Group #21437177 — subgroups

Subgroups — #21437177

Order 7552.

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

normal subgroups only

This group has 356 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	#16838125	C₂	1	2	yes	maximal, complemented
2	#16837796	C₂	1	2	yes	maximal, complemented
3	#16837786	C₂	1	1	yes	maximal, characteristic
4	#16837625	C₂	1	2	yes	maximal, complemented
5	#16837618	C₂	1	2	yes	maximal
6	#16837613	C₂	1	2	yes	maximal, complemented
7	#16837504	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₃₆	C₂²	1	2	yes	—
13	C₂×#11745660	C₂²	1	2	yes	—
14	C₂×#11745660	C₂²	1	2	yes	—
15	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
16	#14273727	C₂²	1	2	yes	—
17	#14273722	—	2	2	no	—
18	#14273722	—	2	2	no	—
19	#14273715	C₂²	1	2	yes	—
20	#14273715	C₂²	1	2	yes	—
21	#14273703	—	2	2	no	—
22	C₄×Dic₁₁₈	C₂²	1	2	yes	—
23	C₄×Dic₁₁₈	C₂²	1	2	yes	—
24	C₄×D₂₃₆	C₂²	1	2	yes	—
25	#14273685	—	2	2	no	—
26	#14273661	C₂²	1	1	yes	characteristic
27	#14273652	C₂²	1	2	yes	—
28	#14273652	C₂²	1	2	yes	—
29	#14273649	C₂²	1	2	yes	—
30	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
32	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
33	#97×C₅₉	C₂²	1	2	yes	—
34	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
36	C₂×D₂₃₆	C₂³	1	2	yes	—
37	C₂×C₄×D₅₉	—	2	2	no	—
38	C₂×Dic₁₁₈	C₂³	1	2	yes	—
39	C₂×Dic₁₁₈	C₂³	1	2	yes	—
40	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂³	1	2	yes	—
42	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
44	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₄²×C₅₉	C₂³	1	2	yes	—
47	(C₂×D₅₉)⋊C₄	—	2	2	no	—
48	(C₂×D₅₉)⋊C₄	—	2	2	no	—
49	(C₂×D₅₉)⋊C₄	—	2	2	no	—
50	(C₂×D₅₉)⋊C₄	—	2	2	no	—
51	(C₂×D₅₉)⋊C₄	—	2	2	no	—
52	C₄⋊Dic₅₉	—	2	2	no	—
53	C₄⋊Dic₅₉	—	2	2	no	—
54	C₄⋊Dic₅₉	C₂³	1	2	yes	—
55	Dic₅₉⋊C₄	—	2	2	no	—
56	Dic₅₉⋊C₄	—	2	2	no	—
57	C₄×Dic₅₉	—	2	2	no	—
58	C₄×Dic₅₉	—	2	2	no	—
59	Dic₂₃₆	—	2	2	no	—
60	Dic₂₃₆	—	4	2	no	—
61	Dic₂₃₆	—	4	1	no	—
62	D₄₇₂	—	4	1	no	—
63	#11745660	—	2	4	no	—
64	#11745660	—	4	2	no	—
65	C₂²×D₅₉	—	2	2	no	—
66	Q₈×C₅₉	—	4	1	no	—
67	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₅₉	—	2	2	no	—
73	C₂×Dic₅₉	—	2	2	no	—
74	C₂×Dic₅₉	—	2	2	no	—
75	D₂₃₆	—	2	2	no	—
76	D₂₃₆	—	2	2	no	—
77	C₄×D₅₉	—	4	4	no	—
78	Dic₁₁₈	—	2	2	no	—
79	Dic₁₁₈	—	2	2	no	—
80	Dic₁₁₈	—	2	2	no	—
81	Dic₁₁₈	—	2	2	no	—
82	C₈×C₅₉	—	2	2	no	—
83	C₈×C₅₉	—	2	1	no	—
84	C₈×C₅₉	#41	1	2	yes	—
85	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
86	C₂×D₅₉	—	4	2	no	—
87	C₂×D₅₉	—	4	2	no	—
88	C₂×D₅₉	—	4	2	no	—
89	C₄×C₅₉	—	2	2	no	—
90	C₄×C₅₉	—	2	1	no	—
91	C₄×C₅₉	—	2	1	no	—
92	C₄×C₅₉	—	2	2	no	—
93	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
94	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
95	Dic₅₉	—	4	2	no	—
96	Dic₅₉	—	4	2	no	—
97	Dic₅₉	—	4	2	no	—
98	#3402	—	59	1	no	maximal, Sylow
99	C₂×C₅₉	#550	1	1	yes	characteristic
100	C₂×C₅₉	#575	1	1	yes	characteristic
101	C₂×C₅₉	#578	1	1	yes	characteristic
