Group #21425577 — subgroups

Subgroups — #21425577

Order 7552.

214 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14273660	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273658	C₂	1	1	yes	maximal, characteristic
3	#16837268	C₂	1	1	yes	maximal, characteristic
4	#16837164	C₂	1	1	yes	maximal, characteristic
5	#16837164	C₂	1	1	yes	maximal, characteristic
6	#16837157	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#389	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	#14273660	—	2	2	no	—
13	#14273658	—	2	2	no	—
14	C₅₉⋊#95	—	2	1	no	—
15	C₅₉⋊#95	—	2	1	no	—
16	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
20	#95×C₅₉	—	2	1	no	—
21	C₂²×C₄×C₅₉	—	2	1	no	—
22	C₂²×C₄×C₅₉	—	2	1	no	—
23	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₉	—	2	1	no	—
27	C₂²×Dic₅₉	—	2	1	no	—
28	C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	4	2	no	—
32	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
33	C₄²×C₅₉	—	2	1	no	—
34	C₄²×C₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	—	2	1	no	—
38	C₄⋊Dic₅₉	—	2	1	no	—
39	C₂×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₅₉⋊C₈	—	2	1	no	—
41	C₂×C₅₉⋊C₈	—	2	1	no	—
42	C₂×C₅₉⋊C₈	—	2	1	no	—
43	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
44	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
45	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	1	no	—
48	C₂×C₄×C₅₉	—	2	1	no	—
49	C₂×C₄×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	—	2	1	no	—
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	4	1	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	1	no	—
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×Q₈	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	—	4	1	no	—
65	C₂×Dic₅₉	—	4	1	no	—
66	C₂×Dic₅₉	—	4	1	no	—
67	C₂×Dic₅₉	—	4	1	no	—
68	C₂×Dic₅₉	—	4	1	no	—
69	C₂×Dic₅₉	—	4	1	no	—
70	C₅₉⋊C₈	—	2	2	no	—
71	C₅₉⋊C₈	—	2	2	no	—
72	C₂²×C₅₉	#121	1	1	yes	characteristic
73	C₂²×C₅₉	#123	1	1	yes	characteristic
74	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
75	C₂²×C₅₉	#135	1	1	yes	characteristic
76	C₂²×C₅₉	#135	1	1	yes	characteristic
77	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₅₉	#137	1	1	yes	characteristic
79	C₄×C₅₉	—	4	2	no	—
80	C₄×C₅₉	—	4	1	no	—
81	C₄×C₅₉	—	4	1	no	—
82	C₄×C₅₉	—	8	1	no	—
83	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
84	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
85	C₄×C₅₉	#125	1	1	yes	characteristic
86	C₄×C₅₉	#125	1	1	yes	characteristic
87	Dic₅₉	—	8	1	no	—
88	Dic₅₉	—	8	1	no	—
89	#2085	—	59	1	no	maximal, Sylow
90	C₂×C₅₉	#398	1	1	yes	characteristic
91	C₂×C₅₉	#463	1	1	yes	characteristic
92	C₂×C₅₉	#464	1	1	yes	characteristic
93	C₂×C₅₉	#482	1	1	yes	characteristic
94	C₂×C₅₉	#485	1	1	yes	characteristic
95	C₂×C₅₉	#499	1	1	yes	characteristic
96	C₂×C₅₉	#501	1	1	yes	characteristic
97	C₂×#107	—	59	1	no	—
98	C₂×#106	—	59	1	no	—
99	#389	D₅₉	1	1	yes	characteristic
100	#382	—	59	1	no	—
101	#340	—	59	1	no	—
102	#340	—	59	1	no	—
103	#336	—	59	1	no	—
104	C₅₉	#2085	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	59	1	no	—
106	C₂²×C₈	—	59	1	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	59	1	no	—
109	C₂×C₄⋊C₄	—	59	1	no	—
110	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
112	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
113	#107	—	118	2	no	—
114	#106	—	118	2	no	—
115	#95	—	2	1	no	—
116	#95	—	118	1	no	—
117	#95	—	118	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	118	1	no	—
121	C₂²×C₄	—	118	1	no	—
122	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
125	C₂×C₈	—	59	2	no	—
126	C₂×C₈	—	59	2	no	—
127	C₂×C₈	—	118	1	no	—
128	C₂×C₈	—	118	1	no	—
129	C₂×C₈	—	118	1	no	—
130	C₂×C₈	—	118	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	118	1	no	—
136	C₄⋊C₄	—	118	1	no	—
137	C₄⋊C₄	—	118	1	no	—
138	C₄⋊C₄	—	118	1	no	—
139	C₄⋊C₄	D₂₃₆	1	2	yes	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	236	1	no	—
156	C₂×C₄	—	236	1	no	—
157	C₂×C₄	—	236	1	no	—
158	C₂×C₄	—	236	1	no	—
159	C₂×C₄	—	236	1	no	—
160	C₂×C₄	—	236	1	no	—
161	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
162	C₂×C₄	#11745685	1	1	yes	characteristic
163	C₂×C₄	#11745687	1	1	yes	characteristic
164	C₂×C₄	#11745687	1	1	yes	characteristic
165	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
166	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
167	C₈	—	118	2	no	—
168	C₈	—	118	2	no	—
169	C₂²	#14273701	1	1	yes	characteristic
170	C₂²	#14273708	1	1	yes	characteristic
171	C₂²	#14273725	1	1	yes	characteristic
172	C₂²	#14273747	1	1	yes	characteristic
173	C₂²	#14273747	1	1	yes	characteristic
174	C₂²	#14273753	1	1	yes	characteristic
175	C₂²	#14273758	1	1	yes	characteristic
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	1	no	—
180	C₄	—	472	1	no	—
181	C₄	—	472	1	no	—
182	C₄	#14273707	1	1	yes	characteristic
183	C₄	#14273707	1	1	yes	characteristic
184	C₄	#14273717	1	1	yes	characteristic
185	C₄	#14273717	1	1	yes	characteristic
186	C₂	#16837332	1	1	yes	characteristic
187	C₂	#16837617	1	1	yes	characteristic
188	C₂	#16837624	1	1	yes	characteristic
189	C₂	#16837723	1	1	yes	characteristic
190	C₂	#16837740	1	1	yes	characteristic
191	C₂	#16837801	1	1	yes	characteristic
192	C₂	#16837808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273660

