

C₂×D₄

C₂²×D₅₉

yes

—

145

C₂²×C₄

—

118

—

146

C₂²×C₄

C₂²×D₅₉

yes

characteristic

147

Q₁₆

—

236

—

148

SD₁₆

—

236

—

149

D₈

—

150

M₁₆

—

236

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²×D₅₉

yes

characteristic

154

C₂×C₈

C₂²×D₅₉

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

118

—

159

C₄⋊C₄

—

118

—

160

C₄⋊C₄

—

118

—

161

C₄⋊C₄

C₂²×D₅₉

yes

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

118

—

165

C₂²⋊C₄

—

118

—

166

C₂²⋊C₄

—

118

—

167

C₂²⋊C₄

—

118

—

168

C₄²

—

169

C₄²

—

118

—

170

C₂³

—

171

C₂³

D₄×D₅₉

yes

characteristic

172

Q₈

—

118

—

173

Q₈

—

118

—

174

Q₈

—

472

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

118

—

184

C₂×C₄

—

118

—

185

C₂×C₄

—

472

—

186

C₂×C₄

D₄×D₅₉

yes

characteristic

187

C₂×C₄

C₂³×D₅₉

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

118

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

C₂×D₄×D₅₉

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

118

—

201

C₄

—

118

—

202

C₄

—

236

—

203

C₄

—

236

—

204

C₄

#14273772

yes

characteristic

205

C₄

#14273774

yes

characteristic

206

C₂

—

207

C₂

—

208

C₂

#16837979

yes

characteristic

209

C₂

#16838203

yes

characteristic

210

C₂

#16838216

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#466	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#466	D₅₉
hypocenter	C₅₉	#3262
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

subgroup

G/H

center

C₂²

C₂×D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#466

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

N	H ≅ G/N	type
1. #16838019	C₂	semidirect
2. #16837696	C₂	semidirect
3. #16837680	C₂	semidirect
4. #16837646	C₂	semidirect
5. #16837643	C₂	semidirect
6. #16837623	C₂	semidirect
7. #16837615	C₂	semidirect
8. #16837428	C₂	semidirect
115. C₅₉	#3262	semidirect

H ≅ G/N

type

1. #16838019

C₂

semidirect

2. #16837696

C₂

semidirect

3. #16837680

C₂

semidirect

4. #16837646

C₂

semidirect

5. #16837643

C₂

semidirect

6. #16837623

C₂

semidirect

7. #16837615

C₂

semidirect

8. #16837428

C₂

semidirect

115. C₅₉

#3262

semidirect

Subgroups — #21435551

Conjugacy classes of subgroups

Complements