Group #21422772 — subgroups

Subgroups — #21422772

Order 7552.

260 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16837953	C₂	1	1	yes	maximal, characteristic
2	#16837424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837148	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837145	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837145	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745689	—	2	1	no	—
9	C₂×#11745659	—	2	1	no	—
10	C₂×C₈×D₅₉	—	2	1	no	—
11	#14273724	C₂²	1	1	yes	characteristic
12	#14273716	—	2	1	no	—
13	#14273702	—	2	1	no	—
14	#14273702	C₄	1	2	yes	—
15	#14273689	—	2	1	no	—
16	#14273689	C₄	1	2	yes	—
17	C₄²×D₅₉	C₂²	1	1	yes	characteristic
18	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₅₉⋊C₈	—	2	1	no	—
21	Dic₅₉⋊C₈	—	2	1	no	—
22	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	#11745689	—	4	2	no	—
27	#11745689	—	4	2	no	—
28	C₂×D₂₃₆	—	2	2	no	—
29	C₂×D₂₃₆	—	2	1	no	—
30	C₂×D₂₃₆	—	2	1	no	—
31	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₅₉	—	2	2	no	—
33	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
35	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₄⋊C₄×C₅₉	—	2	1	no	—
39	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
40	C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	(C₂×D₅₉)⋊C₄	—	2	2	no	—
42	(C₂×D₅₉)⋊C₄	—	2	2	no	—
43	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₅₉	D₄	1	1	yes	characteristic
45	C₄×Dic₅₉	D₄	1	1	yes	characteristic
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	#11745659	—	4	2	no	—
49	C₈×D₅₉	—	4	2	no	—
50	C₂²×D₅₉	—	2	2	no	—
51	C₂²×D₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	Q₈×C₅₉	—	4	1	no	—
53	Q₈×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	D₂₃₆	—	4	2	no	—
62	D₂₃₆	—	4	4	no	—
63	D₂₃₆	—	4	1	no	—
64	D₂₃₆	—	4	1	no	—
65	D₂₃₆	—	4	1	no	—
66	D₂₃₆	—	4	1	no	—
67	C₄×D₅₉	—	2	1	no	—
68	C₄×D₅₉	—	2	1	no	—
69	C₄×D₅₉	—	4	1	no	—
70	C₄×D₅₉	—	4	4	no	—
71	C₄×D₅₉	D₈	1	2	yes	—
72	C₄×D₅₉	SD₁₆	1	2	yes	—
73	C₈×C₅₉	—	4	1	no	—
74	C₈×C₅₉	—	4	1	no	—
75	C₅₉⋊C₈	—	4	1	no	—
76	C₅₉⋊C₈	—	4	1	no	—
77	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×D₅₉	—	2	1	no	—
79	C₂×D₅₉	—	2	1	no	—
80	C₂×D₅₉	—	4	2	no	—
81	C₂×D₅₉	—	4	2	no	—
82	C₂×D₅₉	—	4	2	no	—
83	C₂×D₅₉	#102	1	2	yes	—
84	C₄×C₅₉	—	2	1	no	—
85	C₄×C₅₉	—	4	2	no	—
86	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
87	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
88	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₄×C₅₉	#137	1	1	yes	characteristic
90	Dic₅₉	—	2	1	no	—
91	Dic₅₉	—	2	1	no	—
92	Dic₅₉	#102	1	2	yes	—
93	#1547	—	59	1	no	maximal, Sylow
94	C₂×C₅₉	#413	1	1	yes	characteristic
95	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
