Group #28493740 — subgroups

Subgroups — #28493740

Order 15104.

442 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 153 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	#21425056	C₂	1	2	yes	maximal
2	#21424898	C₂	1	1	yes	maximal, characteristic
3	#21424255	C₂	1	2	yes	maximal
4	C₅₉⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	C₂²	1	1	yes	characteristic
7	C₂×#14273660	—	2	4	no	—
8	C₂×#14273652	—	2	2	no	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
10	#16837272	C₂²	1	1	yes	characteristic
11	#16837164	—	2	4	no	—
12	C₂×#107×C₅₉	—	2	2	no	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#384	C₂²	1	2	yes	—
15	C₂²×Dic₁₁₈	—	2	1	no	—
16	C₂²×C₅₉⋊C₈	D₄	1	4	yes	—
17	#14273727	—	2	2	no	—
18	#14273727	—	2	1	no	—
19	#14273727	Q₈	1	2	yes	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	1	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	#14273660	—	4	8	no	—
24	#14273652	—	4	4	no	—
25	C₄×C₅₉⋊C₈	—	2	2	no	—
26	C₄×C₅₉⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₅₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	#107×C₅₉	—	4	4	no	—
33	C₄×C₈×C₅₉	—	2	2	no	—
34	C₂²×C₄×C₅₉	—	4	2	no	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
37	C₂²×Dic₅₉	—	4	1	no	—
38	C₂²×Dic₅₉	—	4	1	no	—
39	C₂×Dic₁₁₈	—	2	2	no	—
40	C₂×Dic₁₁₈	—	4	2	no	—
41	C₂×Dic₁₁₈	—	4	2	no	—
42	C₂×Dic₁₁₈	—	4	1	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	D₈	1	4	yes	—
46	C₄⋊C₄×C₅₉	—	4	4	no	—
47	C₄⋊C₄×C₅₉	—	4	4	no	—
48	C₄⋊C₄×C₅₉	—	4	4	no	—
49	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅₉	#41	1	2	yes	—
51	C₄²×C₅₉	#41	1	1	yes	characteristic
52	C₄⋊Dic₅₉	—	2	4	no	—
53	C₄⋊Dic₅₉	—	4	2	no	—
54	C₄⋊Dic₅₉	—	4	2	no	—
55	C₄⋊Dic₅₉	—	4	4	no	—
56	C₂×C₅₉⋊C₈	—	2	4	no	—
57	C₂×C₅₉⋊C₈	—	2	4	no	—
58	C₂×C₅₉⋊C₈	Q₁₆	1	8	yes	—
59	C₂³×C₅₉	#127	1	1	yes	characteristic
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	—	8	2	no	—
63	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#123	1	2	yes	—
67	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
69	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
70	C₂×C₄×C₅₉	#137	1	2	yes	—
71	C₂×Dic₅₉	—	4	2	no	—
72	C₂×Dic₅₉	—	8	1	no	—
73	C₂×Dic₅₉	—	8	1	no	—
74	C₂×Dic₅₉	—	8	1	no	—
75	C₂×Dic₅₉	—	8	2	no	—
76	Dic₁₁₈	—	4	4	no	—
77	Dic₁₁₈	—	8	2	no	—
78	C₈×C₅₉	—	2	4	no	—
79	C₅₉⋊C₈	—	2	8	no	—
80	#10674	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	#399	1	1	yes	characteristic
82	C₂²×C₅₉	#493	1	1	yes	characteristic
83	C₂²×C₅₉	#494	1	2	yes	—
84	C₂²×C₅₉	#496	1	2	yes	—
85	C₂²×C₅₉	#497	1	1	yes	characteristic
86	C₄×C₅₉	—	16	2	no	—
87	C₄×C₅₉	#474	1	2	yes	—
88	C₄×C₅₉	#477	1	4	yes	—
89	C₄×C₅₉	#480	1	2	yes	—
90	C₄×C₅₉	#484	1	4	yes	—
91	Dic₅₉	—	8	2	no	—
92	Dic₅₉	—	16	1	no	—
93	#1990	—	59	2	no	—
94	#1956	—	59	1	no	—
95	#1845	—	59	2	no	—
