Group #21424312 — subgroups

Subgroups — #21424312

Order 7552.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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	#16837879	C₂	1	1	yes	maximal, characteristic
2	#16837445	C₂	1	1	yes	maximal, characteristic
3	#16837444	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273645	C₂	1	1	yes	maximal, characteristic
5	#16837308	C₂	1	1	yes	maximal, characteristic
6	#16837162	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#339	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₁₈	—	2	1	no	—
11	C₄×Dic₁₁₈	—	2	1	no	—
12	#14273692	—	2	1	no	—
13	#14273692	—	2	1	no	—
14	#14273686	C₂²	1	1	yes	characteristic
15	#14273683	—	2	2	no	—
16	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
17	#14273661	—	2	1	no	—
18	#14273660	—	2	1	no	—
19	#14273659	—	2	1	no	—
20	#14273658	—	2	1	no	—
21	#14273645	C₄	1	4	yes	complemented
22	C₅₉⋊#95	—	2	2	no	—
23	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	#117×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	—	2	1	no	—
27	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅₉	—	2	1	no	—
29	C₂²×Dic₅₉	D₄	1	2	yes	—
30	C₂×Dic₁₁₈	—	2	1	no	—
31	C₂×Dic₁₁₈	—	2	1	no	—
32	C₂×Dic₁₁₈	—	2	1	no	—
33	C₂×Dic₁₁₈	—	2	1	no	—
34	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
35	C₂×Dic₁₁₈	Q₈	1	2	yes	—
36	M₁₆×C₅₉	—	2	1	no	—
37	M₁₆×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₅₉	—	2	1	no	—
39	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
41	C₂²⋊C₄×C₅₉	—	2	1	no	—
42	C₄²×C₅₉	—	2	1	no	—
43	C₄²×C₅₉	—	2	1	no	—
44	C₄²×C₅₉	—	2	1	no	—
45	C₂²⋊Dic₅₉	—	2	1	no	—
46	C₄⋊Dic₅₉	—	2	1	no	—
47	C₄⋊Dic₅₉	—	2	1	no	—
48	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
49	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
50	Dic₅₉⋊C₄	—	2	2	no	—
51	Dic₅₉⋊C₄	—	2	2	no	—
52	C₄×Dic₅₉	—	2	2	no	—
53	C₄×Dic₅₉	—	2	2	no	—
54	C₄×Dic₅₉	—	2	1	no	—
55	#11745664	—	2	1	no	—
56	#11745664	C₂×C₄	1	2	yes	—
57	C₂×C₅₉⋊C₈	—	2	1	no	—
58	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	1	no	—
61	C₂×C₄×C₅₉	—	2	1	no	—
62	C₂×C₄×C₅₉	—	4	1	no	—
63	C₂×C₄×C₅₉	—	4	1	no	—
64	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
70	C₂×Dic₅₉	—	2	2	no	—
71	C₂×Dic₅₉	—	2	2	no	—
72	C₂×Dic₅₉	—	2	1	no	—
73	C₂×Dic₅₉	—	2	1	no	—
74	C₂×Dic₅₉	—	4	2	no	—
75	C₂×Dic₅₉	—	4	1	no	—
76	C₂×Dic₅₉	C₄⋊C₄	1	4	yes	—
77	Dic₁₁₈	—	2	2	no	—
78	Dic₁₁₈	—	2	2	no	—
79	Dic₁₁₈	—	2	2	no	—
80	Dic₁₁₈	—	2	2	no	—
81	C₈×C₅₉	—	2	2	no	—
82	C₅₉⋊C₈	—	2	2	no	—
83	C₂²×C₅₉	—	2	1	no	—
84	C₂²×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₅₉	#120	1	1	yes	characteristic
87	C₂²×C₅₉	#123	1	1	yes	characteristic
88	C₄×C₅₉	—	4	2	no	—
89	C₄×C₅₉	—	4	1	no	—
90	C₄×C₅₉	—	4	1	no	—
91	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
92	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
93	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
95	Dic₅₉	—	2	4	no	—
96	Dic₅₉	—	4	1	no	—
97	Dic₅₉	—	4	1	no	—
98	Dic₅₉	—	8	1	no	—
99	#1857	—	59	1	no	maximal, Sylow
100	C₂×C₅₉	—	2	1	no	—
