Group #28493741 — subgroups

Subgroups — #28493741

Order 15104.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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	#21425058	C₂	1	2	yes	maximal
2	#21424256	C₂	1	1	yes	maximal, characteristic
3	#21424255	C₂	1	2	yes	maximal
4	C₅₉⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	2	no	—
7	C₂×#14273660	—	2	4	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
9	#16837272	C₂²	1	2	yes	—
10	#16837164	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×#103×C₅₉	—	2	2	no	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#384	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉⋊C₈	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	#14273661	—	4	4	no	—
20	#14273660	—	4	8	no	—
21	C₄×C₅₉⋊C₈	—	2	2	no	—
22	C₄×C₅₉⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₅₉	—	2	1	no	—
24	C₂²×C₈×C₅₉	D₄	1	2	yes	—
25	C₄⋊Q₈×C₅₉	—	2	2	no	—
26	C₄⋊Q₈×C₅₉	—	2	1	no	—
27	C₄⋊Q₈×C₅₉	Q₈	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	1	no	—
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	#103×C₅₉	—	4	4	no	—
33	C₄×C₈×C₅₉	—	2	2	no	—
34	C₂×Q₈×C₅₉	—	2	2	no	—
35	C₂×Q₈×C₅₉	—	4	1	no	—
36	C₂×Q₈×C₅₉	—	4	2	no	—
37	C₂×Q₈×C₅₉	—	4	2	no	—
38	C₂²×C₄×C₅₉	—	4	1	no	—
39	C₂²×C₄×C₅₉	—	4	1	no	—
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
42	C₂²×Dic₅₉	—	4	2	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	—	2	4	no	—
47	C₄⋊C₄×C₅₉	—	4	2	no	—
48	C₄⋊C₄×C₅₉	—	4	2	no	—
49	C₄⋊C₄×C₅₉	—	4	4	no	—
50	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₅₉	#41	1	1	yes	characteristic
52	C₄²×C₅₉	#41	1	2	yes	—
53	C₄⋊Dic₅₉	—	4	4	no	—
54	C₄⋊Dic₅₉	—	4	4	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	Q₈×C₅₉	—	4	4	no	—
61	Q₈×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	—	8	1	no	—
64	C₂×C₄×C₅₉	—	8	1	no	—
65	C₂×C₄×C₅₉	—	8	1	no	—
66	C₂×C₄×C₅₉	—	8	2	no	—
67	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#123	1	2	yes	—
71	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
73	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
74	C₂×C₄×C₅₉	#137	1	2	yes	—
75	C₂×Dic₅₉	—	8	2	no	—
76	C₂×Dic₅₉	—	8	2	no	—
77	C₂×Dic₅₉	—	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	—	8	2	no	—
87	C₄×C₅₉	—	16	1	no	—
88	C₄×C₅₉	#474	1	2	yes	—
89	C₄×C₅₉	#477	1	4	yes	—
90	C₄×C₅₉	#480	1	2	yes	—
91	C₄×C₅₉	#484	1	4	yes	—
92	Dic₅₉	—	16	2	no	—
93	#1990	—	59	2	no	—
94	#1956	D₅₉	1	1	yes	characteristic
95	#1845	—	59	1	no	—
96	#1845	—	59	2	no	—
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	C₂×D₅₉	1	1	yes	characteristic
104	C₂×#107	—	118	4	no	—
105	C₂×#107	—	118	2	no	—
106	C₂×#103	—	2	2	no	—
107	C₂×C₄×C₈	—	59	2	no	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#384	—	59	2	no	—
110	#384	C₂×D₅₉	1	1	yes	characteristic
111	#340	—	118	4	no	—
112	C₅₉	#10674	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	2	1	no	—
114	C₂²×C₈	—	59	4	no	—
115	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	Dic₁₁₈	1	2	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	118	2	no	—
123	C₂×C₄⋊C₄	—	118	2	no	—
124	C₂×C₄⋊C₄	—	118	2	no	—
125	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
126	#107	—	236	8	no	—
127	#107	—	236	4	no	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	118	2	no	—
131	C₄×C₈	—	118	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	236	2	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	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	236	4	no	—
152	C₄⋊C₄	—	236	4	no	—
153	C₄⋊C₄	—	236	4	no	—
154	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
155	C₄²	#11745685	1	1	yes	characteristic
156	C₄²	#11745687	1	2	yes	—
157	C₂³	#14273723	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	472	2	no	—
166	C₂×C₄	—	472	2	no	—
167	C₂×C₄	—	472	2	no	—
168	C₂×C₄	#14273704	1	2	yes	—
169	C₂×C₄	#14273706	1	1	yes	characteristic
170	C₂×C₄	#14273708	1	1	yes	characteristic
171	C₂×C₄	#14273709	1	2	yes	—
172	C₂×C₄	C₂×#11745669	1	2	yes	—
173	C₂×C₄	Q₁₆×D₅₉	1	4	yes	—
174	C₂×C₄	#14273754	1	4	yes	—
175	C₂×C₄	#14273757	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	118	8	no	—
178	C₂²	#16837338	1	1	yes	characteristic
179	C₂²	#16837778	1	1	yes	characteristic
180	C₂²	#16837780	1	2	yes	—
181	C₂²	#16837788	1	2	yes	—
182	C₂²	#16837797	1	1	yes	characteristic
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	944	2	no	—
186	C₄	#16837682	1	2	yes	—
187	C₄	#16837695	1	4	yes	—
188	C₄	#16837713	1	2	yes	—
189	C₄	#16837736	1	4	yes	—
190	C₂	#21423712	1	1	yes	characteristic
191	C₂	#21423760	1	1	yes	characteristic
192	C₂	#21423776	1	2	yes	—
193	C₂	#21425624	1	1	yes	characteristic
194	C₂	#21425632	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425058