96	#1845	D₅₉	1	1	yes	characteristic
97	#1273	—	59	1	no	—
98	C₂×C₅₉	#1713	1	1	yes	characteristic
99	C₂×C₅₉	#1721	1	1	yes	characteristic
100	C₂×C₅₉	#1723	1	2	yes	—
101	C₂×C₅₉	#2093	1	1	yes	characteristic
102	C₂×C₅₉	#2095	1	2	yes	—
103	C₂×C₄⋊Q₈	—	59	1	no	—
104	C₂×#107	—	2	2	no	—
105	C₂×#107	—	118	4	no	—
106	C₂×#103	—	118	2	no	—
107	C₂×C₄×C₈	—	59	2	no	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#384	—	59	1	no	—
110	#384	C₂×D₅₉	1	2	yes	—
111	#340	—	118	4	no	—
112	C₅₉	#10674	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	118	1	no	—
114	C₂²×C₈	—	59	4	no	—
115	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
116	C₄⋊Q₈	—	59	2	no	—
117	C₄⋊Q₈	—	118	2	no	—
118	C₄⋊Q₈	—	118	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	118	2	no	—
123	C₂×C₄⋊C₄	—	118	2	no	—
124	C₂×C₄⋊C₄	—	118	1	no	—
125	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
126	#107	—	4	4	no	—
127	#107	—	236	8	no	—
128	#103	—	236	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	118	2	no	—
131	C₄×C₈	—	118	2	no	—
132	C₂×Q₈	—	118	2	no	—
133	C₂×Q₈	—	236	2	no	—
134	C₂×Q₈	—	236	2	no	—
135	C₂×Q₈	—	236	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	236	1	no	—
138	C₂²×C₄	—	236	1	no	—
139	C₂²×C₄	D₄×D₅₉	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	59	8	no	—
144	C₂×C₈	—	118	4	no	—
145	C₂×C₈	—	118	4	no	—
146	C₂×C₈	#11745669	1	4	yes	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	118	4	no	—
151	C₄⋊C₄	—	236	2	no	—
152	C₄⋊C₄	—	236	4	no	—
153	C₄⋊C₄	—	236	2	no	—
154	C₄²	#11745685	1	2	yes	—
155	C₄²	Q₈×D₅₉	1	1	yes	characteristic
156	C₄²	#11745689	1	1	yes	characteristic
157	C₂³	#14273723	1	1	yes	characteristic
158	Q₈	—	472	2	no	—
159	Q₈	—	236	4	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	472	1	no	—
164	C₂×C₄	—	472	2	no	—
165	C₂×C₄	—	472	1	no	—
166	C₂×C₄	—	472	1	no	—
167	C₂×C₄	—	236	2	no	—
168	C₂×C₄	#14273703	1	1	yes	characteristic
169	C₂×C₄	#14273705	1	2	yes	—
170	C₂×C₄	#14273710	1	1	yes	characteristic
171	C₂×C₄	#14273712	1	2	yes	—
172	C₂×C₄	C₂×#11745669	1	2	yes	—
173	C₂×C₄	Q₁₆×D₅₉	1	4	yes	—
174	C₂×C₄	#14273751	1	4	yes	—
175	C₂×C₄	#14273758	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	118	8	no	—
178	C₂²	#16837337	1	1	yes	characteristic
179	C₂²	#16837778	1	1	yes	characteristic
180	C₂²	#16837780	1	2	yes	—
181	C₂²	#16837790	1	2	yes	—
182	C₂²	#16837794	1	1	yes	characteristic
183	C₄	—	16	2	no	—
184	C₄	—	944	1	no	—
185	C₄	—	472	2	no	—
186	C₄	#16837679	1	2	yes	—
187	C₄	#16837696	1	4	yes	—
188	C₄	#16837715	1	2	yes	—
189	C₄	#16837739	1	4	yes	—
190	C₂	#21423712	1	1	yes	characteristic
191	C₂	#21423763	1	1	yes	characteristic
192	C₂	#21423772	1	2	yes	—
193	C₂	#21425625	1	1	yes	characteristic
194	C₂	#21425631	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425056