101	C₂×C₅₉	—	2	1	no	—
102	C₂×C₅₉	#385	1	1	yes	characteristic
103	C₂×C₅₉	#454	1	1	yes	characteristic
104	C₂×C₅₉	#456	1	1	yes	characteristic
105	#519	—	59	1	no	—
106	#428	—	59	1	no	—
107	#428	—	59	1	no	—
108	C₂×#101	—	59	1	no	—
109	#391	—	59	1	no	—
110	#339	—	59	1	no	—
111	#339	D₅₉	1	1	yes	characteristic
112	C₅₉	#1857	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	59	1	no	—
114	C₂×M₁₆	—	59	1	no	—
115	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
116	C₄×Q₈	—	118	1	no	—
117	C₄×Q₈	—	118	1	no	—
118	C₄×Q₈	—	118	1	no	—
119	C₄×Q₈	—	118	1	no	—
120	#117	—	59	1	no	—
121	#117	—	118	2	no	—
122	#117	C₂×D₅₉	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	59	1	no	—
124	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
125	#107	—	118	1	no	—
126	#107	—	118	1	no	—
127	#106	—	118	1	no	—
128	#106	—	118	1	no	—
129	#101	—	59	4	no	—
130	#95	—	118	2	no	—
131	C₂×Q₈	—	59	2	no	—
132	C₂×Q₈	—	59	2	no	—
133	C₂×Q₈	—	118	1	no	—
134	C₂×Q₈	—	118	1	no	—
135	C₂×Q₈	—	118	1	no	—
136	C₂×Q₈	—	118	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	59	2	no	—
139	C₂²×C₄	—	118	1	no	—
140	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
141	M₁₆	—	2	1	no	—
142	M₁₆	—	59	2	no	—
143	M₁₆	—	118	1	no	—
144	M₁₆	C₄×D₅₉	1	2	yes	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	118	1	no	—
147	C₄⋊C₄	—	59	1	no	—
148	C₄⋊C₄	—	59	1	no	—
149	C₄⋊C₄	—	118	2	no	—
150	C₄⋊C₄	—	118	2	no	—
151	C₄⋊C₄	—	118	1	no	—
152	C₄⋊C₄	—	118	1	no	—
153	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
154	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
155	C₂²⋊C₄	—	2	1	no	—
156	C₂²⋊C₄	—	118	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	118	2	no	—
161	C₄²	—	118	2	no	—
162	C₄²	—	118	1	no	—
163	C₂³	D₄×D₅₉	1	1	yes	characteristic
164	Q₈	—	118	2	no	—
165	Q₈	—	118	2	no	—
166	Q₈	—	118	2	no	—
167	Q₈	—	118	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	59	4	no	—
174	C₂×C₄	—	118	2	no	—
175	C₂×C₄	—	118	2	no	—
176	C₂×C₄	—	118	1	no	—
177	C₂×C₄	—	118	1	no	—
178	C₂×C₄	—	236	2	no	—
179	C₂×C₄	—	236	1	no	—
180	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
182	C₂×C₄	#11745685	1	1	yes	characteristic
183	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
184	C₂×C₄	#11745687	1	1	yes	characteristic
185	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	118	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
191	C₂²	#14273697	1	1	yes	characteristic
192	C₂²	#14273709	1	1	yes	characteristic
193	C₄	—	4	2	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	118	4	no	—
197	C₄	—	472	1	no	—
198	C₄	—	236	1	no	—
199	C₄	—	236	1	no	—
200	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
201	C₄	C₄×D₂₃₆	1	1	yes	characteristic
202	C₄	#14273705	1	1	yes	characteristic
203	C₄	#14273707	1	1	yes	characteristic
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#16837276	1	1	yes	characteristic
207	C₂	#16837576	1	1	yes	characteristic
208	C₂	#16837586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837879

