Group #21435103 — subgroups

Subgroups — #21435103

Order 7552.

390 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16837952	C₂	1	1	yes	maximal, characteristic
2	#16837794	C₂	1	2	yes	maximal, complemented
3	#16837764	C₂	1	2	yes	maximal, complemented
4	#16837612	C₂	1	2	yes	maximal, complemented
5	#16837602	C₂	1	2	yes	maximal, complemented
6	#16837562	C₂	1	4	yes	maximal, complemented
7	#16837459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11745685	C₂²	1	2	yes	—
10	C₂×#11745672	C₂²	1	4	yes	—
11	C₂×#11745671	C₂²	1	4	yes	—
12	#14273727	C₂²	1	1	yes	characteristic
13	#14273724	C₂²	1	1	yes	characteristic
14	#14273715	C₂²	1	2	yes	—
15	#14273701	—	2	1	no	—
16	#14273701	—	2	1	no	—
17	C₄×D₂₃₆	—	2	1	no	—
18	C₄×D₂₃₆	—	2	1	no	—
19	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
20	#14273650	C₂²	1	4	yes	—
21	#14273647	C₂²	1	4	yes	—
22	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
23	#14273638	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
26	#125×C₅₉	—	2	1	no	—
27	C₂²⋊Q₈×C₅₉	—	2	1	no	—
28	C₂²⋊Q₈×C₅₉	C₂²	1	2	yes	—
29	C₄×Q₈×C₅₉	—	2	1	no	—
30	C₄×Q₈×C₅₉	C₂²	1	2	yes	—
31	#117×C₅₉	—	2	1	no	—
32	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
33	C₂×Q₈×C₅₉	—	2	1	no	—
34	C₂×Q₈×C₅₉	C₂³	1	2	yes	—
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	C₂³	1	2	yes	—
45	C₄⋊C₄×C₅₉	—	2	1	no	—
46	C₄⋊C₄×C₅₉	—	2	1	no	—
47	C₄⋊C₄×C₅₉	—	2	1	no	—
48	C₄⋊C₄×C₅₉	—	2	2	no	—
49	C₄⋊C₄×C₅₉	—	2	2	no	—
50	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₅₉	—	2	2	no	—
52	C₄²×C₅₉	—	2	2	no	—
53	C₄²×C₅₉	D₄	1	1	yes	characteristic
54	C₄²×C₅₉	D₄	1	1	yes	characteristic
55	C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄²×C₅₉	C₂³	1	1	yes	characteristic
57	#11745672	—	4	4	no	—
58	#11745671	—	4	4	no	—
59	(C₂×D₅₉)⋊C₄	—	2	2	no	—
60	(C₂×D₅₉)⋊C₄	—	2	2	no	—
61	C₄⋊Dic₅₉	—	2	1	no	—
62	C₄⋊Dic₅₉	—	2	1	no	—
63	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
64	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₅₉	—	2	2	no	—
66	Q₈×C₅₉	—	2	2	no	—
67	Q₈×C₅₉	—	2	2	no	—
68	Q₈×C₅₉	—	4	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	—	2	2	no	—
73	C₂×C₄×C₅₉	—	4	1	no	—
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	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₅₉	—	4	4	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	C₅₉⋊C₈	—	2	4	no	—
88	C₂²×C₅₉	—	2	1	no	—
89	C₂²×C₅₉	—	2	1	no	—
90	C₂²×C₅₉	—	2	1	no	—
91	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×D₅₉	—	4	2	no	—
93	C₂×D₅₉	—	4	2	no	—
94	C₂×D₅₉	—	4	2	no	—
95	C₄×C₅₉	—	2	1	no	—
96	C₄×C₅₉	—	2	1	no	—
97	C₄×C₅₉	—	2	1	no	—
98	C₄×C₅₉	—	2	1	no	—
99	C₄×C₅₉	—	4	2	no	—
100	C₄×C₅₉	—	4	2	no	—
101	C₄×C₅₉	#137	1	2	yes	—
102	C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
103	Dic₅₉	—	4	2	no	—
104	#3216	—	59	1	no	maximal, Sylow
105	C₂×C₅₉	—	4	1	no	—
106	C₂×C₅₉	#535	1	1	yes	characteristic
107	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
108	C₂×C₅₉	#575	1	1	yes	characteristic
109	D₅₉	—	8	2	no	—
110	#533	D₅₉	1	1	yes	characteristic, complemented
111	#532	—	59	1	no	—
112	#497	—	59	2	no	—
113	#489	—	59	2	no	—
114	#462	—	59	2	no	—
115	#460	—	59	2	no	—
116	#452	—	59	4	no	—
117	#432	—	59	1	no	—
118	C₅₉	#3216	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	—	59	2	no	—
120	C₂×Q₁₆	—	59	4	no	—
121	C₂×SD₁₆	—	59	4	no	—
122	C₄⋊Q₈	—	59	1	no	—
123	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
125	#127	—	59	1	no	—
126	#125	—	2	1	no	—
127	#124	—	59	2	no	—
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂²⋊Q₈	C₂×D₅₉	1	2	yes	—
130	#121	—	118	1	no	—
131	#121	—	118	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	C₂×D₅₉	1	2	yes	—
134	C₄×D₄	—	118	1	no	—
135	C₄×D₄	—	118	1	no	—
136	#117	—	2	1	no	—
137	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
138	C₄⋊C₈	—	59	2	no	—
139	#103	—	59	4	no	—
140	#102	—	59	4	no	—
141	C₂²⋊C₈	—	59	2	no	—
142	#97	—	59	2	no	—
143	#41	—	236	4	no	—
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	59	2	no	—
146	C₂×Q₈	C₂²×D₅₉	1	2	yes	—
147	C₂×D₄	—	59	2	no	—
148	C₂×D₄	—	118	2	no	—
149	C₂×D₄	—	118	1	no	—
150	C₂×D₄	—	118	1	no	—
151	C₂²×C₄	—	118	2	no	—
152	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
155	Q₁₆	—	236	4	no	—
156	SD₁₆	—	236	4	no	—
157	C₂×C₈	—	59	4	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	118	1	no	—
164	C₄⋊C₄	—	118	1	no	—
165	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	118	2	no	—
168	C₂²⋊C₄	—	118	2	no	—
169	C₄²	—	2	2	no	—
170	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
171	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
172	C₄²	C₂²×D₅₉	1	1	yes	characteristic
173	C₄²	C₂²×D₅₉	1	1	yes	characteristic
174	C₂³	—	118	2	no	—
175	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	4	2	no	—
179	Q₈	—	118	4	no	—
180	D₄	—	118	4	no	—
181	D₄	—	236	2	no	—
182	D₄	—	236	4	no	—
183	D₄	—	236	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	118	2	no	—
190	C₂×C₄	—	236	4	no	—
191	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
192	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
195	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
196	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
197	C₈	—	118	4	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	236	2	no	—
202	C₂²	—	236	2	no	—
203	C₂²	—	236	2	no	—
204	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	236	2	no	—
212	C₄	#14273758	1	2	yes	—
213	C₄	#14273773	1	2	yes	—
214	C₂	—	4	1	no	—
215	C₂	—	472	2	no	—
216	C₂	#16837966	1	1	yes	characteristic
217	C₂	C₂×#14273758	1	1	yes	characteristic
218	C₂	#16838200	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837952