96	C₂×C₅₉	#417	1	1	yes	characteristic
97	D₅₉	—	2	2	no	—
98	D₅₉	—	8	2	no	—
99	#532	—	59	1	no	—
100	#423	—	59	1	no	—
101	C₂×C₄⋊C₈	—	59	1	no	—
102	#332	—	59	1	no	—
103	#332	D₅₉	1	1	yes	characteristic, complemented
104	#331	—	59	1	no	—
105	#331	—	59	1	no	—
106	C₅₉	#1547	1	1	yes	characteristic, complemented, Sylow
107	C₂×#41	—	118	1	no	—
108	C₂×M₁₆	—	118	1	no	—
109	C₂²×C₈	—	118	1	no	—
110	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
111	#127	—	59	1	no	—
112	#124	—	118	1	no	—
113	#121	—	59	2	no	—
114	#121	—	118	1	no	—
115	C₄×D₄	—	59	2	no	—
116	C₄×D₄	—	118	1	no	—
117	C₂×C₄²	—	59	1	no	—
118	C₄⋊C₈	—	59	1	no	—
119	C₄⋊C₈	—	59	1	no	—
120	C₄⋊C₈	—	118	1	no	—
121	C₄⋊C₈	—	118	1	no	—
122	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
124	#41	—	236	2	no	—
125	#41	—	236	2	no	—
126	C₂×Q₈	—	2	1	no	—
127	C₂×D₄	—	59	2	no	—
128	C₂×D₄	—	118	2	no	—
129	C₂×D₄	—	118	1	no	—
130	C₂×D₄	—	118	1	no	—
131	C₂²×C₄	—	59	1	no	—
132	C₂²×C₄	—	59	1	no	—
133	C₂²×C₄	—	59	1	no	—
134	C₂²×C₄	—	118	2	no	—
135	M₁₆	—	236	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	118	1	no	—
139	C₂×C₈	—	118	1	no	—
140	C₂×C₈	—	236	2	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	C₄×D₅₉	1	2	yes	—
143	C₂²⋊C₄	—	118	2	no	—
144	C₂²⋊C₄	—	118	2	no	—
145	C₄²	—	59	1	no	—
146	C₄²	—	59	1	no	—
147	C₄²	—	59	1	no	—
148	C₄²	C₂²×D₅₉	1	1	yes	characteristic
149	C₂³	—	59	1	no	—
150	C₂³	—	118	2	no	—
151	Q₈	—	4	1	no	—
152	Q₈	—	4	1	no	—
153	D₄	—	236	2	no	—
154	D₄	—	236	4	no	—
155	D₄	—	236	1	no	—
156	D₄	—	236	1	no	—
157	D₄	—	236	1	no	—
158	D₄	—	236	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	59	1	no	—
161	C₂×C₄	—	59	2	no	—
162	C₂×C₄	—	59	2	no	—
163	C₂×C₄	—	59	1	no	—
164	C₂×C₄	—	59	1	no	—
165	C₂×C₄	—	118	1	no	—
166	C₂×C₄	—	118	1	no	—
167	C₂×C₄	—	236	1	no	—
168	C₂×C₄	—	236	4	no	—
169	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
170	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
171	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
172	C₈	—	4	1	no	—
173	C₈	—	4	1	no	—
174	C₈	—	236	1	no	—
175	C₈	—	236	1	no	—
176	C₂²	—	59	2	no	—
177	C₂²	—	118	1	no	—
178	C₂²	—	118	1	no	—
179	C₂²	—	236	2	no	—
180	C₂²	—	236	2	no	—
181	C₂²	—	236	2	no	—
182	C₂²	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	59	2	no	—
186	C₄	—	118	1	no	—
187	C₄	—	118	1	no	—
188	C₄	D₈×D₅₉	1	1	yes	characteristic
189	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
190	C₄	#14273754	1	1	yes	characteristic
191	C₄	#14273759	1	1	yes	characteristic
192	C₂	—	118	2	no	—
193	C₂	—	472	2	no	—
194	C₂	#16837385	1	1	yes	characteristic
195	C₂	#102×D₅₉	1	1	yes	characteristic
196	C₂	#16837402	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837953