C₂

yes

maximal, characteristic

#16837445

C₂

yes

maximal, characteristic

#16837444

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, characteristic

#16837308

C₂

yes

maximal, characteristic

#16837162

C₂

yes

maximal, characteristic

C₅₉×#339

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

#14273692

—

#14273692

—

#14273686

C₂²

yes

characteristic

#14273683

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₅₉⋊#95

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×Dic₁₁₈

Q₈

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

D₄

yes

characteristic

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

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×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1857

—

maximal, Sylow

100

C₂×C₅₉

—

101

C₂×C₅₉

—

102

C₂×C₅₉

#385

yes

characteristic

103

C₂×C₅₉

#454

yes

characteristic

104

C₂×C₅₉

#456

yes

characteristic

105

#519

—

106

#428

—

107

#428

—

108

C₂×#101

—

109

#391

—

110

#339

—

111

#339

D₅₉

yes

characteristic

112

C₅₉

#1857

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂×M₁₆

—

115

C₂×M₁₆

C₂×D₅₉

yes

characteristic

116

C₄×Q₈

—

118

—

117

C₄×Q₈

—

118

—

118

C₄×Q₈

—

118

—

119

C₄×Q₈

—

118

—

120

#117

—

121

#117

—

118

—

122

#117

C₂×D₅₉

yes

characteristic

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂×D₅₉

yes

characteristic

125

#107

—

118

—

126

#107

—

118

—

127

#106

—

118

—

128

#106

—

118

—

129

#101

—

130

#95

—

118

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

118

—

134

C₂×Q₈

—

118

—

135

C₂×Q₈

—

118

—

136

C₂×Q₈

—

118

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

118

—

140

C₂²×C₄

C₂²×D₅₉

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

118

—

144

M₁₆

C₄×D₅₉

yes

—

145

C₂×C₈

—

146

C₂×C₈

—

118

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

118

—

150

C₄⋊C₄

—

118

—

151

C₄⋊C₄

—

118

—

152

C₄⋊C₄

—

118

—

153

C₄⋊C₄

D₂₃₆

yes

characteristic

154

C₄⋊C₄

D₂₃₆

yes

characteristic

155

C₂²⋊C₄

—

156

C₂²⋊C₄

—

118

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

118

—

161

C₄²

—

118

—

162

C₄²

—

118

—

163

C₂³

D₄×D₅₉

yes

characteristic

164

Q₈

—

118

—

165

Q₈

—

118

—

166

Q₈

—

118

—

167

Q₈

—

118

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

118

—

175

C₂×C₄

—

118

—

176

C₂×C₄

—

118

—

177

C₂×C₄

—

118

—

178

C₂×C₄

—

236

—

179

C₂×C₄

—

236

—

180

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

181

C₂×C₄

C₂×D₂₃₆

yes

characteristic

182

C₂×C₄

#11745685

yes

characteristic

183

C₂×C₄

D₄×D₅₉

yes

characteristic

184

C₂×C₄

#11745687

yes

characteristic

185

C₂×C₄

Q₈×D₅₉

yes

characteristic

186

C₈

—

187

C₈

—

118

—

188

C₂²

—

189

C₂²

—

190

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

191

C₂²

#14273697

yes

characteristic

192

C₂²

#14273709

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

118

—

197

C₄

—

472

—

198

C₄

—

236

—

199

C₄

—

236

—

200

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

201

C₄

C₄×D₂₃₆

yes

characteristic

202

C₄

#14273705

yes

characteristic

203

C₄

#14273707

yes

characteristic

204

C₂

—

205

C₂

—

206

C₂

#16837276

yes

characteristic

207

C₂

#16837576

yes

characteristic

208

C₂

#16837586

yes

characteristic

	subgroup	G/H
center	C₂²	#14273697
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#339	C₂
socle	C₂²×C₅₉	#120
hypercenter	#339	D₅₉
hypocenter	C₅₉	#1857
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂²

#14273697

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#339

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
21. #14273645	C₄	semidirect
112. C₅₉	#1857	semidirect

H ≅ G/N

type

21. #14273645

C₄

semidirect

112. C₅₉

#1857

semidirect

Subgroups — #21424312

Conjugacy classes of subgroups

Complements