102	D₅₉	—	8	2	no	—
103	#556	—	59	2	no	—
104	#497	—	59	2	no	—
105	#495	—	59	1	no	—
106	#464	—	59	2	no	—
107	#463	—	59	2	no	—
108	#462	—	59	2	no	—
109	#446	D₅₉	1	1	yes	characteristic, complemented
110	#440	—	59	2	no	—
111	C₄×Q₁₆	—	59	1	no	—
112	C₅₉	#3402	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₁₆	—	59	1	no	—
114	C₂×Q₁₆	—	59	1	no	—
115	C₂×Q₁₆	—	59	2	no	—
116	C₂×SD₁₆	—	59	2	no	—
117	C₂×SD₁₆	—	59	2	no	—
118	C₂×D₈	—	59	1	no	—
119	C₄⋊Q₈	—	59	2	no	—
120	#126	—	118	2	no	—
121	#126	—	118	2	no	—
122	#124	—	59	2	no	—
123	#124	—	59	2	no	—
124	C₂²⋊Q₈	—	118	2	no	—
125	C₄×Q₈	—	59	2	no	—
126	C₄×Q₈	—	59	2	no	—
127	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
128	C₄×D₄	—	59	2	no	—
129	#117	—	118	2	no	—
130	#107	—	59	1	no	—
131	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
132	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
133	#103	—	59	2	no	—
134	#103	—	59	2	no	—
135	#102	—	59	2	no	—
136	#97	C₂×D₅₉	1	2	yes	—
137	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
138	C₂×Q₈	—	59	2	no	—
139	C₂×Q₈	—	59	2	no	—
140	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
141	C₂×D₄	—	59	2	no	—
142	C₂²×C₄	—	118	2	no	—
143	Q₁₆	—	118	2	no	—
144	Q₁₆	—	236	1	no	—
145	Q₁₆	—	236	2	no	—
146	SD₁₆	—	118	4	no	—
147	SD₁₆	—	236	2	no	—
148	D₈	—	236	1	no	—
149	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
150	C₂×C₈	C₂²×D₅₉	1	2	yes	—
151	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
152	C₄⋊C₄	—	59	2	no	—
153	C₄⋊C₄	—	118	2	no	—
154	C₄⋊C₄	—	118	2	no	—
155	C₄⋊C₄	—	118	2	no	—
156	C₄⋊C₄	—	118	2	no	—
157	C₄⋊C₄	D₂₃₆	1	2	yes	—
158	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
159	C₂²⋊C₄	—	118	2	no	—
160	C₂²⋊C₄	—	118	2	no	—
161	C₂²⋊C₄	—	118	2	no	—
162	C₂²⋊C₄	—	118	2	no	—
163	C₂²⋊C₄	—	118	2	no	—
164	C₄²	—	118	2	no	—
165	C₄²	—	118	2	no	—
166	C₄²	C₂²×D₅₉	1	1	yes	characteristic
167	C₄²	C₂²×D₅₉	1	2	yes	—
168	C₂³	—	118	2	no	—
169	Q₈	—	4	1	no	—
170	Q₈	—	118	2	no	—
171	Q₈	—	118	2	no	—
172	Q₈	—	118	2	no	—
173	Q₈	—	118	2	no	—
174	D₄	—	118	2	no	—
175	D₄	—	118	2	no	—
176	C₂×C₄	—	118	2	no	—
177	C₂×C₄	—	118	2	no	—
178	C₂×C₄	—	118	2	no	—
179	C₂×C₄	—	236	4	no	—
180	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
181	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
182	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
183	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
184	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	2	no	—
187	C₈	#11745689	1	2	yes	—
188	C₂²	—	236	2	no	—
189	C₂²	—	236	2	no	—
190	C₂²	—	236	2	no	—
191	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	236	2	no	—
197	C₄	—	236	2	no	—
198	C₄	—	236	2	no	—
199	C₄	C₂×#11745689	1	1	yes	characteristic
200	C₄	#14273773	1	1	yes	characteristic
201	C₂	—	472	2	no	—
202	C₂	#16838074	1	1	yes	characteristic
203	C₂	#16838202	1	1	yes	characteristic
204	C₂	#16838219	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838125

