Group #21425637 — subgroups

Subgroups — #21425637

Order 7552.

254 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂×#14273659	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273658	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273651	C₂	1	1	yes	maximal, characteristic
4	#16837309	C₂	1	1	yes	maximal, characteristic
5	#16837271	C₂	1	1	yes	maximal, characteristic
6	#16837156	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₉⋊C₄	—	2	2	no	—
12	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
13	#14273659	—	2	2	no	—
14	#14273658	—	2	2	no	—
15	#14273651	—	2	2	no	—
16	C₅₉⋊#95	—	2	2	no	—
17	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
20	#103×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	—	2	1	no	—
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	C₂×Q₈×C₅₉	Q₈	1	2	yes	—
27	C₂²×C₄×C₅₉	—	2	1	no	—
28	C₂²×C₄×C₅₉	—	2	1	no	—
29	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₉	—	2	1	no	—
31	C₂²×Dic₅₉	—	2	1	no	—
32	C₂²×Dic₅₉	D₄	1	2	yes	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	C₂×C₈×C₅₉	D₄	1	2	yes	—
36	C₄⋊C₄×C₅₉	—	2	1	no	—
37	C₄⋊C₄×C₅₉	—	2	1	no	—
38	C₄⋊Dic₅₉	—	2	1	no	—
39	C₄⋊Dic₅₉	—	2	1	no	—
40	C₄⋊Dic₅₉	—	2	1	no	—
41	C₄⋊Dic₅₉	—	2	1	no	—
42	Dic₅₉⋊C₄	—	4	4	no	—
43	C₄×Dic₅₉	—	2	1	no	—
44	C₄×Dic₅₉	—	2	1	no	—
45	C₂×C₅₉⋊C₈	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
48	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₅₉	—	2	2	no	—
50	Q₈×C₅₉	—	2	2	no	—
51	Q₈×C₅₉	—	4	1	no	—
52	Q₈×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	4	1	no	—
55	C₂×C₄×C₅₉	—	4	1	no	—
56	C₂×C₄×C₅₉	—	4	1	no	—
57	C₂×C₄×C₅₉	—	4	2	no	—
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
64	C₂×Dic₅₉	—	2	2	no	—
65	C₂×Dic₅₉	—	2	2	no	—
66	C₂×Dic₅₉	—	4	1	no	—
67	C₂×Dic₅₉	—	4	1	no	—
68	C₂×Dic₅₉	—	4	2	no	—
69	C₂×Dic₅₉	—	4	2	no	—
70	C₂×Dic₅₉	SD₁₆	1	4	yes	—
71	C₈×C₅₉	—	2	2	no	—
72	C₅₉⋊C₈	—	2	2	no	—
73	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₅₉	#123	1	1	yes	characteristic
75	C₂²×C₅₉	#127	1	1	yes	characteristic
76	C₂²×C₅₉	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₅₉	#137	1	2	yes	—
78	C₄×C₅₉	—	4	2	no	—
79	C₄×C₅₉	—	8	1	no	—
80	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
82	C₄×C₅₉	#123	1	1	yes	characteristic
83	C₄×C₅₉	#123	1	1	yes	characteristic
84	Dic₅₉	—	2	4	no	—
85	Dic₅₉	—	8	1	no	—
86	Dic₅₉	—	8	1	no	—
87	#2096	—	59	1	no	maximal, Sylow
88	C₂×C₅₉	#399	1	1	yes	characteristic
89	C₂×C₅₉	#475	1	2	yes	—
90	C₂×C₅₉	#483	1	2	yes	—
91	C₂×C₅₉	#492	1	1	yes	characteristic
92	C₂×C₅₉	#497	1	1	yes	characteristic
93	C₂×#106	—	59	1	no	—
94	C₂×#106	—	59	1	no	—
95	C₂×#103	—	59	1	no	—
96	C₂×#103	D₅₉	1	1	yes	characteristic
97	#391	—	59	1	no	—
98	#384	—	59	1	no	—
99	#336	—	59	1	no	—
100	C₅₉	#2096	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
102	C₂²×C₈	—	59	1	no	—
103	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	59	1	no	—
105	C₂×C₄⋊C₄	—	59	1	no	—
106	C₂×C₄⋊C₄	—	118	2	no	—
107	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
108	C₂×C₄²	—	59	1	no	—
109	#106	—	118	2	no	—
110	#106	—	118	2	no	—
111	#103	—	2	2	no	—
112	#103	—	118	2	no	—
113	#95	—	118	2	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	Dic₁₁₈	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	59	2	no	—
123	C₂²×C₄	—	118	1	no	—
124	C₂²×C₄	—	118	1	no	—
125	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	59	2	no	—
129	C₂×C₈	—	118	1	no	—
130	C₂×C₈	—	118	1	no	—
131	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	118	1	no	—
135	C₄⋊C₄	—	118	1	no	—
136	C₄⋊C₄	—	118	1	no	—
137	C₄⋊C₄	—	118	1	no	—
138	C₄⋊C₄	—	236	4	no	—
139	C₄²	—	118	1	no	—
140	C₄²	—	118	1	no	—
141	C₂³	D₄×D₅₉	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	59	4	no	—
152	C₂×C₄	—	118	2	no	—
153	C₂×C₄	—	118	2	no	—
154	C₂×C₄	—	236	1	no	—
155	C₂×C₄	—	236	1	no	—
156	C₂×C₄	—	236	2	no	—
157	C₂×C₄	—	236	2	no	—
158	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
159	C₂×C₄	#11745685	1	1	yes	characteristic
160	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
161	C₂×C₄	#11745687	1	1	yes	characteristic
162	C₂×C₄	#11745687	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	118	2	no	—
166	C₂²	#14273704	1	1	yes	characteristic
167	C₂²	#14273709	1	1	yes	characteristic
168	C₂²	#14273723	1	1	yes	characteristic
169	C₂²	SD₁₆×D₅₉	1	2	yes	—
170	C₂²	#14273759	1	2	yes	—
171	C₄	—	4	2	no	—
172	C₄	—	8	1	no	—
173	C₄	—	118	4	no	—
174	C₄	—	472	1	no	—
175	C₄	—	472	1	no	—
176	C₄	#14273704	1	1	yes	characteristic
177	C₄	#14273706	1	1	yes	characteristic
178	C₄	#14273708	1	1	yes	characteristic
179	C₄	#14273709	1	1	yes	characteristic
180	C₂	#16837338	1	1	yes	characteristic
181	C₂	#16837685	1	2	yes	—
182	C₂	#16837731	1	2	yes	—
183	C₂	#16837773	1	1	yes	characteristic
184	C₂	#16837795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273659

