Group #21435054 — subgroups

Subgroups — #21435054

Order 7552.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16837959	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837952	C₂	1	1	yes	maximal, characteristic
3	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837775	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837753	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837749	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837625	C₂	1	4	yes	maximal, complemented
8	#16837566	C₂	1	4	yes	maximal, complemented
9	C₅₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11745685	C₂²	1	2	yes	—
11	C₂×Dic₂₃₆	C₂²	1	4	yes	—
12	C₂×#11745660	C₂²	1	4	yes	—
13	#14273727	C₂²	1	1	yes	characteristic
14	#14273727	C₂²	1	1	yes	characteristic
15	#14273727	C₂²	1	1	yes	characteristic
16	#14273724	C₂²	1	1	yes	characteristic
17	#14273719	—	2	1	no	—
18	#14273715	C₂²	1	2	yes	—
19	#14273706	—	2	1	no	—
20	#14273706	C₂²	1	2	yes	—
21	#14273701	—	2	1	no	—
22	#14273701	—	2	1	no	—
23	C₄×Dic₁₁₈	—	2	1	no	—
24	C₄×Dic₁₁₈	C₂²	1	2	yes	—
25	C₄×D₂₃₆	—	2	1	no	—
26	C₄×D₂₃₆	—	2	1	no	—
27	#14273686	—	2	1	no	—
28	#14273652	C₂²	1	4	yes	—
29	#14273649	C₂²	1	4	yes	—
30	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
33	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₅₉	—	2	2	no	—
39	#11745685	—	4	4	no	—
40	C₂×D₂₃₆	—	2	1	no	—
41	C₂×D₂₃₆	—	2	1	no	—
42	C₂×D₂₃₆	C₂³	1	2	yes	—
43	C₂×C₄×D₅₉	—	2	2	no	—
44	C₂×Dic₁₁₈	—	2	1	no	—
45	C₂×Dic₁₁₈	C₂³	1	2	yes	—
46	C₂×Dic₁₁₈	C₂³	1	2	yes	—
47	C₂×C₈×C₅₉	C₂³	1	2	yes	—
48	C₂×C₈×C₅₉	C₂³	1	2	yes	—
49	C₄²×C₅₉	D₄	1	1	yes	characteristic
50	C₄²×C₅₉	D₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂³	1	1	yes	characteristic
52	C₄²×C₅₉	C₂³	1	1	yes	characteristic
53	C₂²⋊Dic₅₉	—	2	2	no	—
54	(C₂×D₅₉)⋊C₄	—	2	2	no	—
55	(C₂×D₅₉)⋊C₄	—	2	2	no	—
56	C₄⋊Dic₅₉	—	2	1	no	—
57	C₄⋊Dic₅₉	—	2	1	no	—
58	C₄⋊Dic₅₉	—	2	1	no	—
59	C₄⋊Dic₅₉	—	2	1	no	—
60	C₄⋊Dic₅₉	—	2	1	no	—
61	C₄⋊Dic₅₉	C₂³	1	2	yes	—
62	Dic₅₉⋊C₄	—	2	2	no	—
63	Dic₅₉⋊C₄	—	2	2	no	—
64	C₄×Dic₅₉	—	2	2	no	—
65	Dic₂₃₆	—	4	4	no	—
66	#11745660	—	4	4	no	—
67	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×D₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₅₉	—	4	4	no	—
79	C₂×Dic₅₉	—	2	2	no	—
80	C₂×Dic₅₉	—	2	2	no	—
81	C₂×Dic₅₉	—	2	2	no	—
82	D₂₃₆	—	2	4	no	—
83	D₂₃₆	—	4	2	no	—
84	D₂₃₆	—	4	2	no	—
85	C₄×D₅₉	—	4	4	no	—
86	Dic₁₁₈	—	2	4	no	—
87	Dic₁₁₈	—	2	4	no	—
88	Dic₁₁₈	—	4	2	no	—
89	C₈×C₅₉	—	2	2	no	—
90	C₈×C₅₉	—	2	2	no	—
91	C₂²×C₅₉	—	2	1	no	—
92	C₂²×C₅₉	—	2	1	no	—
93	C₂²×C₅₉	—	2	1	no	—
94	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×D₅₉	—	4	2	no	—
96	C₂×D₅₉	—	4	2	no	—
97	C₂×D₅₉	—	4	2	no	—
98	C₄×C₅₉	—	2	1	no	—
99	C₄×C₅₉	—	2	1	no	—
100	C₄×C₅₉	—	2	1	no	—
101	C₄×C₅₉	#135	1	2	yes	—
102	C₄×C₅₉	#137	1	2	yes	—
103	C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
104	Dic₅₉	—	4	2	no	—
105	Dic₅₉	—	4	2	no	—
106	Dic₅₉	—	4	2	no	—
107	#3210	—	59	1	no	maximal, Sylow
108	C₂×C₅₉	—	4	1	no	—
109	C₂×C₅₉	#535	1	1	yes	characteristic
110	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
