Group #21429132 — subgroups

Subgroups — #21429132

Order 7552.

452 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14273718	C₂	1	2	yes	maximal
2	#16837335	C₂	1	4	yes	maximal
3	#16837301	C₂	1	2	yes	maximal
4	#16837298	C₂	1	2	yes	maximal
5	#16837271	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#384	C₂	1	1	yes	maximal, characteristic
9	#14273718	—	2	8	no	—
10	C₂×C₄⋊Dic₅₉	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₉⋊C₄	C₂²	1	4	yes	—
16	C₂×Dic₅₉⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
18	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
19	C₅₉⋊#95	C₂²	1	2	yes	—
20	C₅₉⋊#95	C₂²	1	2	yes	—
21	C₅₉⋊#95	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	C₂³	1	4	yes	—
27	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₅₉	C₂³	1	2	yes	—
31	C₂²×Dic₅₉	C₂³	1	2	yes	—
32	C₂²×Dic₅₉	C₂³	1	2	yes	—
33	C₂²×Dic₅₉	C₂³	1	2	yes	—
34	C₄⋊C₄×C₅₉	—	2	4	no	—
35	C₄⋊C₄×C₅₉	—	2	4	no	—
36	C₄⋊C₄×C₅₉	—	4	2	no	—
37	C₄²×C₅₉	Q₈	1	4	yes	—
38	C₄⋊Dic₅₉	—	2	2	no	—
39	C₄⋊Dic₅₉	—	2	2	no	—
40	C₄⋊Dic₅₉	—	2	8	no	—
41	C₄⋊Dic₅₉	D₄	1	4	yes	—
42	C₄⋊Dic₅₉	Q₈	1	4	yes	—
43	Dic₅₉⋊C₄	—	2	8	no	—
44	Dic₅₉⋊C₄	—	4	4	no	—
45	C₄×Dic₅₉	—	2	4	no	—
46	C₄×Dic₅₉	—	2	4	no	—
47	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₅₉	—	2	1	no	—
49	C₂×C₄×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	2	4	no	—
51	C₂×C₄×C₅₉	—	2	4	no	—
52	C₂×C₄×C₅₉	—	2	4	no	—
53	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₅₉	#41	1	2	yes	—
59	C₂×C₄×C₅₉	#41	1	2	yes	—
60	C₂×Dic₅₉	—	2	2	no	—
61	C₂×Dic₅₉	—	2	2	no	—
62	C₂×Dic₅₉	—	2	2	no	—
63	C₂×Dic₅₉	—	2	2	no	—
64	C₂×Dic₅₉	—	2	2	no	—
65	C₂×Dic₅₉	—	2	2	no	—
66	C₂×Dic₅₉	—	2	2	no	—
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	—	2	2	no	—
69	C₂×Dic₅₉	—	2	2	no	—
70	C₂×Dic₅₉	C₂×D₄	1	4	yes	—
71	C₂×Dic₅₉	C₂×Q₈	1	4	yes	—
72	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₅₉	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₅₉	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
76	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
77	C₂²×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₅₉	—	2	2	no	—
80	C₄×C₅₉	—	4	4	no	—
81	C₄×C₅₉	#125	1	4	yes	complemented
82	C₄×C₅₉	C₄⋊Q₈	1	4	yes	—
83	Dic₅₉	—	2	4	no	—
84	Dic₅₉	—	2	4	no	—
85	Dic₅₉	—	4	2	no	—
86	Dic₅₉	—	4	2	no	—
87	#2552	—	59	1	no	maximal, Sylow
88	C₂×C₅₉	C₂×#125	1	1	yes	characteristic
89	C₂×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₅₉	#544	1	1	yes	characteristic
91	C₂×C₅₉	#547	1	1	yes	characteristic
92	C₂×C₅₉	#550	1	1	yes	characteristic
93	C₂×C₅₉	#554	1	1	yes	characteristic
