Group #21433583 — subgroups

Subgroups — #21433583

Order 7552.

480 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14273757	C₂	1	2	yes	maximal
2	C₂×#14273727	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837612	C₂	1	4	yes	maximal, complemented
4	#16837607	C₂	1	4	yes	maximal, complemented
5	#16837426	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837410	C₂	1	2	yes	maximal, complemented
7	C₄×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
9	#14273757	—	2	8	no	—
10	C₂×#11745672	C₂²	1	4	yes	—
11	C₂×#11745670	C₂²	1	4	yes	—
12	C₂×#11745664	C₂²	1	2	yes	—
13	#14273727	—	2	1	no	—
14	#14273727	—	2	1	no	—
15	#14273727	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	C₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
18	#14273650	C₂²	1	8	yes	—
19	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₅₉	C₂²	1	2	yes	—
21	C₄×D₄×C₅₉	—	2	2	no	—
22	C₄×D₄×C₅₉	C₂²	1	2	yes	—
23	#117×C₅₉	—	2	1	no	—
24	#117×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	—	2	1	no	—
26	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
27	#41×C₅₉	—	2	2	no	—
28	#41×C₅₉	D₄	1	4	yes	—
29	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×D₄×C₅₉	—	2	1	no	—
31	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	—	2	1	no	—
33	C₂²×C₄×C₅₉	—	2	2	no	—
34	C₂²×C₄×C₅₉	D₄	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	2	2	no	—
37	C₂×Dic₁₁₈	—	2	4	no	—
38	C₂×Dic₁₁₈	—	2	4	no	—
39	C₂×Dic₁₁₈	—	2	2	no	—
40	C₂×Dic₁₁₈	C₂³	1	4	yes	—
41	C₄⋊C₄×C₅₉	—	2	2	no	—
42	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
43	C₂²⋊C₄×C₅₉	—	2	1	no	—
44	C₂²⋊C₄×C₅₉	—	2	2	no	—
45	C₄²×C₅₉	—	2	1	no	—
46	C₄²×C₅₉	—	2	2	no	—
47	C₄²×C₅₉	D₄	1	1	yes	characteristic
48	C₄²×C₅₉	D₄	1	1	yes	characteristic
49	C₄²×C₅₉	C₂³	1	2	yes	—
50	#11745672	—	2	8	no	—
51	#11745670	—	2	8	no	—
52	C₄⋊Dic₅₉	—	2	4	no	—
53	C₄⋊Dic₅₉	—	2	4	no	—
54	#11745664	—	2	4	no	—
55	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
56	C₂³×C₅₉	—	2	1	no	—
57	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
58	Q₈×C₅₉	—	2	1	no	—
59	Q₈×C₅₉	C₂×D₄	1	2	yes	—
60	D₄×C₅₉	—	2	4	no	—
61	D₄×C₅₉	—	2	1	no	—
62	D₄×C₅₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₅₉	—	2	1	no	—
64	C₂×C₄×C₅₉	—	2	4	no	—
65	C₂×C₄×C₅₉	—	2	4	no	—
66	C₂×C₄×C₅₉	—	2	2	no	—
67	C₂×C₄×C₅₉	—	4	4	no	—
68	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₅₉	#41	1	2	yes	—
73	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₅₉	—	2	4	no	—
75	C₂×Dic₅₉	—	4	4	no	—
76	Dic₁₁₈	—	2	4	no	—
77	Dic₁₁₈	—	2	4	no	—
78	Dic₁₁₈	—	4	4	no	—
79	C₅₉⋊C₈	—	2	4	no	—
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	—	2	2	no	—
82	C₂²×C₅₉	—	4	2	no	—
83	C₂²×C₅₉	#121	1	2	yes	—
84	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
85	C₄×C₅₉	—	2	2	no	—
86	C₄×C₅₉	—	2	2	no	—
87	C₄×C₅₉	—	4	2	no	—
88	C₄×C₅₉	#121	1	2	yes	—
89	C₄×C₅₉	#137	1	4	yes	complemented