111	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
112	D₅₉	—	8	2	no	—
113	#533	—	59	1	no	—
114	#532	—	59	1	no	—
115	#494	—	59	1	no	—
116	#492	—	59	1	no	—
117	#487	—	59	1	no	—
118	#486	—	59	1	no	—
119	#464	—	59	4	no	—
120	#452	—	59	4	no	—
121	#431	D₅₉	1	1	yes	characteristic, complemented
122	C₅₉	#3210	1	1	yes	characteristic, complemented, Sylow
123	C₂×#41	—	59	2	no	—
124	C₂×Q₁₆	—	59	4	no	—
125	C₂×SD₁₆	—	59	4	no	—
126	C₄⋊Q₈	—	59	1	no	—
127	C₄⋊Q₈	—	59	1	no	—
128	C₄⋊Q₈	—	59	1	no	—
129	#127	—	59	1	no	—
130	#125	—	118	1	no	—
131	#124	—	59	2	no	—
132	C₂²⋊Q₈	—	59	2	no	—
133	C₂²⋊Q₈	—	118	1	no	—
134	#121	—	118	1	no	—
135	#121	—	118	1	no	—
136	C₄×Q₈	—	59	2	no	—
137	C₄×Q₈	—	118	1	no	—
138	C₄×D₄	—	118	1	no	—
139	C₄×D₄	—	118	1	no	—
140	#117	—	118	1	no	—
141	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
142	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
143	#103	—	59	4	no	—
144	#102	—	59	4	no	—
145	C₂²⋊C₈	C₂×D₅₉	1	2	yes	—
146	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
147	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
148	#41	—	236	4	no	—
149	C₂×Q₈	—	59	2	no	—
150	C₂×Q₈	—	59	2	no	—
151	C₂×Q₈	—	118	1	no	—
152	C₂×D₄	—	59	2	no	—
153	C₂×D₄	—	118	2	no	—
154	C₂×D₄	—	118	1	no	—
155	C₂×D₄	—	118	1	no	—
156	C₂²×C₄	—	118	2	no	—
157	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
158	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
160	Q₁₆	—	236	4	no	—
161	SD₁₆	—	236	4	no	—
162	C₂×C₈	C₂²×D₅₉	1	2	yes	—
163	C₂×C₈	C₂²×D₅₉	1	2	yes	—
164	C₄⋊C₄	—	59	2	no	—
165	C₄⋊C₄	—	118	1	no	—
166	C₄⋊C₄	—	118	1	no	—
167	C₄⋊C₄	—	118	1	no	—
168	C₄⋊C₄	—	118	1	no	—
169	C₄⋊C₄	—	118	2	no	—
170	C₄⋊C₄	—	118	2	no	—
171	C₄⋊C₄	—	118	1	no	—
172	C₂²⋊C₄	—	118	2	no	—
173	C₂²⋊C₄	—	118	2	no	—
174	C₂²⋊C₄	—	118	2	no	—
175	C₄²	—	118	2	no	—
176	C₄²	D₂₃₆	1	1	yes	characteristic
177	C₄²	D₂₃₆	1	1	yes	characteristic
178	C₄²	C₂²×D₅₉	1	1	yes	characteristic
179	C₄²	C₂²×D₅₉	1	1	yes	characteristic
180	C₂³	—	118	2	no	—
181	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
182	Q₈	—	118	4	no	—
183	Q₈	—	118	4	no	—
184	Q₈	—	236	2	no	—
185	D₄	—	118	4	no	—
186	D₄	—	236	4	no	—
187	D₄	—	236	2	no	—
188	D₄	—	236	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	118	2	no	—
193	C₂×C₄	—	118	2	no	—
194	C₂×C₄	—	118	2	no	—
195	C₂×C₄	—	236	4	no	—
196	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
197	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
198	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
199	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
200	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	236	2	no	—
208	C₂²	—	236	2	no	—
209	C₂²	—	236	2	no	—
210	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	236	2	no	—
215	C₄	—	236	2	no	—
216	C₄	—	236	2	no	—
217	C₄	#14273749	1	2	yes	—
218	C₄	#14273760	1	2	yes	—
219	C₄	#14273773	1	2	yes	—
220	C₂	—	4	1	no	—
221	C₂	—	472	2	no	—
222	C₂	#16837971	1	1	yes	characteristic
223	C₂	C₂×#14273749	1	1	yes	characteristic
224	C₂	C₂×#14273760	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic, complemented