C₂

yes

maximal

#21424898

C₂

yes

maximal, characteristic

#21424255

C₂

yes

maximal

C₅₉⋊#1273

C₂

yes

maximal, characteristic

C₅₉×#1845

C₂

yes

maximal, characteristic

C₂×#14273727

C₂²

yes

characteristic

C₂×#14273660

—

C₂×#14273652

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837272

C₂²

yes

characteristic

#16837164

—

C₂×#107×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#384

C₂²

yes

—

C₂²×Dic₁₁₈

—

C₂²×C₅₉⋊C₈

D₄

yes

—

#14273727

—

#14273727

—

#14273727

Q₈

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273660

—

#14273652

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#107×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄²×C₅₉

#41

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₁₆

yes

—

C₂³×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#123

yes

characteristic

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#496

yes

—

C₂²×C₅₉

#497

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#474

yes

—

C₄×C₅₉

#477

yes

—

C₄×C₅₉

#480

yes

—

C₄×C₅₉

#484

yes

—

Dic₅₉

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

#1273

—

C₂×C₅₉

yes

C₂×C₅₉

yes

100

C₂×C₅₉

#1723

yes

—

101

C₂×C₅₉

#2093

yes

characteristic

102

C₂×C₅₉

#2095

yes

—

103

C₂×C₄⋊Q₈

—

104

C₂×#107

—

105

C₂×#107

—

118

—

106

C₂×#103

—

118

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

109

#384

—

110

#384

C₂×D₅₉

yes

—

111

#340

—

118

—

112

C₅₉

#10674

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

118

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₅₉

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

—

118

C₄⋊Q₈

—

118

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

118

—

123

C₂×C₄⋊C₄

—

118

—

124

C₂×C₄⋊C₄

—

118

—

125

C₂×C₄²

C₂²×D₅₉

yes

characteristic

126

#107

—

127

#107

—

236

—

128

#103

—

236

—

129

C₄×C₈

—

130

C₄×C₈

—

118

—

131

C₄×C₈

—

118

—

132

C₂×Q₈

—

118

—

133

C₂×Q₈

—

236

—

134

C₂×Q₈

—

236

—

135

C₂×Q₈

—

236

—

136

C₂²×C₄

—

137

C₂²×C₄

—

236

—

138

C₂²×C₄

—

236

—

139

C₂²×C₄

D₄×D₅₉

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

118

—

145

C₂×C₈

—

118

—

146

C₂×C₈

#11745669

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

118

—

151

C₄⋊C₄

—

236

—

152

C₄⋊C₄

—

236

—

153

C₄⋊C₄

—

236

—

154

C₄²

#11745685

yes

—

155

C₄²

Q₈×D₅₉

yes

characteristic

156

C₄²

#11745689

yes

characteristic

157

C₂³

#14273723

yes

characteristic

158

Q₈

—

472

—

159

Q₈

—

236

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

472

—

164

C₂×C₄

—

472

—

165

C₂×C₄

—

472

—

166

C₂×C₄

—

472

—

167

C₂×C₄

—

236

—

168

C₂×C₄

#14273703

yes

characteristic

169

C₂×C₄

#14273705

yes

—

170

C₂×C₄

#14273710

yes

characteristic

171

C₂×C₄

#14273712

yes

—

172

C₂×C₄

C₂×#11745669

yes

—

173

C₂×C₄

Q₁₆×D₅₉

yes

—

174

C₂×C₄

#14273751

yes

—

175

C₂×C₄

#14273758

yes

—

176

C₈

—

177

C₈

—

118

—

178

C₂²

#16837337

yes

characteristic

179

C₂²

#16837778

yes

characteristic

180

C₂²

#16837780

yes

—

181

C₂²

#16837790

yes

—

182

C₂²

#16837794

yes

characteristic

183

C₄

—

184

C₄

—

944

—

185

C₄

—

472

—

186

C₄

#16837679

yes

—

187

C₄

#16837696

yes

—

188

C₄

#16837715

yes

—

189

C₄

#16837739

yes

—

190

C₂

#21423712

yes

characteristic

191

C₂

#21423763

yes

characteristic

192

C₂

#21423772

yes

—

193

C₂

#21425625

yes

characteristic

194

C₂

#21425631

yes

—

	subgroup	G/H
center	C₂³	#14273723
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1845	C₂
socle, prime radical	C₂³×C₅₉	#127
hypercenter	#1845	D₅₉
hypocenter	C₅₉	#10674
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14273723

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1845

C₂

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
112. C₅₉	#10674	semidirect

H ≅ G/N

type

112. C₅₉

#10674

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21425056	2
#21424898	2
#21424255	2
C₅₉⋊#1273	2
C₅₉×#1845	2
#10674	59

subgroup

index

#21425056

#21424898

#21424255

C₅₉⋊#1273

C₅₉×#1845

#10674

Subgroups — #28493740

Conjugacy classes of subgroups

Complements

Maximal subgroups