90	C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
92	Dic₅₉	—	4	4	no	—
93	#3039	—	59	1	no	maximal, Sylow
94	C₂×C₅₉	—	2	2	no	—
95	C₂×C₅₉	—	4	2	no	—
96	C₂×C₅₉	C₂×#121	1	1	yes	characteristic
97	C₂×C₅₉	C₂×#137	1	2	yes	—
98	C₂×#137	—	59	2	no	—
99	C₂×C₄⋊Q₈	—	59	1	no	—
100	C₄×#41	D₅₉	1	1	yes	characteristic
101	#462	—	59	4	no	—
102	#461	—	59	4	no	—
103	#423	—	59	1	no	—
104	#419	—	59	2	no	—
105	C₅₉	#3039	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
107	C₂²×Q₈	—	59	2	no	—
108	#137	—	118	8	no	—
109	C₂×Q₁₆	—	59	4	no	—
110	C₂×SD₁₆	—	59	4	no	—
111	C₂×M₁₆	—	59	2	no	—
112	C₄⋊Q₈	—	59	4	no	—
113	C₄⋊Q₈	—	118	1	no	—
114	C₄⋊Q₈	—	118	1	no	—
115	C₄×Q₈	C₂×D₅₉	1	2	yes	—
116	C₄×D₄	—	2	2	no	—
117	C₄×D₄	C₂×D₅₉	1	2	yes	—
118	#117	—	2	1	no	—
119	#117	C₂×D₅₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	118	2	no	—
121	C₂×C₄²	—	2	1	no	—
122	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
123	C₄⋊C₈	—	59	4	no	—
124	#103	—	59	8	no	—
125	#41	—	2	2	no	—
126	#41	D₂₃₆	1	4	yes	—
127	C₂×Q₈	—	59	4	no	—
128	C₂×Q₈	—	118	2	no	—
129	C₂×Q₈	—	118	4	no	—
130	C₂×Q₈	—	118	4	no	—
131	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
132	C₂×D₄	—	2	1	no	—
133	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	118	2	no	—
137	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
138	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
139	Q₁₆	—	118	8	no	—
140	SD₁₆	—	118	8	no	—
141	M₁₆	—	118	4	no	—
142	C₂×C₈	—	59	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	118	4	no	—
145	C₄⋊C₄	—	118	4	no	—
146	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
147	C₂²⋊C₄	—	2	1	no	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
152	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
153	C₄²	C₂²×D₅₉	1	2	yes	—
154	C₂³	—	2	1	no	—
155	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
156	Q₈	—	2	1	no	—
157	Q₈	—	118	4	no	—
158	Q₈	—	118	4	no	—
159	Q₈	—	236	4	no	—
160	Q₈	C₂×D₂₃₆	1	2	yes	—
161	D₄	—	2	4	no	—
162	D₄	—	2	1	no	—
163	D₄	C₂×D₂₃₆	1	2	yes	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	118	4	no	—
170	C₂×C₄	—	236	4	no	—
171	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
172	C₂×C₄	#11745685	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
175	C₂×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
176	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
177	C₈	—	118	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	2	no	—
181	C₂²	#14273701	1	2	yes	—
182	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	236	4	no	—
187	C₄	#14273701	1	2	yes	—
188	C₄	#14273757	1	4	yes	complemented
189	C₄	C₂²×D₂₃₆	1	1	yes	characteristic
190	C₄	C₂×#11745685	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	4	2	no	—
193	C₂	C₂×#14273701	1	1	yes	characteristic
194	C₂	C₂×#14273757	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273757