C₂

yes

maximal, characteristic

C₂×#14273658

C₂

yes

maximal, characteristic

C₂×#14273651

C₂

yes

maximal, characteristic

#16837309

C₂

yes

maximal, characteristic

#16837271

C₂

yes

maximal, characteristic

#16837156

C₂

yes

maximal, characteristic

C₂×#103×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

—

—

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

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×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

SD₁₆

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂²⋊Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×SD₁₆

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

#483

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

D₅₉

yes

—

—

—

100

C₅₉

#2096

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

C₂×D₅₉

yes

characteristic

102

C₂²×C₈

—

103

C₂²×C₈

C₂×D₅₉

yes

characteristic

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

118

—

107

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

108

C₂×C₄²

—

109

#106

—

118

—

110

#106

—

118

—

111

#103

—

112

#103

—

118

—

113

#95

—

118

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

Dic₁₁₈

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

118

—

124

C₂²×C₄

—

118

—

125

C₂²×C₄

C₂²×D₅₉

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

118

—

130

C₂×C₈

—

118

—

131

C₂×C₈

C₂²⋊D₅₉

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

118

—

135

C₄⋊C₄

—

118

—

136

C₄⋊C₄

—

118

—

137

C₄⋊C₄

—

118

—

138

C₄⋊C₄

—

236

—

139

C₄²

—

118

—

140

C₄²

—

118

—

141

C₂³

D₄×D₅₉

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

118

—

153

C₂×C₄

—

118

—

154

C₂×C₄

—

236

—

155

C₂×C₄

—

236

—

156

C₂×C₄

—

236

—

157

C₂×C₄

—

236

—

158

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

159

C₂×C₄

#11745685

yes

characteristic

160

C₂×C₄

D₄×D₅₉

yes

characteristic

161

C₂×C₄

#11745687

yes

characteristic

162

C₂×C₄

#11745687

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

164

C₈

—

165

C₈

—

118

—

166

C₂²

#14273704

yes

characteristic

167

C₂²

#14273709

yes

characteristic

168

C₂²

#14273723

yes

characteristic

169

C₂²

SD₁₆×D₅₉

yes

—

170

C₂²

#14273759

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

118

—

174

C₄

—

472

—

175

C₄

—

472

—

176

C₄

#14273704

yes

characteristic

177

C₄

#14273706

yes

characteristic

178

C₄

#14273708

yes

characteristic

179

C₄

#14273709

yes

characteristic

180

C₂

#16837338

yes

characteristic

181

C₂

#16837685

yes

—

182

C₂

#16837731

yes

—

183

C₂

#16837773

yes

characteristic

184

C₂

#16837795

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#103×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#103	D₅₉
hypocenter	C₅₉	#2096
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#103×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
100. C₅₉	#2096	semidirect

H ≅ G/N

type

100. C₅₉

#2096

semidirect

Subgroups — #21425637

Conjugacy classes of subgroups

Complements