Group #21422476 — subgroups

Subgroups — #21422476

Order 7552.

276 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₄×#11745685	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837132	C₂	1	2	yes	maximal, complemented
4	#16837129	C₂	1	2	yes	maximal, complemented
5	C₅₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745685	C₄	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₈	—	2	1	no	—
8	C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
9	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
10	C₄×D₂₃₆	C₄	1	1	yes	characteristic
11	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
12	C₄×C₂²⋊D₅₉	—	2	1	no	—
13	C₄×C₂²⋊D₅₉	—	2	1	no	—
14	#14273686	C₄	1	1	yes	characteristic
15	#14273684	—	2	1	no	—
16	C₄²×D₅₉	—	2	1	no	—
17	C₄×C₅₉⋊C₈	—	2	1	no	—
18	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
21	C₂²⋊C₈×C₅₉	—	2	1	no	—
22	C₄×C₈×C₅₉	—	2	1	no	—
23	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂×C₂²⋊D₅₉	—	2	1	no	—
27	#11745685	—	2	2	no	—
28	#11745685	C₈	1	4	yes	complemented
29	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₉	—	2	1	no	—
31	C₂×C₄×D₅₉	—	2	1	no	—
32	C₂×C₄×D₅₉	—	2	1	no	—
33	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	—	2	2	no	—
35	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₅₉	D₄	1	1	yes	characteristic
37	C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₄²×C₅₉	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₅₉	—	2	1	no	—
40	(C₂×D₅₉)⋊C₄	—	2	1	no	—
41	(C₂×D₅₉)⋊C₄	—	2	1	no	—
42	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
44	Dic₅₉⋊C₄	—	2	1	no	—
45	Dic₅₉⋊C₄	—	2	1	no	—
46	C₄×Dic₅₉	—	2	1	no	—
47	C₄×Dic₅₉	—	2	1	no	—
48	C₄×Dic₅₉	—	2	1	no	—
49	C₂×C₅₉⋊C₈	—	2	4	no	—
50	C₂×C₅₉⋊C₈	—	2	2	no	—
51	C₂³×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×D₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×C₈	1	2	yes	—
59	C₂×C₄×C₅₉	M₁₆	1	2	yes	—
60	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²⋊D₅₉	—	2	4	no	—
64	C₂×Dic₅₉	—	2	1	no	—
65	C₂×Dic₅₉	—	2	1	no	—
66	C₂×Dic₅₉	—	2	1	no	—
67	D₂₃₆	—	2	1	no	—
68	D₂₃₆	C₂×C₈	1	2	yes	complemented
69	C₄×D₅₉	—	2	4	no	—
70	C₄×D₅₉	—	4	2	no	—
71	C₄×D₅₉	—	4	2	no	—
72	Dic₁₁₈	—	2	1	no	—
73	Dic₁₁₈	C₂×C₈	1	2	yes	complemented
74	C₈×C₅₉	—	4	2	no	—
75	C₅₉⋊C₈	—	2	4	no	—
76	C₂²×C₅₉	—	2	1	no	—
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	C₂²⋊C₈	1	2	yes	—
79	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×D₅₉	—	2	2	no	—
81	C₂×D₅₉	—	4	1	no	—
82	C₂×D₅₉	—	4	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	—	2	1	no	—
85	C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
86	C₄×C₅₉	#104	1	2	yes	—
87	C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
88	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
89	C₄×C₅₉	#135	1	2	yes	—
90	Dic₅₉	—	2	2	no	—
91	Dic₅₉	—	4	1	no	—
92	Dic₅₉	—	4	1	no	—
93	#1476	—	59	1	no	maximal, Sylow
94	C₂×C₅₉	—	2	2	no	—
95	C₂×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₅₉	#416	1	1	yes	characteristic