C₂

yes

maximal, characteristic

C₂×#14273658

C₂

yes

maximal, characteristic

#16837268

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal, characteristic

#16837157

C₂

yes

maximal, characteristic

C₅₉×#389

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

#14273660

—

#14273658

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×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×Q₈

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

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₄⋊Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

D₅₉

yes

characteristic

100

#382

—

101

#340

—

102

#340

—

103

#336

—

104

C₅₉

#2085

yes

characteristic, complemented, Sylow

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

112

C₂×C₄²

C₂×D₅₉

yes

characteristic

113

#107

—

118

—

114

#106

—

118

—

115

#95

—

116

#95

—

118

—

117

#95

—

118

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

118

—

121

C₂²×C₄

—

118

—

122

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

123

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

124

C₂²×C₄

C₂²×D₅₉

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

118

—

128

C₂×C₈

—

118

—

129

C₂×C₈

—

118

—

130

C₂×C₈

—

118

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

118

—

136

C₄⋊C₄

—

118

—

137

C₄⋊C₄

—

118

—

138

C₄⋊C₄

—

118

—

139

C₄⋊C₄

D₂₃₆

yes

—

140

C₄²

—

141

C₄²

—

142

C₂³

C₂×C₂²⋊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

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

236

—

156

C₂×C₄

—

236

—

157

C₂×C₄

—

236

—

158

C₂×C₄

—

236

—

159

C₂×C₄

—

236

—

160

C₂×C₄

—

236

—

161

C₂×C₄

C₂×D₂₃₆

yes

characteristic

162

C₂×C₄

#11745685

yes

characteristic

163

C₂×C₄

#11745687

yes

characteristic

164

C₂×C₄

#11745687

yes

characteristic

165

C₂×C₄

Q₈×D₅₉

yes

characteristic

166

C₂×C₄

Q₈×D₅₉

yes

characteristic

167

C₈

—

118

—

168

C₈

—

118

—

169

C₂²

#14273701

yes

characteristic

170

C₂²

#14273708

yes

characteristic

171

C₂²

#14273725

yes

characteristic

172

C₂²

#14273747

yes

characteristic

173

C₂²

#14273747

yes

characteristic

174

C₂²

#14273753

yes

characteristic

175

C₂²

#14273758

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

472

—

181

C₄

—

472

—

182

C₄

#14273707

yes

characteristic

183

C₄

#14273707

yes

characteristic

184

C₄

#14273717

yes

characteristic

185

C₄

#14273717

yes

characteristic

186

C₂

#16837332

yes

characteristic

187

C₂

#16837617

yes

characteristic

188

C₂

#16837624

yes

characteristic

189

C₂

#16837723

yes

characteristic

190

C₂

#16837740

yes

characteristic

191

C₂

#16837801

yes

characteristic

192

C₂

#16837808

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#389	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#389	D₅₉
hypocenter	C₅₉	#2085
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#389

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
104. C₅₉	#2085	semidirect

H ≅ G/N

type

104. C₅₉

#2085

semidirect

Subgroups — #21425577

Conjugacy classes of subgroups

Complements