C₂

yes

maximal, characteristic

#16837794

C₂

yes

maximal, complemented

#16837764

C₂

yes

maximal, complemented

#16837612

C₂

yes

maximal, complemented

#16837602

C₂

yes

maximal, complemented

#16837562

C₂

yes

maximal, complemented

#16837459

C₂

yes

maximal, characteristic, complemented

C₅₉×#533

C₂

yes

maximal, characteristic, complemented

C₂×#11745685

C₂²

yes

—

C₂×#11745672

C₂²

yes

—

C₂×#11745671

C₂²

yes

—

#14273727

C₂²

yes

characteristic

#14273724

C₂²

yes

characteristic

#14273715

C₂²

yes

—

#14273701

—

#14273701

—

C₄×D₂₃₆

—

C₄×D₂₃₆

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273650

C₂²

yes

—

#14273647

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273638

C₂²

yes

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#125×C₅₉

—

C₂²⋊Q₈×C₅₉

—

C₂²⋊Q₈×C₅₉

C₂²

yes

—

C₄×Q₈×C₅₉

—

C₄×Q₈×C₅₉

C₂²

yes

—

#117×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂³

yes

—

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

C₂³

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

#11745672

—

#11745671

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

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

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

100

C₄×C₅₉

—

101

C₄×C₅₉

#137

yes

—

102

C₄×C₅₉

H(5,𝔽₂)

