Group #21422549 — subgroups

Subgroups — #21422549

Order 7552.

232 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 75 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	#16837956	C₂	1	1	yes	maximal, characteristic
2	#16837424	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837134	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837132	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745659	—	2	1	no	—
9	C₂×C₈×D₅₉	—	2	1	no	—
10	#14273726	C₄	1	1	yes	characteristic
11	#14273725	C₄	1	1	yes	characteristic
12	#14273717	—	2	1	no	—
13	#14273707	C₄	1	1	yes	characteristic
14	#14273705	—	2	1	no	—
15	#14273703	C₄	1	1	yes	characteristic
16	C₄×Dic₁₁₈	—	2	1	no	—
17	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
18	#14273684	—	2	1	no	—
19	C₄²×D₅₉	C₂²	1	1	yes	characteristic
20	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
21	Dic₅₉⋊C₈	—	2	1	no	—
22	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₅₉	—	2	1	no	—
24	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
30	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
31	C₂×Dic₁₁₈	—	2	1	no	—
32	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₄⋊C₄×C₅₉	—	2	1	no	—
36	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₅₉	—	2	1	no	—
38	C₄²×C₅₉	C₂³	1	1	yes	characteristic
39	(C₂×D₅₉)⋊C₄	—	2	1	no	—
40	(C₂×D₅₉)⋊C₄	—	2	1	no	—
41	C₄⋊Dic₅₉	—	2	1	no	—
42	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
43	Dic₅₉⋊C₄	—	2	1	no	—
44	Dic₅₉⋊C₄	—	2	1	no	—
45	Dic₅₉⋊C₄	—	2	1	no	—
46	Dic₅₉⋊C₄	—	2	1	no	—
47	Dic₅₉⋊C₄	—	2	1	no	—
48	C₄×Dic₅₉	—	2	1	no	—
49	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₅₉	D₄	1	1	yes	characteristic
51	C₄×Dic₅₉	D₄	1	1	yes	characteristic
52	C₂×C₅₉⋊C₈	—	2	1	no	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	#11745659	—	4	2	no	—
55	C₈×D₅₉	—	2	4	no	—
56	C₂²×D₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	Q₈×C₅₉	—	2	1	no	—
58	Q₈×C₅₉	—	2	1	no	—
59	C₂×C₄×C₅₉	—	2	1	no	—
60	C₂×C₄×C₅₉	—	2	1	no	—
61	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₅₉	—	2	1	no	—
65	C₂×Dic₅₉	—	2	1	no	—
66	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₄×D₅₉	—	2	1	no	—
70	C₄×D₅₉	—	2	1	no	—
71	C₄×D₅₉	—	2	2	no	—
72	C₄×D₅₉	SD₁₆	1	2	yes	—
73	C₄×D₅₉	Q₁₆	1	2	yes	—
74	Dic₁₁₈	—	2	1	no	—
75	Dic₁₁₈	—	2	1	no	—
76	Dic₁₁₈	—	4	2	no	—
77	C₈×C₅₉	—	2	2	no	—
78	C₈×C₅₉	—	4	1	no	—
79	C₅₉⋊C₈	—	2	2	no	—
80	C₅₉⋊C₈	—	4	1	no	—
81	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×D₅₉	—	2	1	no	—
83	C₂×D₅₉	—	2	1	no	—
84	C₂×D₅₉	#103	1	2	yes	—
85	C₄×C₅₉	—	2	1	no	—
86	C₄×C₅₉	—	4	1	no	—
87	C₄×C₅₉	—	4	1	no	—
88	C₄×C₅₉	#131	1	1	yes	characteristic
89	C₄×C₅₉	#131	1	1	yes	characteristic
90	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
92	Dic₅₉	—	2	1	no	—
93	Dic₅₉	—	4	1	no	—
94	Dic₅₉	—	4	1	no	—
95	Dic₅₉	#103	1	2	yes	—
96	Dic₅₉	#104	1	2	yes	—
97	#1493	—	59	1	no	maximal, Sylow
98	C₂×C₅₉	#408	1	1	yes	characteristic
