Group #21424853 — subgroups

Subgroups — #21424853

Order 7552.

256 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 107 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×#14273661	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273660	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#16837271	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×Dic₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837164	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#340	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₉⋊C₄	—	2	1	no	—
12	C₂×Dic₅₉⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₅₉	—	2	1	no	—
14	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
15	#14273661	—	2	2	no	—
16	#14273660	—	2	2	no	—
17	Dic₅₉⋊C₈	C₄	1	4	yes	complemented
18	C₅₉⋊#95	—	2	1	no	—
19	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
27	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
28	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	—	2	1	no	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₄⋊C₄×C₅₉	—	2	1	no	—
33	C₄⋊C₄×C₅₉	—	2	1	no	—
34	C₄⋊C₄×C₅₉	—	2	1	no	—
35	C₄⋊C₄×C₅₉	Q₈	1	2	yes	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	—	2	1	no	—
38	C₄⋊Dic₅₉	—	2	1	no	—
39	C₄⋊Dic₅₉	D₄	1	2	yes	—
40	Dic₅₉⋊C₄	—	4	2	no	—
41	Dic₅₉⋊C₄	—	4	2	no	—
42	C₄×Dic₅₉	—	4	2	no	—
43	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
44	C₄×Dic₅₉	Q₈	1	2	yes	—
45	C₂×C₅₉⋊C₈	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
48	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	—	2	2	no	—
50	C₂×C₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	—	4	1	no	—
55	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
61	C₂×Dic₅₉	—	2	1	no	—
62	C₂×Dic₅₉	—	2	1	no	—
63	C₂×Dic₅₉	—	2	2	no	—
64	C₂×Dic₅₉	—	4	1	no	—
65	C₂×Dic₅₉	—	4	1	no	—
66	C₂×Dic₅₉	—	4	1	no	—
67	C₂×Dic₅₉	—	4	1	no	—
68	C₂×Dic₅₉	—	4	1	no	—
69	C₂×Dic₅₉	C₄⋊C₄	1	2	yes	—
70	C₂×Dic₅₉	C₄⋊C₄	1	2	yes	—
71	C₂×Dic₅₉	D₈	1	2	yes	—
72	C₂×Dic₅₉	Q₁₆	1	2	yes	—
73	C₈×C₅₉	—	2	2	no	—
74	C₅₉⋊C₈	—	2	2	no	—
75	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₅₉	#121	1	1	yes	characteristic
77	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
79	C₂²×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
80	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²×C₅₉	#137	1	1	yes	characteristic
82	C₄×C₅₉	—	4	2	no	—
83	C₄×C₅₉	—	8	1	no	—
84	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
85	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
86	C₄×C₅₉	#125	1	1	yes	characteristic
87	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
88	Dic₅₉	—	2	2	no	—
89	Dic₅₉	—	4	2	no	—
90	Dic₅₉	—	8	1	no	—
91	Dic₅₉	#107	1	4	yes	complemented
92	#1947	—	59	1	no	maximal, Sylow
93	C₂×C₅₉	#389	1	1	yes	characteristic
94	C₂×C₅₉	C₂×#107	1	1	yes	characteristic
95	C₂×C₅₉	#428	1	1	yes	characteristic
96	C₂×C₅₉	#459	1	1	yes	characteristic
97	C₂×C₅₉	#462	1	1	yes	characteristic
98	C₂×C₅₉	#474	1	1	yes	characteristic
99	C₂×C₅₉	#477	1	1	yes	characteristic
100	C₂×#107	—	59	1	no	—
101	C₂×#107	—	59	1	no	—
102	C₂×C₄⋊C₈	—	59	1	no	—
103	#384	—	59	1	no	—
104	C₄×C₄⋊C₄	—	59	1	no	—
105	#340	—	59	1	no	—
106	#340	D₅₉	1	1	yes	characteristic
107	C₅₉	#1947	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	—	59	1	no	—
109	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
110	C₂×C₄⋊C₄	—	59	1	no	—
111	C₂×C₄⋊C₄	—	59	1	no	—
112	C₂×C₄⋊C₄	—	118	1	no	—
113	C₂×C₄⋊C₄	—	118	1	no	—
114	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
116	C₂×C₄²	—	59	1	no	—
117	C₂×C₄²	—	118	1	no	—
118	#107	—	118	2	no	—
119	#107	—	118	2	no	—
120	C₄⋊C₈	—	59	4	no	—
121	#95	—	118	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	59	1	no	—
125	C₂²×C₄	—	59	1	no	—
126	C₂²×C₄	—	118	1	no	—
127	C₂²×C₄	—	118	1	no	—
128	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	59	2	no	—
132	C₂×C₈	—	118	1	no	—
133	C₂×C₈	—	118	1	no	—
134	C₂×C₈	C₄×D₅₉	1	2	yes	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	59	2	no	—
139	C₄⋊C₄	—	118	1	no	—
140	C₄⋊C₄	—	118	1	no	—
141	C₄⋊C₄	—	118	1	no	—
142	C₄⋊C₄	—	236	2	no	—
143	C₄⋊C₄	—	236	2	no	—
144	C₄⋊C₄	Dic₁₁₈	1	2	yes	—
145	C₄²	—	59	2	no	—
146	C₄²	—	59	2	no	—
147	C₄²	—	236	2	no	—
148	C₂³	D₄×D₅₉	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	59	2	no	—
156	C₂×C₄	—	59	2	no	—
157	C₂×C₄	—	59	2	no	—
158	C₂×C₄	—	59	2	no	—
159	C₂×C₄	—	118	1	no	—
160	C₂×C₄	—	118	1	no	—
161	C₂×C₄	—	118	2	no	—
162	C₂×C₄	—	236	1	no	—
163	C₂×C₄	—	236	1	no	—
164	C₂×C₄	—	236	1	no	—
165	C₂×C₄	—	236	1	no	—
166	C₂×C₄	—	236	1	no	—
167	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
169	C₂×C₄	#11745685	1	1	yes	characteristic
170	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
171	C₂×C₄	#11745687	1	1	yes	characteristic
172	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	118	2	no	—
175	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
176	C₂²	#14273700	1	1	yes	characteristic
177	C₂²	#14273704	1	1	yes	characteristic
178	C₂²	D₈×D₅₉	1	1	yes	characteristic
179	C₂²	Q₁₆×D₅₉	1	1	yes	characteristic
180	C₂²	#14273754	1	1	yes	characteristic
181	C₂²	#14273756	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	59	4	no	—
185	C₄	—	118	2	no	—
186	C₄	—	472	1	no	—
187	C₄	—	236	2	no	—
188	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
189	C₄	C₄×Dic₁₁₈	1	1	yes	characteristic
190	C₄	#14273717	1	1	yes	characteristic
191	C₄	#14273726	1	1	yes	characteristic
192	C₂	#16837299	1	1	yes	characteristic
193	C₂	#107×D₅₉	1	1	yes	characteristic
194	C₂	#16837441	1	1	yes	characteristic
195	C₂	#16837596	1	1	yes	characteristic
196	C₂	#16837611	1	1	yes	characteristic
197	C₂	#16837680	1	1	yes	characteristic
198	C₂	#16837695	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal, characteristic