yes

—

103

Dic₅₉

—

104

#3216

—

maximal, Sylow

105

C₂×C₅₉

—

106

C₂×C₅₉

#535

yes

characteristic

107

C₂×C₅₉

C₂×#137

yes

characteristic

108

C₂×C₅₉

#575

yes

characteristic

109

D₅₉

—

110

#533

D₅₉

yes

characteristic, complemented

111

#532

—

112

#497

—

113

#489

—

114

#462

—

115

#460

—

116

#452

—

117

#432

—

118

C₅₉

#3216

yes

characteristic, complemented, Sylow

119

C₂×#41

—

120

C₂×Q₁₆

—

121

C₂×SD₁₆

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

124

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

125

#127

—

126

#125

—

127

#124

—

128

C₂²⋊Q₈

—

129

C₂²⋊Q₈

C₂×D₅₉

yes

—

130

#121

—

118

—

131

#121

—

118

—

132

C₄×Q₈

—

133

C₄×Q₈

C₂×D₅₉

yes

—

134

C₄×D₄

—

118

—

135

C₄×D₄

—

118

—

136

#117

—

137

C₂×C₄²

C₂×D₅₉

yes

characteristic

138

C₄⋊C₈

—

139

#103

—

140

#102

—

141

C₂²⋊C₈

—

142

#97

—

143

#41

—

236

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

C₂²×D₅₉

yes

—

147

C₂×D₄

—

148

C₂×D₄

—

118

—

149

C₂×D₄

—

118

—

150

C₂×D₄

—

118

—

151

C₂²×C₄

—

118

—

152

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

153

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

154

C₂²×C₄

C₂²×D₅₉

yes

characteristic

155

Q₁₆

—

236

—

156

SD₁₆

—

236

—

157

C₂×C₈

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

118

—

164

C₄⋊C₄

—

118

—

165

C₄⋊C₄

C₂²×D₅₉

yes

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

118

—

168

C₂²⋊C₄

—

118

—

169

C₄²

—

170

C₄²

C₂²⋊D₅₉

yes

characteristic

171

C₄²

C₂²⋊D₅₉

yes

characteristic

172

C₄²

C₂²×D₅₉

yes

characteristic

173

C₄²

C₂²×D₅₉

yes

characteristic

174

C₂³

—

118

—

175

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

176

Q₈

—

177

Q₈

—

178

Q₈

—

179

Q₈

—

118

—

180

D₄

—

118

—

181

D₄

—

236

—

182

D₄

—

236

—

183

D₄

—

236

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

118

—

190

C₂×C₄

—

236

—

191

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

192

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

193

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

194

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

195

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

196

C₂×C₄

C₂³×D₅₉

yes

characteristic

197

C₈

—

118

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

236

—

202

C₂²

—

236

—

203

C₂²

—

236

—

204

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

236

—

212

C₄

#14273758

yes

—

213

C₄

#14273773

yes

—

214

C₂

—

215

C₂

—

472

—

216

C₂

#16837966

yes

characteristic

217

C₂

C₂×#14273758

yes

characteristic

218

C₂

#16838200

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#533	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#533	D₅₉
hypocenter	C₅₉	#3216
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#533

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #16837794	C₂	semidirect
3. #16837764	C₂	semidirect
4. #16837612	C₂	semidirect
5. #16837602	C₂	semidirect
6. #16837562	C₂	semidirect
7. #16837459	C₂	semidirect
8. C₅₉×#533	C₂	semidirect
110. #533	D₅₉	semidirect
118. C₅₉	#3216	semidirect

H ≅ G/N

type

2. #16837794

C₂

semidirect

3. #16837764

C₂

semidirect

4. #16837612

C₂

semidirect

5. #16837602

C₂

semidirect

6. #16837562

C₂

semidirect

7. #16837459

C₂

semidirect

8. C₅₉×#533

C₂

semidirect

110. #533

D₅₉

semidirect

118. C₅₉

#3216

semidirect

Subgroups — #21435103

Conjugacy classes of subgroups

Complements