97	C₂×C₅₉	C₂×#104	1	1	yes	characteristic
98	D₅₉	—	4	2	no	—
99	C₄×#41	—	59	1	no	—
100	#431	D₅₉	1	1	yes	characteristic, complemented
101	C₂×C₄×C₈	—	59	1	no	—
102	Q₈⋊C₈	—	59	2	no	—
103	D₄⋊C₈	—	59	2	no	—
104	C₅₉	#1476	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	59	1	no	—
106	C₂²×C₈	—	118	1	no	—
107	C₄×Q₈	—	59	1	no	—
108	C₄×Q₈	—	59	1	no	—
109	C₄×D₄	—	59	1	no	—
110	C₄×D₄	—	59	1	no	—
111	C₄×D₄	—	118	1	no	—
112	C₄×D₄	—	118	1	no	—
113	#117	—	59	1	no	—
114	#117	—	118	1	no	—
115	C₂×C₄²	—	118	1	no	—
116	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
117	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
118	C₂²⋊C₈	—	2	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	59	2	no	—
121	C₄×C₈	—	118	1	no	—
122	#41	—	59	4	no	—
123	#41	—	118	2	no	—
124	C₂×Q₈	—	59	1	no	—
125	C₂×D₄	—	59	1	no	—
126	C₂×D₄	—	118	1	no	—
127	C₂²×C₄	—	118	1	no	—
128	C₂²×C₄	—	118	1	no	—
129	C₂²×C₄	—	118	1	no	—
130	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
131	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	118	4	no	—
135	C₂×C₈	—	118	2	no	—
136	C₄⋊C₄	—	59	1	no	—
137	C₄⋊C₄	—	59	1	no	—
138	C₄⋊C₄	—	118	1	no	—
139	C₄⋊C₄	—	118	1	no	—
140	C₂²⋊C₄	—	118	1	no	—
141	C₂²⋊C₄	—	118	1	no	—
142	C₂²⋊C₄	—	118	1	no	—
143	C₄²	—	118	1	no	—
144	C₄²	—	118	1	no	—
145	C₄²	—	118	1	no	—
146	C₄²	C₄×D₅₉	1	1	yes	characteristic
147	C₄²	D₂₃₆	1	1	yes	characteristic
148	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
149	C₄²	C₂²×D₅₉	1	1	yes	characteristic
150	C₂³	—	118	1	no	—
151	C₂³	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
152	Q₈	—	59	2	no	—
153	Q₈	—	118	1	no	—
154	D₄	—	59	2	no	—
155	D₄	—	118	4	no	—
156	D₄	—	118	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	118	1	no	—
160	C₂×C₄	—	118	4	no	—
161	C₂×C₄	—	118	1	no	—
162	C₂×C₄	—	118	1	no	—
163	C₂×C₄	—	236	2	no	—
164	C₂×C₄	—	236	2	no	—
165	C₂×C₄	C₈×D₅₉	1	2	yes	—
166	C₂×C₄	#11745659	1	2	yes	—
167	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
168	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
169	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
171	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
172	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
173	C₈	—	4	2	no	—
174	C₈	—	118	4	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	118	2	no	—
178	C₂²	—	236	1	no	—
179	C₂²	—	236	1	no	—
180	C₂²	(C₂×D₅₉)⋊C₈	1	2	yes	—
181	C₂²	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	118	2	no	—
185	C₄	—	236	1	no	—
186	C₄	—	236	1	no	—
187	C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
188	C₄	#14273653	1	2	yes	—
189	C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
190	C₄	C₂×#11745659	1	1	yes	characteristic
191	C₄	#14273747	1	2	yes	—
192	C₂	—	2	2	no	—
193	C₂	—	236	2	no	—
194	C₂	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
195	C₂	#16837396	1	1	yes	characteristic
196	C₂	C₂×#14273653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11745685

