Group #21425363 — subgroups

Subgroups — #21425363

Order 7552.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×#11745672	C₂	1	1	yes	maximal, characteristic
2	#16837429	C₂	1	1	yes	maximal, characteristic
3	#16837410	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273646	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273645	C₂	1	1	yes	maximal, characteristic
6	#16837166	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#419	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₂×#11745672	—	2	2	no	—
10	C₂×#11745672	—	2	2	no	—
11	C₂×#11745672	—	2	2	no	—
12	C₂×#11745664	C₂²	1	1	yes	characteristic
13	C₂×#11745664	C₂²	1	1	yes	characteristic
14	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
16	#14273650	—	2	2	no	—
17	#14273646	—	2	2	no	—
18	#14273645	—	2	2	no	—
19	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	#117×C₅₉	C₂²	1	1	yes	characteristic
22	#103×C₅₉	—	2	2	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	—	2	1	no	—
27	C₂×Q₈×C₅₉	—	2	1	no	—
28	C₂×Q₈×C₅₉	D₄	1	2	yes	—
29	C₂²×C₄×C₅₉	—	2	1	no	—
30	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₉	—	2	1	no	—
32	C₂×Dic₁₁₈	—	2	1	no	—
33	C₂×Dic₁₁₈	—	2	1	no	—
34	C₂×Dic₁₁₈	—	2	1	no	—
35	C₂×Dic₁₁₈	—	2	1	no	—
36	C₂×Dic₁₁₈	—	2	1	no	—
37	C₂×Dic₁₁₈	D₄	1	2	yes	—
38	M₁₆×C₅₉	—	2	1	no	—
39	M₁₆×C₅₉	—	2	1	no	—
40	C₂×C₈×C₅₉	—	2	1	no	—
41	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
42	C₂²⋊C₄×C₅₉	—	2	1	no	—
43	C₄²×C₅₉	—	2	1	no	—
44	#11745672	—	2	8	no	—
45	#11745664	—	2	1	no	—
46	#11745664	—	2	1	no	—
47	#11745664	—	2	1	no	—
48	#11745664	—	2	1	no	—
49	C₂×C₅₉⋊C₈	—	2	1	no	—
50	C₂×C₅₉⋊C₈	—	2	1	no	—
51	C₂×C₅₉⋊C₈	—	2	1	no	—
52	C₂×C₅₉⋊C₈	—	2	1	no	—
53	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
54	C₂³×C₅₉	#41	1	1	yes	characteristic
55	Q₈×C₅₉	—	2	2	no	—
56	Q₈×C₅₉	—	2	2	no	—
57	Q₈×C₅₉	—	4	1	no	—
58	Q₈×C₅₉	—	4	1	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	4	1	no	—
62	C₂×C₄×C₅₉	—	4	1	no	—
63	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
69	C₂×Dic₅₉	—	2	2	no	—
70	C₂×Dic₅₉	—	4	1	no	—
71	C₂×Dic₅₉	—	4	1	no	—
72	Dic₁₁₈	—	2	2	no	—
73	Dic₁₁₈	—	2	2	no	—
74	Dic₁₁₈	—	4	1	no	—
75	Dic₁₁₈	—	4	1	no	—
76	C₈×C₅₉	—	4	1	no	—
77	C₅₉⋊C₈	—	2	2	no	—
78	C₅₉⋊C₈	—	4	1	no	—
79	C₅₉⋊C₈	—	4	1	no	—
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	#121	1	1	yes	characteristic
82	C₂²×C₅₉	#123	1	1	yes	characteristic
83	C₂²×C₅₉	#123	1	1	yes	characteristic
84	C₄×C₅₉	—	4	2	no	—
85	C₄×C₅₉	—	4	2	no	—
86	C₄×C₅₉	#120	1	1	yes	characteristic
87	C₄×C₅₉	#121	1	1	yes	characteristic
88	C₄×C₅₉	#121	1	1	yes	characteristic
89	C₄×C₅₉	#124	1	1	yes	characteristic
90	Dic₅₉	—	4	2	no	—
91	#2047	—	59	1	no	maximal, Sylow
92	C₂×C₅₉	—	2	1	no	—
93	C₂×C₅₉	—	2	1	no	—
94	C₂×C₅₉	#394	1	1	yes	characteristic
95	C₂×C₅₉	#473	1	2	yes	—
96	C₂²×Q₁₆	—	59	1	no	—
97	#424	—	59	1	no	—
98	#419	—	59	1	no	—
99	#419	D₅₉	1	1	yes	characteristic
100	C₂×#101	—	59	1	no	—
101	C₂×#101	—	59	1	no	—
102	#341	—	59	1	no	—
103	C₅₉	#2047	1	1	yes	characteristic, complemented, Sylow
104	C₂²×Q₈	—	59	1	no	—
105	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
106	C₂×Q₁₆	—	118	2	no	—
107	C₂×Q₁₆	—	118	2	no	—
108	C₂×Q₁₆	—	118	2	no	—
109	C₂×M₁₆	—	59	1	no	—
110	C₂×M₁₆	—	59	1	no	—
111	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
112	C₂²×C₈	—	59	1	no	—
113	#117	C₂×D₅₉	1	1	yes	characteristic
114	C₄⋊C₈	—	118	2	no	—
115	#103	—	2	2	no	—
116	#103	—	118	2	no	—
117	#101	—	118	2	no	—
118	#101	—	118	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	2	1	no	—
123	C₂×Q₈	—	2	1	no	—
124	C₂×Q₈	—	59	2	no	—
125	C₂×Q₈	—	118	1	no	—
126	C₂×Q₈	—	118	1	no	—
127	C₂×Q₈	—	118	1	no	—
128	C₂×Q₈	—	118	1	no	—
129	C₂×Q₈	—	118	1	no	—
130	C₂×Q₈	D₂₃₆	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	118	1	no	—
133	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
134	Q₁₆	—	118	8	no	—
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	118	1	no	—
138	M₁₆	—	118	1	no	—
139	M₁₆	—	118	1	no	—
140	M₁₆	—	118	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	59	2	no	—
143	C₂×C₈	—	118	1	no	—
144	C₂×C₈	—	118	1	no	—
145	C₂×C₈	—	118	1	no	—
146	C₂×C₈	—	118	1	no	—
147	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
148	C₂²⋊C₄	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₂³	#11745687	1	1	yes	characteristic
151	Q₈	—	2	2	no	—
152	Q₈	—	2	2	no	—
153	Q₈	—	4	1	no	—
154	Q₈	—	4	1	no	—
155	Q₈	—	118	2	no	—
156	Q₈	—	118	2	no	—
157	Q₈	—	236	1	no	—
158	Q₈	—	236	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	1	no	—
163	C₂×C₄	—	118	2	no	—
164	C₂×C₄	—	236	1	no	—
165	C₂×C₄	—	236	1	no	—
166	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
167	C₂×C₄	#11745685	1	1	yes	characteristic
168	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
169	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
170	C₂×C₄	#11745687	1	1	yes	characteristic
171	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
172	C₈	—	4	1	no	—
173	C₈	—	118	2	no	—
174	C₈	—	236	1	no	—
175	C₈	—	236	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#14273699	1	1	yes	characteristic
178	C₂²	#14273709	1	1	yes	characteristic
179	C₂²	#14273711	1	1	yes	characteristic
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	236	2	no	—
183	C₄	#14273697	1	1	yes	characteristic
184	C₄	#14273698	1	1	yes	characteristic
185	C₄	#14273701	1	1	yes	characteristic
186	C₄	#14273713	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#16837318	1	1	yes	characteristic
190	C₂	#16837675	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745672