99	C₂×C₅₉	C₂×#103	1	1	yes	characteristic
100	C₂×C₅₉	C₂×#104	1	1	yes	characteristic
101	D₅₉	—	2	2	no	—
102	#533	—	59	1	no	—
103	#423	—	59	1	no	—
104	C₂×C₄×C₈	—	59	1	no	—
105	Q₈⋊C₈	—	59	1	no	—
106	Q₈⋊C₈	—	59	1	no	—
107	Q₈⋊C₈	—	59	1	no	—
108	Q₈⋊C₈	D₅₉	1	1	yes	characteristic, complemented
109	C₅₉	#1493	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	118	1	no	—
111	C₂²×C₈	—	118	1	no	—
112	C₄⋊Q₈	—	59	1	no	—
113	C₄⋊Q₈	—	59	1	no	—
114	#125	—	118	1	no	—
115	C₂²⋊Q₈	—	59	1	no	—
116	C₂²⋊Q₈	—	59	1	no	—
117	C₂²⋊Q₈	—	118	1	no	—
118	C₄×Q₈	—	59	1	no	—
119	C₄×Q₈	—	118	1	no	—
120	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
121	#117	—	118	1	no	—
122	C₂×C₄²	—	59	1	no	—
123	C₄⋊C₈	—	59	1	no	—
124	C₄⋊C₈	—	118	1	no	—
125	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
126	C₄×C₈	—	59	1	no	—
127	C₄×C₈	—	118	1	no	—
128	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
129	C₂×Q₈	—	59	1	no	—
130	C₂×Q₈	—	118	1	no	—
131	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
132	C₂²×C₄	—	59	1	no	—
133	C₂²×C₄	—	59	1	no	—
134	C₂²×C₄	—	59	1	no	—
135	M₁₆	—	236	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	118	4	no	—
139	C₂×C₈	—	118	1	no	—
140	C₂×C₈	—	118	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	59	1	no	—
143	C₄⋊C₄	—	118	1	no	—
144	C₄⋊C₄	—	118	1	no	—
145	C₄⋊C₄	—	118	1	no	—
146	C₄⋊C₄	—	118	1	no	—
147	C₄⋊C₄	—	118	1	no	—
148	C₄⋊C₄	—	118	1	no	—
149	C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
150	C₂²⋊C₄	—	118	1	no	—
151	C₂²⋊C₄	—	118	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	59	1	no	—
154	C₄²	—	59	1	no	—
155	C₄²	—	59	1	no	—
156	C₄²	—	118	1	no	—
157	C₄²	C₂²×D₅₉	1	1	yes	characteristic
158	C₂³	—	59	1	no	—
159	Q₈	—	2	1	no	—
160	Q₈	—	2	1	no	—
161	Q₈	—	118	1	no	—
162	Q₈	—	118	1	no	—
163	Q₈	—	236	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	59	1	no	—
167	C₂×C₄	—	59	1	no	—
168	C₂×C₄	—	59	2	no	—
169	C₂×C₄	—	59	2	no	—
170	C₂×C₄	—	59	1	no	—
171	C₂×C₄	—	118	2	no	—
172	C₂×C₄	—	118	1	no	—
173	C₂×C₄	—	118	1	no	—
174	C₂×C₄	—	118	1	no	—
175	C₂×C₄	—	118	1	no	—
176	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
177	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
178	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	4	1	no	—
181	C₈	—	118	2	no	—
182	C₈	—	236	1	no	—
183	C₂²	—	59	2	no	—
184	C₂²	—	118	1	no	—
185	C₂²	—	118	1	no	—
186	C₂²	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	1	no	—
190	C₄	—	59	2	no	—
191	C₄	—	59	2	no	—
192	C₄	—	118	1	no	—
193	C₄	—	236	1	no	—
194	C₄	—	236	1	no	—
195	C₄	#14273733	1	1	yes	characteristic
196	C₄	#14273735	1	1	yes	characteristic
197	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
198	C₄	Q₁₆×D₅₉	1	1	yes	characteristic
199	C₂	—	118	2	no	—
200	C₂	#16837367	1	1	yes	characteristic
201	C₂	#103×D₅₉	1	1	yes	characteristic
202	C₂	#104×D₅₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837956

