Group #21437178 — subgroups

Subgroups — #21437178

Order 7552.

356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16838121	C₂	1	2	yes	maximal, complemented
2	#16837793	C₂	1	2	yes	maximal, complemented
3	#16837621	C₂	1	2	yes	maximal, complemented
4	#16837615	C₂	1	2	yes	maximal
5	#16837610	C₂	1	2	yes	maximal, complemented
6	#16837532	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837503	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#495	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
11	C₂×#11745672	C₂²	1	2	yes	—
12	C₂×#11745670	C₂²	1	2	yes	—
13	#14273725	C₂²	1	2	yes	—
14	#14273720	—	2	2	no	—
15	#14273720	—	2	2	no	—
16	#14273714	C₂²	1	2	yes	—
17	#14273704	—	2	2	no	—
18	Q₈×Dic₅₉	C₂²	1	2	yes	—
19	C₄×Dic₁₁₈	C₂²	1	2	yes	—
20	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
21	D₄×Dic₅₉	C₂²	1	2	yes	—
22	#14273683	—	2	2	no	—
23	#14273661	C₂²	1	1	yes	characteristic
24	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₅₉⋊C₈	C₂²	1	2	yes	—
26	#14273652	C₂²	1	2	yes	—
27	#14273651	C₂²	1	2	yes	—
28	#14273648	C₂²	1	2	yes	—
29	#14273637	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×SD₁₆×C₅₉	C₂²	1	2	yes	—
32	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
33	#124×C₅₉	C₂²	1	2	yes	—
34	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₅₉	C₂³	1	2	yes	—
36	C₂×D₄×C₅₉	C₂³	1	2	yes	—
37	C₂²×Dic₅₉	—	2	2	no	—
38	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
39	C₂×Dic₁₁₈	C₂³	1	2	yes	—
40	Q₁₆×C₅₉	—	4	1	no	—
41	SD₁₆×C₅₉	—	2	4	no	—
42	D₈×C₅₉	—	4	1	no	—
43	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₂²⋊C₄×C₅₉	—	2	2	no	—
46	C₂²⋊C₄×C₅₉	—	2	2	no	—
47	C₄²×C₅₉	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₅₉	—	2	2	no	—
49	C₂²⋊Dic₅₉	—	2	2	no	—
50	C₂²⋊Dic₅₉	—	2	2	no	—
51	#11745672	—	4	2	no	—
52	#11745670	—	4	2	no	—
53	C₄⋊Dic₅₉	—	2	2	no	—
54	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
55	C₄⋊Dic₅₉	C₂³	1	2	yes	—
56	Dic₅₉⋊C₄	—	2	2	no	—
57	Dic₅₉⋊C₄	—	2	2	no	—
58	Dic₅₉⋊C₄	—	2	2	no	—
59	Dic₅₉⋊C₄	D₄	1	2	yes	—
60	C₄×Dic₅₉	—	2	2	no	—
61	C₄×Dic₅₉	—	2	2	no	—
62	C₄×Dic₅₉	C₂³	1	2	yes	—
63	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
64	Dic₂₃₆	—	2	2	no	—
65	C₂³×C₅₉	—	2	2	no	—
66	Q₈×C₅₉	—	2	2	no	—
67	Q₈×C₅₉	—	2	2	no	—
68	D₄×C₅₉	—	2	2	no	—
69	D₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₅₉	—	2	2	no	—
75	C₂×Dic₅₉	—	2	2	no	—
76	C₂×Dic₅₉	—	4	4	no	—
77	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
78	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
79	Dic₁₁₈	—	2	2	no	—
80	Dic₁₁₈	—	2	2	no	—
81	Dic₁₁₈	—	4	1	no	—
82	C₈×C₅₉	—	2	1	no	—
83	C₈×C₅₉	#41	1	2	yes	—
84	C₅₉⋊C₈	—	2	2	no	—
85	C₂²×C₅₉	—	4	2	no	—
86	C₂²×C₅₉	—	4	2	no	—
87	C₂²×C₅₉	—	4	2	no	—
88	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₅₉	—	2	1	no	—
90	C₄×C₅₉	—	2	1	no	—
91	C₄×C₅₉	—	4	2	no	—
92	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