C₂

yes

maximal, characteristic

C₂×C₄×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#16837132

C₂

yes

maximal, complemented

#16837129

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₄

yes

characteristic

C₂²×C₅₉⋊C₈

—

C₄×Dic₁₁₈

C₄

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

characteristic

C₄×D₂₃₆

C₄

yes

characteristic

C₄×D₂₃₆

C₂²

yes

characteristic

C₄×C₂²⋊D₅₉

—

C₄×C₂²⋊D₅₉

—

#14273686

C₄

yes

characteristic

#14273684

—

C₄²×D₅₉

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×C₂²⋊D₅₉

—

#11745685

—

#11745685

C₈

yes

complemented

C₂×D₂₃₆

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

C₄⋊Dic₅₉

C₂×C₄

yes

characteristic

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂²⋊C₄

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₈

yes

—

C₂×C₄×C₅₉

M₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

C₂×C₈

yes

complemented

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

C₂×C₈

yes

complemented

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²⋊C₈

yes

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊C₈

yes

—

C₄×C₅₉

#104

yes

—

C₄×C₅₉

C₂²×C₈

yes

characteristic

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#135

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1476

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×C₂²⋊C₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

C₄×#41

—

100

#431

D₅₉

yes

characteristic, complemented

101

C₂×C₄×C₈

—

102

Q₈⋊C₈

—

103

D₄⋊C₈

—

104

C₅₉

#1476

yes

characteristic, complemented, Sylow

105

C₂×#41

—

106

C₂²×C₈

—

118

—

107

C₄×Q₈

—

108

C₄×Q₈

—

109

C₄×D₄

—

110

C₄×D₄

—

111

C₄×D₄

—

118

—

112

C₄×D₄

—

118

—

113

#117

—

114

#117

—

118

—

115

C₂×C₄²

—

118

—

116

C₂×C₄²

C₂×D₅₉

yes

characteristic

117

C₄⋊C₈

C₂×D₅₉

yes

—

118

C₂²⋊C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

118

—

122

#41

—

123

#41

—

118

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

C₂×D₄

—

118

—

127

C₂²×C₄

—

118

—

128

C₂²×C₄

—

118

—

129

C₂²×C₄

—

118

—

130

C₂²×C₄

C₄×D₅₉

yes

characteristic

131

C₂²×C₄

D₂₃₆

yes

characteristic

132

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

118

—

135

C₂×C₈

—

118

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

118

—

139

C₄⋊C₄

—

118

—

140

C₂²⋊C₄

—

118

—

141

C₂²⋊C₄

—

118

—

142

C₂²⋊C₄

—

118

—

143

C₄²

—

118

—

144

C₄²

—

118

—

145

C₄²

—

118

—

146

C₄²

C₄×D₅₉

yes

characteristic

147

C₄²

D₂₃₆

yes

characteristic

148

C₄²

C₂²⋊D₅₉

yes

characteristic

149

C₄²

C₂²×D₅₉

yes

characteristic

150

C₂³

—

118

—

151

C₂³

(C₂×D₅₉)⋊C₄

yes

characteristic

152

Q₈

—

153

Q₈

—

118

—

154

D₄

—

155

D₄

—

118

—

156

D₄

—

118

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

118

—

160

C₂×C₄

—

118

—

161

C₂×C₄

—

118

—

162

C₂×C₄

—

118

—

163

C₂×C₄

—

236

—

164

C₂×C₄

—

236

—

165

C₂×C₄

C₈×D₅₉

yes

—

166

C₂×C₄

#11745659

yes

—

167

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

168

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

169

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

170

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

171

C₂×C₄

C₂×D₂₃₆

yes

characteristic

172

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

173

C₈

—

174

C₈

—

118

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

118

—

178

C₂²

—

236

—

179

C₂²

—

236

—

180

C₂²

(C₂×D₅₉)⋊C₈

yes

—

181

C₂²

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

118

—

185

C₄

—

236

—

186

C₄

—

236

—

187

C₄

(C₂×D₅₉)⋊C₈

yes

—

188

C₄

#14273653

yes

—

189

C₄

C₂×C₈×D₅₉

yes

characteristic

190

C₄

C₂×#11745659

yes

characteristic

191

C₄

#14273747

yes

—

192

C₂

—

193

C₂

—

236

—

194

C₂

C₂×(C₂×D₅₉)⋊C₈

yes

characteristic

195

C₂

#16837396

yes

characteristic

196

C₂

C₂×#14273653

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂²×C₈
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#1476
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₅₉

commutator

C₄×C₅₉

C₂²×C₈

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
2. C₂×C₄×C₅₉⋊C₈	C₂	semidirect
3. #16837132	C₂	semidirect
4. #16837129	C₂	semidirect
5. C₅₉×#431	C₂	semidirect
28. #11745685	C₈	semidirect
68. D₂₃₆	C₂×C₈	semidirect
73. Dic₁₁₈	C₂×C₈	semidirect
100. #431	D₅₉	semidirect
104. C₅₉	#1476	semidirect

H ≅ G/N

type

2. C₂×C₄×C₅₉⋊C₈

C₂

semidirect

3. #16837132

C₂

semidirect

4. #16837129

C₂

semidirect

5. C₅₉×#431

C₂

semidirect

28. #11745685

C₈

semidirect

68. D₂₃₆

C₂×C₈

semidirect

73. Dic₁₁₈

C₂×C₈

semidirect

100. #431

D₅₉

semidirect

104. C₅₉

#1476

semidirect

Subgroups — #21422476

Conjugacy classes of subgroups

Complements