C₂

yes

maximal

#21424256

C₂

yes

maximal, characteristic

#21424255

C₂

yes

maximal

C₅₉⋊#1273

C₂

yes

maximal, characteristic

C₅₉×#1956

C₂

yes

maximal, characteristic

C₂×#14273661

—

C₂×#14273660

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837272

C₂²

yes

—

#16837164

—

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₂×#103×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#384

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273660

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

Q₈

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#103×C₅₉

—

C₄×C₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×Q₈

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₁₆

yes

—

C₂³×C₅₉

#127

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

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

—

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

—

C₄×C₅₉

#474

yes

—

C₄×C₅₉

#477

yes

—

C₄×C₅₉

#480

yes

—

C₄×C₅₉

#484

yes

—

Dic₅₉

—

#1990

—

#1956

D₅₉

yes

—

—

—

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

C₂×D₅₉

yes

characteristic

104

C₂×#107

—

118

—

105

C₂×#107

—

118

—

106

C₂×#103

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

109

#384

—

110

#384

C₂×D₅₉

yes

characteristic

111

#340

—

118

—

112

C₅₉

#10674

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₅₉

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

Dic₁₁₈

yes

—

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

—

236

—

127

#107

—

236

—

128

#103

—

129

C₄×C₈

—

130

C₄×C₈

—

118

—

131

C₄×C₈

—

118

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

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

—

151

C₄⋊C₄

—

236

—

152

C₄⋊C₄

—

236

—

153

C₄⋊C₄

—

236

—

154

C₄²

C₂×Dic₁₁₈

yes

characteristic

155

C₄²

#11745685

yes

characteristic

156

C₄²

#11745687

yes

—

157

C₂³

#14273723

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

472

—

166

C₂×C₄

—

472

—

167

C₂×C₄

—

472

—

168

C₂×C₄

#14273704

yes

—

169

C₂×C₄

#14273706

yes

characteristic

170

C₂×C₄

#14273708

yes

characteristic

171

C₂×C₄

#14273709

yes

—

172

C₂×C₄

C₂×#11745669

yes

—

173

C₂×C₄

Q₁₆×D₅₉

yes

—

174

C₂×C₄

#14273754

yes

—

175

C₂×C₄

#14273757

yes

—

176

C₈

—

177

C₈

—

118

—

178

C₂²

#16837338

yes

characteristic

179

C₂²

#16837778

yes

characteristic

180

C₂²

#16837780

yes

—

181

C₂²

#16837788

yes

—

182

C₂²

#16837797

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

944

—

186

C₄

#16837682

yes

—

187

C₄

#16837695

yes

—

188

C₄

#16837713

yes

—

189

C₄

#16837736

yes

—

190

C₂

#21423712

yes

characteristic

191

C₂

#21423760

yes

characteristic

192

C₂

#21423776

yes

—

193

C₂

#21425624

yes

characteristic

194

C₂

#21425632

yes

—

	subgroup	G/H
center	C₂³	#14273723
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1956	C₂
socle, prime radical	C₂³×C₅₉	#127
hypercenter	#1956	D₅₉
hypocenter	C₅₉	#10674
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#14273723

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1956

C₂

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

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
#21425058	2
#21424256	2
#21424255	2
C₅₉⋊#1273	2
C₅₉×#1956	2
#10674	59

subgroup

index

#21425058

#21424256

#21424255

C₅₉⋊#1273

C₅₉×#1956

#10674

Subgroups — #28493741

Conjugacy classes of subgroups

Complements

Maximal subgroups