93	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
94	Dic₅₉	—	2	2	no	—
95	Dic₅₉	—	2	2	no	—
96	Dic₅₉	—	4	2	no	—
97	Dic₅₉	—	4	2	no	—
98	#3402	—	59	1	no	maximal, Sylow
99	C₂×C₅₉	—	8	2	no	—
100	C₂×C₅₉	#550	1	1	yes	characteristic
101	C₂×C₅₉	#575	1	1	yes	characteristic
102	C₂×C₅₉	#578	1	1	yes	characteristic
103	#556	—	59	2	no	—
104	#497	—	59	2	no	—
105	#495	D₅₉	1	1	yes	characteristic
106	#464	—	59	2	no	—
107	#463	—	59	2	no	—
108	#462	—	59	2	no	—
109	#446	—	59	1	no	—
110	#440	—	59	2	no	—
111	C₄×Q₁₆	—	59	1	no	—
112	C₅₉	#3402	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₁₆	—	59	2	no	—
114	C₂×Q₁₆	—	59	1	no	—
115	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
116	C₂×SD₁₆	—	59	2	no	—
117	C₂×SD₁₆	C₂×D₅₉	1	2	yes	—
118	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
119	C₄⋊Q₈	—	59	2	no	—
120	#126	—	118	2	no	—
121	#126	—	118	2	no	—
122	#124	—	59	2	no	—
123	#124	C₂×D₅₉	1	2	yes	—
124	C₂²⋊Q₈	—	118	2	no	—
125	C₄×Q₈	—	59	1	no	—
126	C₄×Q₈	—	59	2	no	—
127	C₄×Q₈	—	59	2	no	—
128	C₄×D₄	—	59	2	no	—
129	#117	—	118	2	no	—
130	#107	—	59	1	no	—
131	C₄⋊C₈	—	59	2	no	—
132	C₄⋊C₈	—	59	1	no	—
133	#103	—	59	2	no	—
134	#103	—	59	2	no	—
135	#102	—	59	2	no	—
136	#97	—	59	2	no	—
137	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
138	C₂×Q₈	—	59	1	no	—
139	C₂×Q₈	—	59	2	no	—
140	C₂×Q₈	C₂²×D₅₉	1	2	yes	—
141	C₂×D₄	C₂²×D₅₉	1	2	yes	—
142	C₂²×C₄	—	118	2	no	—
143	Q₁₆	—	4	1	no	—
144	Q₁₆	—	118	2	no	—
145	Q₁₆	—	236	2	no	—
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	236	2	no	—
148	D₈	—	4	1	no	—
149	C₂×C₈	—	59	2	no	—
150	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
151	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
152	C₄⋊C₄	—	59	2	no	—
153	C₄⋊C₄	—	59	1	no	—
154	C₄⋊C₄	—	59	2	no	—
155	C₄⋊C₄	—	118	2	no	—
156	C₄⋊C₄	—	118	2	no	—
157	C₄⋊C₄	—	118	2	no	—
158	C₄⋊C₄	—	118	2	no	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	118	2	no	—
162	C₂²⋊C₄	—	118	2	no	—
163	C₂²⋊C₄	—	118	2	no	—
164	C₄²	—	59	2	no	—
165	C₄²	—	118	2	no	—
166	C₄²	—	118	2	no	—
167	C₄²	C₂²×D₅₉	1	1	yes	characteristic
168	C₂³	—	2	2	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	118	2	no	—
172	Q₈	—	118	2	no	—
173	Q₈	—	236	1	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	59	2	no	—
178	C₂×C₄	—	59	2	no	—
179	C₂×C₄	—	118	2	no	—
180	C₂×C₄	—	118	2	no	—
181	C₂×C₄	—	236	4	no	—
182	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
183	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	118	2	no	—
187	C₈	#11745687	1	2	yes	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	118	2	no	—
196	C₄	—	118	2	no	—
197	C₄	—	236	2	no	—
198	C₄	—	236	2	no	—
199	C₄	C₂×#11745687	1	1	yes	characteristic
200	C₄	#14273772	1	1	yes	characteristic
201	C₂	—	8	2	no	—
202	C₂	#16838072	1	1	yes	characteristic
203	C₂	#16838199	1	1	yes	characteristic
204	C₂	#16838216	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838121