C₂

yes

maximal, characteristic

#16837424

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×D₅₉

C₂

yes

maximal, characteristic, complemented

#16837148

C₂

yes

maximal, characteristic, complemented

#16837145

C₂

yes

maximal, characteristic, complemented

#16837145

C₂

yes

maximal, characteristic, complemented

C₅₉×#332

C₂

yes

maximal, characteristic, complemented

C₂×#11745689

—

C₂×#11745659

—

C₂×C₈×D₅₉

—

#14273724

C₂²

yes

—

—

C₄

yes

—

#14273689

—

#14273689

C₄

yes

—

C₄²×D₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

Dic₅₉⋊C₈

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

#11745689

—

#11745689

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

—

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

#11745659

—

C₈×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

C₂²⋊C₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

D₈

yes

—

C₄×D₅₉

SD₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

#102

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

#102

yes

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

D₅₉

—

#532

—

100

#423

—

101

C₂×C₄⋊C₈

—

102

#332

—

103

#332

D₅₉

yes

characteristic, complemented

104

#331

—

105

#331

—

106

C₅₉

#1547

yes

characteristic, complemented, Sylow

107

C₂×#41

—

118

—

108

C₂×M₁₆

—

118

—

109

C₂²×C₈

—

118

—

110

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

111

#127

—

112

#124

—

118

—

113

#121

—

114

#121

—

118

—

115

C₄×D₄

—

116

C₄×D₄

—

118

—

117

C₂×C₄²

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

118

—

121

C₄⋊C₈

—

118

—

122

C₄⋊C₈

C₂×D₅₉

yes

characteristic

123

C₄⋊C₈

C₂×D₅₉

yes

characteristic

124

#41

—

236

—

125

#41

—

236

—

126

C₂×Q₈

—

127

C₂×D₄

—

128

C₂×D₄

—

118

—

129

C₂×D₄

—

118

—

130

C₂×D₄

—

118

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

118

—

135

M₁₆

—

236

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

118

—

139

C₂×C₈

—

118

—

140

C₂×C₈

—

236

—

141

C₄⋊C₄

—

142

C₄⋊C₄

C₄×D₅₉

yes

—

143

C₂²⋊C₄

—

118

—

144

C₂²⋊C₄

—

118

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

C₂²×D₅₉

yes

characteristic

149

C₂³

—

150

C₂³

—

118

—

151

Q₈

—

152

Q₈

—

153

D₄

—

236

—

154

D₄

—

236

—

155

D₄

—

236

—

156

D₄

—

236

—

157

D₄

—

236

—

158

D₄

—

236

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

118

—

166

C₂×C₄

—

118

—

167

C₂×C₄

—

236

—

168

C₂×C₄

—

236

—

169

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

170

C₂×C₄

D₄×D₅₉

yes

characteristic

171

C₂×C₄

D₄×D₅₉

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

236

—

175

C₈

—

236

—

176

C₂²

—

177

C₂²

—

118

—

178

C₂²

—

118

—

179

C₂²

—

236

—

180

C₂²

—

236

—

181

C₂²

—

236

—

182

C₂²

C₂²⋊C₄×D₅₉

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

118

—

187

C₄

—

118

—

188

C₄

D₈×D₅₉

yes

characteristic

189

C₄

SD₁₆×D₅₉

yes

characteristic

190

C₄

#14273754

yes

characteristic

191

C₄

#14273759

yes

characteristic

192

C₂

—

118

—

193

C₂

—

472

—

194

C₂

#16837385

yes

characteristic

195

C₂

#102×D₅₉

yes

characteristic

196

C₂

#16837402

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#332	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#332	D₅₉
hypocenter	C₅₉	#1547
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#332

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
2. #16837424	C₂	semidirect
3. C₄⋊C₈×D₅₉	C₂	semidirect
4. #16837148	C₂	semidirect
5. #16837145	C₂	semidirect
6. #16837145	C₂	semidirect
7. C₅₉×#332	C₂	semidirect
103. #332	D₅₉	semidirect
106. C₅₉	#1547	semidirect

H ≅ G/N

type

2. #16837424

C₂

semidirect

3. C₄⋊C₈×D₅₉

C₂

semidirect

4. #16837148

C₂

semidirect

5. #16837145

C₂

semidirect

6. #16837145

C₂

semidirect

7. C₅₉×#332

C₂

semidirect

103. #332

D₅₉

semidirect

106. C₅₉

#1547

semidirect

Subgroups — #21422772

Conjugacy classes of subgroups

Complements