C₂

yes

maximal, complemented

#16837796

C₂

yes

maximal, complemented

#16837786

C₂

yes

maximal, characteristic

#16837625

C₂

yes

maximal, complemented

#16837618

C₂

yes

maximal

#16837613

C₂

yes

maximal, complemented

#16837504

C₂

yes

maximal, complemented

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic, complemented

C₅₉×#446

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₂²

yes

—

C₂×#11745660

C₂²

yes

—

C₂×#11745660

C₂²

yes

—

C₂×D₄₇₂

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273722

—

#14273722

—

#14273715

C₂²

yes

—

#14273715

C₂²

yes

—

#14273703

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

—

#14273685

—

#14273661

C₂²

yes

characteristic

#14273652

C₂²

yes

—

#14273652

C₂²

yes

—

#14273649

C₂²

yes

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

#97×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×D₂₃₆

C₂³

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×Dic₁₁₈

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

#11745660

—

#11745660

—

C₂²×D₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#41

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#41

yes

characteristic

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

100

C₂×C₅₉

#575

yes

characteristic

101

C₂×C₅₉

#578

yes

characteristic

102

D₅₉

—

103

#556

—

104

#497

—

105

#495

—

106

#464

—

107

#463

—

108

#462

—

109

#446

D₅₉

yes

characteristic, complemented

110

#440

—

111

C₄×Q₁₆

—

112

C₅₉

#3402

yes

characteristic, complemented, Sylow

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂×SD₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₄⋊Q₈

—

120

#126

—

118

—

121

#126

—

118

—

122

#124

—

123

#124

—

124

C₂²⋊Q₈

—

118

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

C₂×D₅₉

yes

characteristic

128

C₄×D₄

—

129

#117

—

118

—

130

#107

—

131

C₄⋊C₈

C₂×D₅₉

yes

—

132

C₄⋊C₈

C₂×D₅₉

yes

characteristic

133

#103

—

134

#103

—

135

#102

—

136

#97

C₂×D₅₉

yes

—

137

C₄×C₈

C₂×D₅₉

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

D₂₃₆

yes

characteristic

141

C₂×D₄

—

142

C₂²×C₄

—

118

—

143

Q₁₆

—

118

—

144

Q₁₆

—

236

—

145

Q₁₆

—

236

—

146

SD₁₆

—

118

—

147

SD₁₆

—

236

—

148

D₈

—

236

—

149

C₂×C₈

C₂²×D₅₉

yes

characteristic

150

C₂×C₈

C₂²×D₅₉

yes

—

151

C₂×C₈

C₂²×D₅₉

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

118

—

154

C₄⋊C₄

—

118

—

155

C₄⋊C₄

—

118

—

156

C₄⋊C₄

—

118

—

157

C₄⋊C₄

D₂₃₆

yes

—

158

C₄⋊C₄

D₂₃₆

yes

characteristic

159

C₂²⋊C₄

—

118

—

160

C₂²⋊C₄

—

118

—

161

C₂²⋊C₄

—

118

—

162

C₂²⋊C₄

—

118

—

163

C₂²⋊C₄

—

118

—

164

C₄²

—

118

—

165

C₄²

—

118

—

166

C₄²

C₂²×D₅₉

yes

characteristic

167

C₄²

C₂²×D₅₉

yes

—

168

C₂³

—

118

—

169

Q₈

—

170

Q₈

—

118

—

171

Q₈

—

118

—

172

Q₈

—

118

—

173

Q₈

—

118

—

174

D₄

—

118

—

175

D₄

—

118

—

176

C₂×C₄

—

118

—

177

C₂×C₄

—

118

—

178

C₂×C₄

—

118

—

179

C₂×C₄

—

236

—

180

C₂×C₄

C₂×D₂₃₆

yes

characteristic

181

C₂×C₄

C₂×D₂₃₆

yes

—

182

C₂×C₄

C₂×D₂₃₆

yes

characteristic

183

C₂×C₄

C₂×D₂₃₆

yes

—

184

C₂×C₄

C₂³×D₅₉

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

#11745689

yes

—

188

C₂²

—

236

—

189

C₂²

—

236

—

190

C₂²

—

236

—

191

C₂²

C₂²×D₂₃₆

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

236

—

197

C₄

—

236

—

198

C₄

—

236

—

199

C₄

C₂×#11745689

yes

characteristic

200

C₄

#14273773

yes

characteristic

201

C₂

—

472

—

202

C₂

#16838074

yes

characteristic

203

C₂

#16838202

yes

characteristic

204

C₂

#16838219

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#446	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#446	D₅₉
hypocenter	C₅₉	#3402
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#446

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #16838125	C₂	semidirect
2. #16837796	C₂	semidirect
4. #16837625	C₂	semidirect
6. #16837613	C₂	semidirect
7. #16837504	C₂	semidirect
8. C₄×Dic₂₃₆	C₂	semidirect
9. C₅₉×#446	C₂	semidirect
109. #446	D₅₉	semidirect
112. C₅₉	#3402	semidirect

H ≅ G/N

type

1. #16838125

C₂

semidirect

2. #16837796

C₂

semidirect

4. #16837625

C₂

semidirect

6. #16837613

C₂

semidirect

7. #16837504

C₂

semidirect

8. C₄×Dic₂₃₆

C₂

semidirect

9. C₅₉×#446

C₂

semidirect

109. #446

D₅₉

semidirect

112. C₅₉

#3402

semidirect

Subgroups — #21437177

Conjugacy classes of subgroups

Complements