#16837952

C₂

yes

maximal, characteristic

#16837782

C₂

yes

maximal, characteristic, complemented

#16837775

C₂

yes

maximal, characteristic, complemented

#16837753

C₂

yes

maximal, characteristic, complemented

#16837749

C₂

yes

maximal, characteristic, complemented

#16837625

C₂

yes

maximal, complemented

#16837566

C₂

yes

maximal, complemented

C₅₉×#431

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

C₂×Dic₂₃₆

C₂²

yes

—

C₂×#11745660

C₂²

yes

—

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273724

C₂²

yes

characteristic

#14273719

—

#14273715

C₂²

yes

—

#14273706

—

#14273706

C₂²

yes

—

#14273701

—

#14273701

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×D₂₃₆

—

C₄×D₂₃₆

—

#14273686

—

#14273652

C₂²

yes

—

#14273649

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

#11745685

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂³

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

#11745660

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

100

C₄×C₅₉

—

101

C₄×C₅₉

#135

yes

—

102

C₄×C₅₉

#137

yes

—

103

C₄×C₅₉

H(5,𝔽₂)

yes

—

104

Dic₅₉

—

105

Dic₅₉

—

106

Dic₅₉

—

107

#3210

—

maximal, Sylow

108

C₂×C₅₉

—

109

C₂×C₅₉

#535

yes

characteristic

110

C₂×C₅₉

C₂×#135

yes

characteristic

111

C₂×C₅₉

C₂×#137

yes

characteristic

112

D₅₉

—

113

#533

—

114

#532

—

115

#494

—

116

#492

—

117

#487

—

118

#486

—

119

#464

—

120

#452

—

121

#431

D₅₉

yes

characteristic, complemented

122

C₅₉

#3210

yes

characteristic, complemented, Sylow

123

C₂×#41

—

124

C₂×Q₁₆

—

125

C₂×SD₁₆

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

#127

—

130

#125

—

118

—

131

#124

—

132

C₂²⋊Q₈

—

133

C₂²⋊Q₈

—

118

—

134

#121

—

118

—

135

#121

—

118

—

136

C₄×Q₈

—

137

C₄×Q₈

—

118

—

138

C₄×D₄

—

118

—

139

C₄×D₄

—

118

—

140

#117

—

118

—

141

C₂×C₄²

C₂×D₅₉

yes

characteristic

142

C₄⋊C₈

C₂×D₅₉

yes

—

143

#103

—

144

#102

—

145

C₂²⋊C₈

C₂×D₅₉

yes

—

146

C₄×C₈

C₂×D₅₉

yes

characteristic

147

C₄×C₈

C₂×D₅₉

yes

characteristic

148

#41

—

236

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

118

—

152

C₂×D₄

—

153

C₂×D₄

—

118

—

154

C₂×D₄

—

118

—

155

C₂×D₄

—

118

—

156

C₂²×C₄

—

118

—

157

C₂²×C₄

D₂₃₆

yes

characteristic

158

C₂²×C₄

D₂₃₆

yes

characteristic

159

C₂²×C₄

C₂²×D₅₉

yes

characteristic

160

Q₁₆

—

236

—