C₂×#14273660

C₂

yes

maximal, characteristic

C₂×Dic₅₉⋊C₈

C₂

yes

maximal, characteristic

#16837271

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₅₉

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal, characteristic

C₅₉×#340

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273661

—

#14273660

—

Dic₅₉⋊C₈

C₄

yes

complemented

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

Q₈

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

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×Q₈

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

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₂×Dic₅₉

D₈

yes

—

C₂×Dic₅₉

Q₁₆

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#121

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂²×C₅₉

C₂×Q₁₆

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

C₄×C₅₉

#125

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

100

—

101

C₂×#107

—

102

C₂×C₄⋊C₈

—

103

#384

—

104

C₄×C₄⋊C₄

—

105

#340

—

106

#340

D₅₉

yes

characteristic

107

C₅₉

#1947

yes

characteristic, complemented, Sylow

108

C₂²×C₈

—

109

C₂²×C₈

C₂×D₅₉

yes

characteristic

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

118

—

113

C₂×C₄⋊C₄

—

118

—

114

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

115

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

118

—

118

#107

—

118

—

119

#107

—

118

—

120

C₄⋊C₈

—

121

#95

—

118

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

118

—

127

C₂²×C₄

—

118

—

128

C₂²×C₄

C₂²×D₅₉

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

118

—

133

C₂×C₈

—

118

—

134

C₂×C₈

C₄×D₅₉

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

118

—

140

C₄⋊C₄

—

118

—

141

C₄⋊C₄

—

118

—

142

C₄⋊C₄

—

236

—

143

C₄⋊C₄

—

236

—

144

C₄⋊C₄

Dic₁₁₈

yes

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

236

—

148

C₂³

D₄×D₅₉

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

118

—

160

C₂×C₄

—

118

—

161

C₂×C₄

—

118

—

162

C₂×C₄

—

236

—

163

C₂×C₄

—

236

—

164

C₂×C₄

—

236

—

165

C₂×C₄

—

236

—

166

C₂×C₄

—

236

—

167

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

168

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

169

C₂×C₄

#11745685

yes

characteristic

170

C₂×C₄

D₄×D₅₉

yes

characteristic

171

C₂×C₄

#11745687

yes

characteristic

172

C₂×C₄

Q₈×D₅₉

yes

characteristic

173

C₈

—

174

C₈

—

118

—

175

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

176

C₂²

#14273700

yes

characteristic

177

C₂²

#14273704

yes

characteristic

178

C₂²

D₈×D₅₉

yes

characteristic

179

C₂²

Q₁₆×D₅₉

yes

characteristic

180

C₂²

#14273754

yes

characteristic

181

C₂²

#14273756

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

118

—

186

C₄

—

472

—

187

C₄

—

236

—

188

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

189

C₄

C₄×Dic₁₁₈

yes

characteristic

190

C₄

#14273717

yes

characteristic

191

C₄

#14273726

yes

characteristic

192

C₂

#16837299

yes

characteristic

193

C₂

#107×D₅₉

yes

characteristic

194

C₂

#16837441

yes

characteristic

195

C₂

#16837596

yes

characteristic

196

C₂

#16837611

yes

characteristic

197

C₂

#16837680

yes

characteristic

198

C₂

#16837695

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#340	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#340	D₅₉
hypocenter	C₅₉	#1947
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#340

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
17. Dic₅₉⋊C₈	C₄	semidirect
91. Dic₅₉	#107	semidirect
107. C₅₉	#1947	semidirect

H ≅ G/N

type

17. Dic₅₉⋊C₈

C₄

semidirect

91. Dic₅₉

#107

semidirect

107. C₅₉

#1947

semidirect

Subgroups — #21424853

Conjugacy classes of subgroups

Complements