C₂

yes

maximal, characteristic

#16837424

C₂

yes

maximal, characteristic

C₄×C₈×D₅₉

C₂

yes

maximal, characteristic

#16837134

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₅₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

#16837132

C₂

yes

maximal, characteristic, complemented

C₅₉×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×#11745659

—

C₂×C₈×D₅₉

—

#14273726

C₄

yes

characteristic

#14273725

C₄

yes

characteristic

#14273717

—

#14273707

C₄

yes

characteristic

#14273705

—

#14273703

C₄

yes

characteristic

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

#14273684

—

C₄²×D₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₅₉

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

#11745659

—

C₈×D₅₉

—

C₂²×D₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×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×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

SD₁₆

yes

—

C₄×D₅₉

Q₁₆

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

#103

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

#103

yes

—

Dic₅₉

#104

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

100

C₂×C₅₉

C₂×#104

yes

characteristic

101

D₅₉

—

102

#533

—

103

#423

—

104

C₂×C₄×C₈

—

105

Q₈⋊C₈

—

106

Q₈⋊C₈

—

107

Q₈⋊C₈

—

108

Q₈⋊C₈

D₅₉

yes

characteristic, complemented

109

C₅₉

#1493

yes

characteristic, complemented, Sylow

110

C₂×M₁₆

—

118

—

111

C₂²×C₈

—

118

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

#125

—

118

—

115

C₂²⋊Q₈

—

116

C₂²⋊Q₈

—

117

C₂²⋊Q₈

—

118

—

118

C₄×Q₈

—

119

C₄×Q₈

—

118

—

120

C₄×Q₈

C₂×D₅₉

yes

characteristic

121

#117

—

118

—

122

C₂×C₄²

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

118

—

125

C₄⋊C₈

C₂×D₅₉

yes

characteristic

126

C₄×C₈

—

127

C₄×C₈

—

118

—

128

C₄×C₈

C₂×D₅₉

yes

characteristic

129

C₂×Q₈

—

130

C₂×Q₈

—

118

—

131

C₂×Q₈

C₄×D₅₉

yes

characteristic

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

M₁₆

—

236

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

118

—

139

C₂×C₈

—

118

—

140

C₂×C₈

—

118

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

118

—

144

C₄⋊C₄

—

118

—

145

C₄⋊C₄

—

118

—

146

C₄⋊C₄

—

118

—

147

C₄⋊C₄

—

118

—

148

C₄⋊C₄

—

118

—

149

C₄⋊C₄

C₄×D₅₉

yes

characteristic

150

C₂²⋊C₄

—

118

—

151

C₂²⋊C₄

—

118

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

118

—

157

C₄²

C₂²×D₅₉

yes

characteristic

158

C₂³

—

159

Q₈

—

160

Q₈

—

161

Q₈

—

118

—

162

Q₈

—

118

—

163

Q₈

—

236

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

118

—

172

C₂×C₄

—

118

—

173

C₂×C₄

—

118

—

174

C₂×C₄

—

118

—

175

C₂×C₄

—

118

—

176

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

177

C₂×C₄

D₄×D₅₉

yes

characteristic

178

C₂×C₄

D₄×D₅₉

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

118

—

182

C₈

—

236

—

183

C₂²

—

184

C₂²

—

118

—

185

C₂²

—

118

—

186

C₂²

C₂²⋊C₄×D₅₉

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

118

—

193

C₄

—

236

—

194

C₄

—

236

—

195

C₄

#14273733

yes

characteristic

196

C₄

#14273735

yes

characteristic

197

C₄

SD₁₆×D₅₉

yes

characteristic

198

C₄

Q₁₆×D₅₉

yes

characteristic

199

C₂

—

118

—

200

C₂

#16837367

yes

characteristic

201

C₂

#103×D₅₉

yes

characteristic

202

C₂

#104×D₅₉

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×Q₈⋊C₈	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	Q₈⋊C₈	D₅₉
hypocenter	C₅₉	#1493
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×Q₈⋊C₈

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

hypercenter

Q₈⋊C₈

D₅₉

hypocenter

C₅₉

#1493

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
4. #16837134	C₂	semidirect
5. Q₈⋊(C₅₉⋊C₈)	C₂	semidirect
6. #16837132	C₂	semidirect
7. C₅₉×Q₈⋊C₈	C₂	semidirect
108. Q₈⋊C₈	D₅₉	semidirect
109. C₅₉	#1493	semidirect

H ≅ G/N

type

4. #16837134

C₂

semidirect

5. Q₈⋊(C₅₉⋊C₈)

C₂

semidirect

6. #16837132

C₂

semidirect

7. C₅₉×Q₈⋊C₈

C₂

semidirect

108. Q₈⋊C₈

D₅₉

semidirect

109. C₅₉

#1493

semidirect

Subgroups — #21422549

Conjugacy classes of subgroups

Complements