C₂

yes

maximal, complemented

#16837793

C₂

yes

maximal, complemented

#16837621

C₂

yes

maximal, complemented

#16837615

C₂

yes

maximal

#16837610

C₂

yes

maximal, complemented

#16837532

C₂

yes

maximal, characteristic, complemented

#16837503

C₂

yes

maximal, complemented

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic, complemented

C₅₉×#495

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745672

C₂²

yes

—

C₂×#11745670

C₂²

yes

—

#14273725

C₂²

yes

—

#14273720

—

#14273720

—

#14273714

C₂²

yes

—

#14273704

—

Q₈×Dic₅₉

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

characteristic

D₄×Dic₅₉

C₂²

yes

—

#14273683

—

#14273661

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

C₂²

yes

—

#14273652

C₂²

yes

—

#14273651

C₂²

yes

—

#14273648

C₂²

yes

—

#14273637

C₂²

yes

—

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×SD₁₆×C₅₉

C₂²

yes

—

C₂×D₈×C₅₉

C₂²

yes

characteristic

#124×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

C₂³

yes

—

C₂×D₄×C₅₉

C₂³

yes

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

D₄

yes

characteristic

C₂×Dic₁₁₈

C₂³

yes

—

Q₁₆×C₅₉

—

SD₁₆×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

#11745672

—

#11745670

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

D₄

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂³

yes

—

C₂×C₅₉⋊C₈

C₂³

yes

—

Dic₂₃₆

—

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×D₄

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#41

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#41

yes

characteristic

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#3402

—

maximal, Sylow

C₂×C₅₉

—

100

C₂×C₅₉

#550

yes

characteristic

101

C₂×C₅₉

#575

yes

characteristic

102

C₂×C₅₉

#578

yes

characteristic

103

#556

—

104

#497

—

105

#495

D₅₉

yes

characteristic

106

#464

—

107

#463

—

108

#462

—

109

#446

—

110

#440

—

111

C₄×Q₁₆

—

112

C₅₉

#3402

yes

characteristic, complemented, Sylow

113

C₂×Q₁₆

—

114

C₂×Q₁₆

—

115

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

116

C₂×SD₁₆

—

117

C₂×SD₁₆

C₂×D₅₉

yes

—

118

C₂×D₈

C₂×D₅₉

yes

characteristic

119

C₄⋊Q₈

—

120

#126

—

118

—

121

#126

—

118

—

122

#124

—

123

#124

C₂×D₅₉

yes

—

124

C₂²⋊Q₈

—

118

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

—

118

—

130

#107

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

#103

—

134

#103

—

135

#102

—

136

#97

—

137

C₄×C₈

C₂×D₅₉

yes

characteristic

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

C₂²×D₅₉

yes

—

141

C₂×D₄

C₂²×D₅₉

yes

—

142

C₂²×C₄

—

118

—

143

Q₁₆

—

144

Q₁₆

—

118

—

145

Q₁₆

—

236

—

146

SD₁₆

—

147

SD₁₆

—

236

—

148

D₈

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₅₉

yes

characteristic

151

C₂×C₈

C₂²×D₅₉

yes

characteristic

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

118

—

156

C₄⋊C₄

—

118

—

157

C₄⋊C₄

—

118

—

158

C₄⋊C₄

—

118

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

118

—

162

C₂²⋊C₄

—

118

—

163

C₂²⋊C₄

—

118

—

164

C₄²

—

165

C₄²

—

118

—

166

C₄²

—

118

—

167

C₄²

C₂²×D₅₉

yes

characteristic

168

C₂³

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

118

—

172

Q₈

—

118

—

173

Q₈

—

236

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

118

—

180

C₂×C₄

—

118

—

181

C₂×C₄

—

236

—

182

C₂×C₄

D₄×D₅₉

yes

characteristic

183

C₂×C₄

D₄×D₅₉

yes

characteristic

184

C₂×C₄

C₂³×D₅₉

yes

characteristic

185

C₈

—

186

C₈

—

118

—

187

C₈

#11745687

yes

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

C₂×D₄×D₅₉

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

118

—

196

C₄

—

118

—

197

C₄

—

236

—

198

C₄

—

236

—

199

C₄

C₂×#11745687

yes

characteristic

200

C₄

#14273772

yes

characteristic

201

C₂

—

202

C₂

#16838072

yes

characteristic

203

C₂

#16838199

yes

characteristic

204

C₂

#16838216

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#495	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#495	D₅₉
hypocenter	C₅₉	#3402
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

subgroup

G/H

center

C₂²

C₂×D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#495

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #495

N	H ≅ G/N	type
1. #16838121	C₂	semidirect
2. #16837793	C₂	semidirect
3. #16837621	C₂	semidirect
5. #16837610	C₂	semidirect
6. #16837532	C₂	semidirect
7. #16837503	C₂	semidirect
8. C₄×Dic₂₃₆	C₂	semidirect
112. C₅₉	#3402	semidirect

H ≅ G/N

type

1. #16838121

C₂

semidirect

2. #16837793

C₂

semidirect

3. #16837621

C₂

semidirect

5. #16837610

C₂

semidirect

6. #16837532

C₂

semidirect

7. #16837503

C₂

semidirect

8. C₄×Dic₂₃₆

C₂

semidirect

112. C₅₉

#3402

semidirect

Subgroups — #21437178

Conjugacy classes of subgroups

Complements