161

SD₁₆

—

236

—

162

C₂×C₈

C₂²×D₅₉

yes

—

163

C₂×C₈

C₂²×D₅₉

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

118

—

166

C₄⋊C₄

—

118

—

167

C₄⋊C₄

—

118

—

168

C₄⋊C₄

—

118

—

169

C₄⋊C₄

—

118

—

170

C₄⋊C₄

—

118

—

171

C₄⋊C₄

—

118

—

172

C₂²⋊C₄

—

118

—

173

C₂²⋊C₄

—

118

—

174

C₂²⋊C₄

—

118

—

175

C₄²

—

118

—

176

C₄²

D₂₃₆

yes

characteristic

177

C₄²

D₂₃₆

yes

characteristic

178

C₄²

C₂²×D₅₉

yes

characteristic

179

C₄²

C₂²×D₅₉

yes

characteristic

180

C₂³

—

118

—

181

C₂³

C₂×D₂₃₆

yes

characteristic

182

Q₈

—

118

—

183

Q₈

—

118

—

184

Q₈

—

236

—

185

D₄

—

118

—

186

D₄

—

236

—

187

D₄

—

236

—

188

D₄

—

236

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

118

—

193

C₂×C₄

—

118

—

194

C₂×C₄

—

118

—

195

C₂×C₄

—

236

—

196

C₂×C₄

C₂×D₂₃₆

yes

characteristic

197

C₂×C₄

C₂×D₂₃₆

yes

characteristic

198

C₂×C₄

C₂×D₂₃₆

yes

characteristic

199

C₂×C₄

C₂×D₂₃₆

yes

characteristic

200

C₂×C₄

C₂×D₂₃₆

yes

characteristic

201

C₂×C₄

C₂³×D₅₉

yes

characteristic

202

C₈

—

203

C₈

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

236

—

208

C₂²

—

236

—

209

C₂²

—

236

—

210

C₂²

C₂²×D₂₃₆

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

236

—

215

C₄

—

236

—

216

C₄

—

236

—

217

C₄

#14273749

yes

—

218

C₄

#14273760

yes

—

219

C₄

#14273773

yes

—

220

C₂

—

221

C₂

—

472

—

222

C₂

#16837971

yes

characteristic

223

C₂

C₂×#14273749

yes

characteristic

224

C₂

C₂×#14273760

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#431	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#431	D₅₉
hypocenter	C₅₉	#3210
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#431

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
1. #16837959	C₂	semidirect
3. #16837782	C₂	semidirect
4. #16837775	C₂	semidirect
5. #16837753	C₂	semidirect
6. #16837749	C₂	semidirect
7. #16837625	C₂	semidirect
8. #16837566	C₂	semidirect
9. C₅₉×#431	C₂	semidirect
121. #431	D₅₉	semidirect
122. C₅₉	#3210	semidirect

H ≅ G/N

type

1. #16837959

C₂

semidirect

3. #16837782

C₂

semidirect

4. #16837775

C₂

semidirect

5. #16837753

C₂

semidirect

6. #16837749

C₂

semidirect

7. #16837625

C₂

semidirect

8. #16837566

C₂

semidirect

9. C₅₉×#431

C₂

semidirect

121. #431

D₅₉

semidirect

122. C₅₉

#3210

semidirect

Subgroups — #21435054

Conjugacy classes of subgroups

Complements