Group #21422798 — subgroups

Subgroups — #21422798

Order 7552.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16837950	C₂	1	1	yes	maximal, characteristic
2	#16837458	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837454	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837149	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837147	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837146	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11745687	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273714	—	2	1	no	—
11	#14273698	—	2	1	no	—
12	#14273698	C₄	1	2	yes	—
13	D₄×Dic₅₉	—	2	1	no	—
14	D₄×Dic₅₉	C₄	1	2	yes	—
15	C₄²×D₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
18	(C₂×D₅₉)⋊C₈	—	2	1	no	—
19	(C₂×D₅₉)⋊C₈	—	2	1	no	—
20	#14273637	—	2	1	no	—
21	C₈×Dic₅₉	—	2	1	no	—
22	#127×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×D₄×C₅₉	—	2	1	no	—
26	C₂×D₄×C₅₉	—	2	1	no	—
27	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊D₅₉	—	2	2	no	—
29	C₂²×Dic₅₉	—	2	2	no	—
30	#11745687	—	4	2	no	—
31	#11745687	—	4	2	no	—
32	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
34	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
35	C₂×Dic₁₁₈	—	2	1	no	—
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₄²×C₅₉	C₂³	1	1	yes	characteristic
39	C₂²⋊Dic₅₉	—	2	2	no	—
40	C₂²⋊Dic₅₉	—	2	2	no	—
41	C₄⋊Dic₅₉	—	2	1	no	—
42	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
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	C₂³×C₅₉	—	2	2	no	—
49	C₂²×D₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	D₄×C₅₉	—	4	2	no	—
51	D₄×C₅₉	—	4	1	no	—
52	D₄×C₅₉	—	4	1	no	—
53	D₄×C₅₉	—	4	1	no	—
54	D₄×C₅₉	—	4	1	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	C₂²⋊D₅₉	—	4	4	no	—
59	C₂×Dic₅₉	—	2	2	no	—
60	C₂×Dic₅₉	—	4	4	no	—
61	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄×D₅₉	—	2	1	no	—
65	C₄×D₅₉	—	2	1	no	—
66	C₄×D₅₉	—	2	1	no	—
67	C₄×D₅₉	—	2	1	no	—
68	C₄×D₅₉	—	4	1	no	—
69	Dic₁₁₈	—	4	1	no	—
70	Dic₁₁₈	—	4	1	no	—
71	C₈×C₅₉	—	4	1	no	—
72	C₈×C₅₉	—	4	1	no	—
73	C₅₉⋊C₈	—	4	1	no	—
74	C₅₉⋊C₈	—	4	1	no	—
75	C₂²×C₅₉	—	4	2	no	—
76	C₂²×C₅₉	—	4	2	no	—
77	C₂²×C₅₉	—	4	2	no	—
78	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×D₅₉	—	2	1	no	—
80	C₂×D₅₉	—	2	1	no	—
81	C₂×D₅₉	—	2	1	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	#135	1	1	yes	characteristic
84	C₄×C₅₉	#135	1	1	yes	characteristic
85	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
87	Dic₅₉	—	2	1	no	—
88	Dic₅₉	—	2	1	no	—
89	Dic₅₉	—	2	1	no	—
90	Dic₅₉	—	4	2	no	—
91	#1553	—	59	1	no	maximal, Sylow
92	C₂×C₅₉	—	8	2	no	—
93	C₂×C₅₉	#413	1	1	yes	characteristic