94	C₂×C₅₉	#557	1	1	yes	characteristic
95	C₂×#125	—	59	2	no	—
96	#398	—	59	4	no	—
97	#389	—	59	2	no	—
98	#389	—	59	2	no	—
99	#384	—	59	2	no	—
100	#384	D₅₉	1	1	yes	characteristic
101	C₄×C₄⋊C₄	—	59	1	no	—
102	C₄×C₄⋊C₄	—	59	1	no	—
103	C₅₉	#2552	1	1	yes	characteristic, complemented, Sylow
104	#125	—	118	8	no	—
105	C₂×C₄⋊C₄	—	59	1	no	—
106	C₂×C₄⋊C₄	—	59	1	no	—
107	C₂×C₄⋊C₄	—	59	4	no	—
108	C₂×C₄⋊C₄	—	59	4	no	—
109	C₂×C₄⋊C₄	—	59	1	no	—
110	C₂×C₄⋊C₄	—	59	1	no	—
111	C₂×C₄⋊C₄	—	59	4	no	—
112	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
113	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
115	C₂×C₄²	—	59	2	no	—
116	C₂×C₄²	—	59	2	no	—
117	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
118	#95	—	59	2	no	—
119	#95	—	59	2	no	—
120	#95	—	59	4	no	—
121	C₂²×C₄	—	59	2	no	—
122	C₂²×C₄	—	59	2	no	—
123	C₂²×C₄	—	59	2	no	—
124	C₂²×C₄	—	59	2	no	—
125	C₂²×C₄	C₂²×D₅₉	1	4	yes	—
126	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	59	4	no	—
133	C₄⋊C₄	—	59	4	no	—
134	C₄⋊C₄	—	118	8	no	—
135	C₄⋊C₄	—	118	8	no	—
136	C₄⋊C₄	—	118	2	no	—
137	C₄⋊C₄	—	118	2	no	—
138	C₄⋊C₄	—	236	4	no	—
139	C₄²	—	118	4	no	—
140	C₄²	—	118	4	no	—
141	C₄²	Dic₁₁₈	1	4	yes	—
142	C₂³	C₂³×D₅₉	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	59	4	no	—
149	C₂×C₄	—	59	4	no	—
150	C₂×C₄	—	118	2	no	—
151	C₂×C₄	—	118	2	no	—
152	C₂×C₄	—	118	2	no	—
153	C₂×C₄	—	118	2	no	—
154	C₂×C₄	—	118	2	no	—
155	C₂×C₄	—	118	2	no	—
156	C₂×C₄	—	118	2	no	—
157	C₂×C₄	—	118	2	no	—
158	C₂×C₄	—	118	2	no	—
159	C₂×C₄	—	118	2	no	—
160	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
162	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
163	C₂×C₄	D₄×D₅₉	1	2	yes	—
164	C₂×C₄	#11745687	1	2	yes	—
165	C₂×C₄	Q₈×D₅₉	1	2	yes	—
166	C₂×C₄	#11745689	1	2	yes	—
167	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₅₉	1	1	yes	characteristic
169	C₂²	C₂²×Dic₁₁₈	1	1	yes	characteristic
170	C₂²	C₂×#11745687	1	1	yes	characteristic
171	C₂²	C₂×#11745689	1	1	yes	characteristic
172	C₂²	#14273772	1	1	yes	characteristic
173	C₂²	#14273775	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	4	no	—
176	C₄	—	118	4	no	—
177	C₄	—	118	4	no	—
178	C₄	—	236	2	no	—
179	C₄	—	236	2	no	—
180	C₄	#14273718	1	4	yes	complemented
181	C₄	#14273726	1	4	yes	—
182	C₂	C₂×#14273718	1	1	yes	characteristic
183	C₂	C₂×#14273726	1	1	yes	characteristic
184	C₂	#16838033	1	1	yes	characteristic
185	C₂	#16838055	1	1	yes	characteristic
186	C₂	#16838072	1	1	yes	characteristic
187	C₂	#16838113	1	1	yes	characteristic
188	C₂	#16838127	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273718