C₂

yes

maximal

C₂×#14273727

C₂

yes

maximal, characteristic, complemented

#16837612

C₂

yes

maximal, complemented

#16837607

C₂

yes

maximal, complemented

#16837426

C₂

yes

maximal, characteristic, complemented

#16837410

C₂

yes

maximal, complemented

C₄×#41×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

#14273757

—

C₂×#11745672

C₂²

yes

—

C₂×#11745670

C₂²

yes

—

C₂×#11745664

C₂²

yes

—

#14273727

—

#14273727

—

#14273727

C₂²

yes

—

C₂×C₄⋊Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273650

C₂²

yes

—

C₂×#41×C₅₉

C₂²

yes

characteristic

C₄×Q₈×C₅₉

C₂²

yes

—

C₄×D₄×C₅₉

—

C₄×D₄×C₅₉

C₂²

yes

—

#117×C₅₉

—

#117×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

#41×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

C₂³

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂³

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

—

#11745672

—

#11745670

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

C₂×D₄

yes

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

C₂×D₄

yes

—

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×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#121

yes

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#121

yes

—

C₄×C₅₉

#137

yes

complemented

C₄×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

C₂×#41

yes

characteristic

Dic₅₉

—

#3039

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

C₂×#121

yes

characteristic

C₂×C₅₉

C₂×#137

yes

—

C₂×#137

—

C₂×C₄⋊Q₈

—

100

C₄×#41

D₅₉

yes

characteristic

101

#462

—

102

#461

—

103

#423

—

104

#419

—

105

C₅₉

#3039

yes

characteristic, complemented, Sylow

106

C₂×#41

C₂×D₅₉

yes

characteristic

107

C₂²×Q₈

—

108

#137

—

118

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×M₁₆

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

118

—

114

C₄⋊Q₈

—

118

—

115

C₄×Q₈

C₂×D₅₉

yes

—

116

C₄×D₄

—

117

C₄×D₄

C₂×D₅₉

yes

—

118

#117

—

119

#117

C₂×D₅₉

yes

characteristic

120

C₂×C₄⋊C₄

—

118

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂×D₅₉

yes

characteristic

123

C₄⋊C₈

—

124

#103

—

125

#41

—

126

#41

D₂₃₆

yes

—

127

C₂×Q₈

—

128

C₂×Q₈

—

118

—

129

C₂×Q₈

—

118

—

130

C₂×Q₈

—

118

—

131

C₂×Q₈

C₂²×D₅₉

yes

characteristic

132

C₂×D₄

—

133

C₂×D₄

C₂²×D₅₉

yes

characteristic

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

118

—

137

C₂²×C₄

C₂²⋊D₅₉

yes

—

138

C₂²×C₄

C₂²×D₅₉

yes

characteristic

139

Q₁₆

—

118

—

140

SD₁₆

—

118

—

141

M₁₆

—

118

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

118

—

145

C₄⋊C₄

—

118

—

146

C₄⋊C₄

C₂²×D₅₉

yes

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂²⋊D₅₉

yes

characteristic

152

C₄²

C₂²⋊D₅₉

yes

characteristic

153

C₄²

C₂²×D₅₉

yes

—

154

C₂³

—

155

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

156

Q₈

—

157

Q₈

—

118

—

158

Q₈

—

118

—

159

Q₈

—

236

—

160

Q₈

C₂×D₂₃₆

yes

—

161

D₄

—

162

D₄

—

163

D₄

C₂×D₂₃₆

yes

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

118

—

170

C₂×C₄

—

236

—

171

C₂×C₄

C₂×D₂₃₆

yes

—

172

C₂×C₄

#11745685

yes

—

173

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

174

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

175

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

176

C₂×C₄

C₂³×D₅₉

yes

characteristic

177

C₈

—

118

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

#14273701

yes

—

182

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

236

—

187

C₄

#14273701

yes

—

188

C₄

#14273757

yes

complemented

189

C₄

C₂²×D₂₃₆

yes

characteristic

190

C₄

C₂×#11745685

yes

characteristic

191

C₂

—

192

C₂

—

193

C₂

C₂×#14273701

yes

characteristic

194

C₂

C₂×#14273757

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₄×#41×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₄×#41	D₅₉
hypocenter	C₅₉	#3039
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₄×#41×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

C₄×#41

D₅₉

hypocenter

C₅₉

#3039

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

N	H ≅ G/N	type
2. C₂×#14273727	C₂	semidirect
3. #16837612	C₂	semidirect
4. #16837607	C₂	semidirect
5. #16837426	C₂	semidirect
6. #16837410	C₂	semidirect
89. C₄×C₅₉	#137	semidirect
105. C₅₉	#3039	semidirect
188. C₄	#14273757	semidirect

H ≅ G/N

type

2. C₂×#14273727

C₂

semidirect

3. #16837612

C₂

semidirect

4. #16837607

C₂

semidirect

5. #16837426

C₂

semidirect

6. #16837410

C₂

semidirect

89. C₄×C₅₉

#137

semidirect

105. C₅₉

#3039

semidirect

188. C₄

#14273757

semidirect

Subgroups — #21433583

Conjugacy classes of subgroups

Complements