94	C₂×C₅₉	#416	1	1	yes	characteristic
95	C₂×C₅₉	#418	1	1	yes	characteristic
96	D₅₉	—	4	1	no	—
97	#532	—	59	1	no	—
98	#432	—	59	1	no	—
99	#431	—	59	1	no	—
100	#332	—	59	1	no	—
101	#332	—	59	1	no	—
102	#331	—	59	1	no	—
103	#331	D₅₉	1	1	yes	characteristic, complemented
104	C₅₉	#1553	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	118	1	no	—
106	C₄⋊Q₈	—	59	1	no	—
107	#127	C₂×D₅₉	1	1	yes	characteristic
108	#124	—	118	1	no	—
109	#121	—	59	2	no	—
110	#121	—	118	1	no	—
111	C₄×D₄	—	59	2	no	—
112	C₄×D₄	—	118	1	no	—
113	C₂×C₄²	—	59	1	no	—
114	C₄⋊C₈	—	59	1	no	—
115	C₄⋊C₈	—	59	1	no	—
116	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
117	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
118	C₂²⋊C₈	—	118	1	no	—
119	C₂²⋊C₈	—	118	1	no	—
120	#97	—	118	1	no	—
121	C₄×C₈	—	118	1	no	—
122	#41	—	236	2	no	—
123	#41	—	236	2	no	—
124	C₂×Q₈	—	118	1	no	—
125	C₂×D₄	—	2	1	no	—
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	—	118	2	no	—
128	C₂×D₄	C₄×D₅₉	1	2	yes	—
129	C₂²×C₄	—	59	1	no	—
130	C₂²×C₄	—	59	1	no	—
131	C₂²×C₄	—	59	1	no	—
132	C₂²×C₄	—	118	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	118	1	no	—
136	C₂×C₈	—	118	1	no	—
137	C₄⋊C₄	—	59	2	no	—
138	C₄⋊C₄	—	118	1	no	—
139	C₂²⋊C₄	—	118	2	no	—
140	C₂²⋊C₄	—	118	2	no	—
141	C₄²	—	59	1	no	—
142	C₄²	—	59	1	no	—
143	C₄²	—	59	1	no	—
144	C₄²	C₂²×D₅₉	1	1	yes	characteristic
145	C₂³	—	2	2	no	—
146	C₂³	—	59	1	no	—
147	Q₈	—	236	1	no	—
148	Q₈	—	236	1	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	236	4	no	—
155	C₂×C₄	—	59	1	no	—
156	C₂×C₄	—	59	1	no	—
157	C₂×C₄	—	59	1	no	—
158	C₂×C₄	—	118	1	no	—
159	C₂×C₄	—	118	1	no	—
160	C₂×C₄	—	118	1	no	—
161	C₂×C₄	—	118	1	no	—
162	C₂×C₄	—	118	2	no	—
163	C₂×C₄	—	236	1	no	—
164	C₂×C₄	—	236	4	no	—
165	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
166	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
167	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
168	C₈	—	4	1	no	—
169	C₈	—	4	1	no	—
170	C₈	—	236	1	no	—
171	C₈	—	236	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	118	1	no	—
176	C₂²	—	118	1	no	—
177	C₂²	—	118	1	no	—
178	C₂²	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	118	1	no	—
181	C₄	—	118	1	no	—
182	C₄	—	118	1	no	—
183	C₄	—	236	2	no	—
184	C₄	#14273746	1	1	yes	characteristic
185	C₄	#14273748	1	1	yes	characteristic
186	C₄	#14273751	1	1	yes	characteristic
187	C₄	#14273754	1	1	yes	characteristic
188	C₂	—	8	2	no	—
189	C₂	—	236	1	no	—
190	C₂	#16837383	1	1	yes	characteristic
191	C₂	#16837395	1	1	yes	characteristic
192	C₂	#16837405	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837950