C₂

yes

maximal

#16837335

C₂

yes

maximal

#16837301

C₂

yes

maximal

#16837298

C₂

yes

maximal

#16837271

C₂

yes

maximal

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

C₅₉×#384

C₂

yes

maximal, characteristic

#14273718

—

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

C₂²

yes

—

C₂×Dic₅₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₅₉⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₂²×Dic₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

C₄⋊Dic₅₉

Q₈

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂⁴

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₄

yes

—

C₂×Dic₅₉

C₂×Q₈

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂²×C₅₉

C₂²×Q₈

yes

characteristic

C₂²×C₅₉

C₂²×Q₈

yes

characteristic

C₂²×C₅₉

C₂×#41

yes

characteristic

C₂²×C₅₉

C₂×#41

yes

characteristic

C₂²×C₅₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₅₉

2₋¹⁺⁴

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#125

yes

complemented

C₄×C₅₉

C₄⋊Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×C₄⋊Q₈

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

—

—

—

100

#384

D₅₉

yes

characteristic

101

C₄×C₄⋊C₄

—

102

C₄×C₄⋊C₄

—

103

C₅₉

#2552

yes

characteristic, complemented, Sylow

104

#125

—

118

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

113

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

114

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×D₅₉

yes

characteristic

118

#95

—

119

#95

—

120

#95

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²×D₅₉

yes

—

126

C₂²×C₄

C₂²×D₅₉

yes

characteristic

127

C₂²×C₄

C₂²×D₅₉

yes

characteristic

128

C₂²×C₄

C₂²×D₅₉

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

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

C₄²

Dic₁₁₈

yes

—

142

C₂³

C₂³×D₅₉

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

118

—

151

C₂×C₄

—

118

—

152

C₂×C₄

—

118

—

153

C₂×C₄

—

118

—

154

C₂×C₄

—

118

—

155

C₂×C₄

—

118

—

156

C₂×C₄

—

118

—

157

C₂×C₄

—

118

—

158

C₂×C₄

—

118

—

159

C₂×C₄

—

118

—

160

C₂×C₄

C₂×Dic₁₁₈

yes

—

161

C₂×C₄

C₂×Dic₁₁₈

yes

—

162

C₂×C₄

C₂×Dic₁₁₈

yes

—

163

C₂×C₄

D₄×D₅₉

yes

—

164

C₂×C₄

#11745687

yes

—

165

C₂×C₄

Q₈×D₅₉

yes

—

166

C₂×C₄

#11745689

yes

—

167

C₂²

C₂×D₄×D₅₉

yes

characteristic

168

C₂²

C₂×Q₈×D₅₉

yes

characteristic

169

C₂²

C₂²×Dic₁₁₈

yes

characteristic

170

C₂²

C₂×#11745687

yes

characteristic

171

C₂²

C₂×#11745689

yes

characteristic

172

C₂²

#14273772

yes

characteristic

173

C₂²

#14273775

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

118

—

177

C₄

—

118

—

178

C₄

—

236

—

179

C₄

—

236

—

180

C₄

#14273718

yes

complemented

181

C₄

#14273726

yes

—

182

C₂

C₂×#14273718

yes

characteristic

183

C₂

C₂×#14273726

yes

characteristic

184

C₂

#16838033

yes

characteristic

185

C₂

#16838055

yes

characteristic

186

C₂

#16838072

yes

characteristic

187

C₂

#16838113

yes

characteristic

188

C₂

#16838127

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator, socle	C₂³×C₅₉	C₂⁴
Fitting	C₅₉×#384	C₂
hypercenter	#384	D₅₉
hypocenter	C₅₉	#2552
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₉

commutator, socle

C₂³×C₅₉

C₂⁴

Fitting

C₅₉×#384

C₂

D₅₉

C₅₉

G ▷ C₂³×C₅₉ ▷ C₅₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
81. C₄×C₅₉	#125	semidirect
103. C₅₉	#2552	semidirect
180. C₄	#14273718	semidirect

H ≅ G/N

type

81. C₄×C₅₉

#125

semidirect

103. C₅₉

#2552

semidirect

180. C₄

#14273718

semidirect

Subgroups — #21429132

Conjugacy classes of subgroups

Complements