C₂

yes

maximal, characteristic

#16837429

C₂

yes

maximal, characteristic

#16837410

C₂

yes

maximal, characteristic

C₂×#14273646

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, characteristic

#16837166

C₂

yes

maximal, characteristic

C₅₉×#419

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

#14273650

—

#14273646

—

#14273645

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

—

—

—

—

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

#41

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×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#2047

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#394

yes

characteristic

C₂×C₅₉

#473

yes

—

C₂²×Q₁₆

—

#424

—

#419

—

#419

D₅₉

yes

characteristic

100

C₂×#101

—

101

C₂×#101

—

102

#341

—

103

C₅₉

#2047

yes

characteristic, complemented, Sylow

104

C₂²×Q₈

—

105

C₂²×Q₈

C₂×D₅₉

yes

characteristic

106

C₂×Q₁₆

—

118

—

107

C₂×Q₁₆

—

118

—

108

C₂×Q₁₆

—

118

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂×M₁₆

C₂×D₅₉

yes

characteristic

112

C₂²×C₈

—

113

#117

C₂×D₅₉

yes

characteristic

114

C₄⋊C₈

—

118

—

115

#103

—

116

#103

—

118

—

117

#101

—

118

—

118

#101

—

118

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

118

—

126

C₂×Q₈

—

118

—

127

C₂×Q₈

—

118

—

128

C₂×Q₈

—

118

—

129

C₂×Q₈

—

118

—

130

C₂×Q₈

D₂₃₆

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

118

—

133

C₂²×C₄

C₂²×D₅₉

yes

characteristic

134

Q₁₆

—

118

—

135

M₁₆

—

136

M₁₆

—

137

M₁₆

—

118

—

138

M₁₆

—

118

—

139

M₁₆

—

118

—

140

M₁₆

—

118

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

118

—

144

C₂×C₈

—

118

—

145

C₂×C₈

—

118

—

146

C₂×C₈

—

118

—

147

C₄⋊C₄

C₂²⋊D₅₉

yes

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₂³

#11745687

yes

characteristic

151

Q₈

—

152

Q₈

—

153

Q₈

—

154

Q₈

—

155

Q₈

—

118

—

156

Q₈

—

118

—

157

Q₈

—

236

—

158

Q₈

—

236

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

118

—

164

C₂×C₄

—

236

—

165

C₂×C₄

—

236

—

166

C₂×C₄

C₂×D₂₃₆

yes

characteristic

167

C₂×C₄

#11745685

yes

characteristic

168

C₂×C₄

D₄×D₅₉

yes

characteristic

169

C₂×C₄

D₄×D₅₉

yes

characteristic

170

C₂×C₄

#11745687

yes

characteristic

171

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

172

C₈

—

173

C₈

—

118

—

174

C₈

—

236

—

175

C₈

—

236

—

176

C₂²

—

177

C₂²

#14273699

yes

characteristic

178

C₂²

#14273709

yes

characteristic

179

C₂²

#14273711

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

236

—

183

C₄

#14273697

yes

characteristic

184

C₄

#14273698

yes

characteristic

185

C₄

#14273701

yes

characteristic

186

C₄

#14273713

yes

characteristic

187

C₂

—

188

C₂

—

189

C₂

#16837318

yes

characteristic

190

C₂

#16837675

yes

—

	subgroup	G/H
center	C₂²	#14273699
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#419	C₂
socle	C₂²×C₅₉	#121
hypercenter	#419	D₅₉
hypocenter	C₅₉	#2047
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14273699

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#419

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
103. C₅₉	#2047	semidirect

H ≅ G/N

type

103. C₅₉

#2047

semidirect

Subgroups — #21425363

Conjugacy classes of subgroups

Complements