C₂

yes

maximal, characteristic

#16837458

C₂

yes

maximal, characteristic, complemented

#16837454

C₂

yes

maximal, characteristic, complemented

#16837149

C₂

yes

maximal, characteristic, complemented

#16837147

C₂

yes

maximal, characteristic, complemented

#16837146

C₂

yes

maximal, characteristic, complemented

C₅₉×#331

C₂

yes

maximal, characteristic, complemented

C₂×#11745687

—

#14273727

C₂²

yes

—

—

C₄

yes

—

D₄×Dic₅₉

—

D₄×Dic₅₉

C₄

yes

—

C₄²×D₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

(C₂×D₅₉)⋊C₈

—

(C₂×D₅₉)⋊C₈

—

#14273637

—

C₈×Dic₅₉

—

#127×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂×C₄

yes

—

C₂×C₂²⋊D₅₉

—

C₂²×Dic₅₉

—

#11745687

—

#11745687

—

C₂×C₄×D₅₉

C₂×C₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×C₄×D₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂²×D₅₉

C₂²⋊C₄

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1553

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

D₅₉

—

#532

—

#432

—

#431

—

100

#332

—

101

#332

—

102

#331

—

103

#331

D₅₉

yes

characteristic, complemented

104

C₅₉

#1553

yes

characteristic, complemented, Sylow

105

C₂×#41

—

118

—

106

C₄⋊Q₈

—

107

#127

C₂×D₅₉

yes

characteristic

108

#124

—

118

—

109

#121

—

110

#121

—

118

—

111

C₄×D₄

—

112

C₄×D₄

—

118

—

113

C₂×C₄²

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

C₂×D₅₉

yes

characteristic

117

C₄⋊C₈

C₂×D₅₉

yes

characteristic

118

C₂²⋊C₈

—

118

—

119

C₂²⋊C₈

—

118

—

120

#97

—

118

—

121

C₄×C₈

—

118

—

122

#41

—

236

—

123

#41

—

236

—

124

C₂×Q₈

—

118

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

118

—

128

C₂×D₄

C₄×D₅₉

yes

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

118

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

118

—

136

C₂×C₈

—

118

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

118

—

139

C₂²⋊C₄

—

118

—

140

C₂²⋊C₄

—

118

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

C₂²×D₅₉

yes

characteristic

145

C₂³

—

146

C₂³

—

147

Q₈

—

236

—

148

Q₈

—

236

—

149

D₄

—

150

D₄

—

151

D₄

—

152

D₄

—

153

D₄

—

154

D₄

—

236

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

118

—

159

C₂×C₄

—

118

—

160

C₂×C₄

—

118

—

161

C₂×C₄

—

118

—

162

C₂×C₄

—

118

—

163

C₂×C₄

—

236

—

164

C₂×C₄

—

236

—

165

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

166

C₂×C₄

D₄×D₅₉

yes

characteristic

167

C₂×C₄

D₄×D₅₉

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

236

—

171

C₈

—

236

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

118

—

176

C₂²

—

118

—

177

C₂²

—

118

—

178

C₂²

C₂²⋊C₄×D₅₉

yes

characteristic

179

C₄

—

180

C₄

—

118

—

181

C₄

—

118

—

182

C₄

—

118

—

183

C₄

—

236

—

184

C₄

#14273746

yes

characteristic

185

C₄

#14273748

yes

characteristic

186

C₄

#14273751

yes

characteristic

187

C₄

#14273754

yes

characteristic

188

C₂

—

189

C₂

—

236

—

190

C₂

#16837383

yes

characteristic

191

C₂

#16837395

yes

characteristic

192

C₂

#16837405

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#331	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₄
hypercenter	#331	D₅₉
hypocenter	C₅₉	#1553
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₅₉×#331

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #16837458	C₂	semidirect
3. #16837454	C₂	semidirect
4. #16837149	C₂	semidirect
5. #16837147	C₂	semidirect
6. #16837146	C₂	semidirect
7. C₅₉×#331	C₂	semidirect
103. #331	D₅₉	semidirect
104. C₅₉	#1553	semidirect

H ≅ G/N

type

2. #16837458

C₂

semidirect

3. #16837454

C₂

semidirect

4. #16837149

C₂

semidirect

5. #16837147

C₂

semidirect

6. #16837146

C₂

semidirect

7. C₅₉×#331

C₂

semidirect

103. #331

D₅₉

semidirect

104. C₅₉

#1553

semidirect

Subgroups — #21422798

Conjugacy classes of subgroups

Complements