Group #28503186 — subgroups

Subgroups — #28503186

Order 15104.

204 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 63 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	#21426325	C₂	1	1	yes	maximal, characteristic
2	#21425998	C₂	1	1	yes	maximal, characteristic
3	#21423487	C₂	1	1	yes	maximal, characteristic
4	#21423467	C₂	1	1	yes	maximal, characteristic
5	#21422169	C₂	1	1	yes	maximal, characteristic
6	#21422146	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1364	C₂	1	1	yes	maximal, characteristic
8	#16837782	C₂²	1	1	yes	characteristic
9	#16837688	—	2	1	no	—
10	#16837609	—	2	1	no	—
11	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
12	Dic₅₉⋊C₁₆	—	2	1	no	—
13	#16837205	—	2	1	no	—
14	#16837205	—	2	1	no	—
15	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₆×Dic₅₉	—	2	1	no	—
17	#16837155	C₂²	1	1	yes	characteristic
18	#16837148	—	2	1	no	—
19	#16837147	—	2	1	no	—
20	C₅₉×#500	C₂²	1	1	yes	characteristic
21	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₅₉×#335	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₃₆	—	2	1	no	—
24	C₂×Dic₂₃₆	—	2	1	no	—
25	C₂×#11745672	—	4	1	no	—
26	#14273727	—	2	1	no	—
27	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
28	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₅₉⋊C₁₆	D₄	1	1	yes	characteristic
30	#14273659	—	4	1	no	—
31	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
32	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
33	Dic₅₉⋊C₈	—	2	1	no	—
34	#14273652	—	4	1	no	—
35	#14273651	—	4	1	no	—
36	C₈×Dic₅₉	—	2	1	no	—
37	C₄⋊Q₈×C₅₉	—	2	1	no	—
38	C₂×C₁₆×C₅₉	—	2	1	no	—
39	#107×C₅₉	—	2	1	no	—
40	#107×C₅₉	—	2	1	no	—
41	C₄⋊C₈×C₅₉	—	2	1	no	—
42	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₅₉	—	4	1	no	—
44	C₂×Dic₁₁₈	—	4	1	no	—
45	C₂×Dic₁₁₈	D₈	1	1	yes	characteristic
46	C₂×C₈×C₅₉	—	4	1	no	—
47	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₅₉	—	4	1	no	—
50	C₄⋊C₄×C₅₉	—	4	1	no	—
51	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
52	#11745672	—	8	2	no	—
53	C₄⋊Dic₅₉	—	4	1	no	—
54	C₄⋊Dic₅₉	—	4	1	no	—
55	C₄⋊Dic₅₉	D₈	1	1	yes	characteristic
56	Dic₅₉⋊C₄	—	2	1	no	—
57	C₄×Dic₅₉	—	2	1	no	—
58	C₂×C₅₉⋊C₈	—	2	1	no	—
59	C₂×C₅₉⋊C₈	—	4	1	no	—
60	Dic₂₃₆	—	4	1	no	—
61	Dic₂₃₆	—	4	1	no	—
62	Dic₂₃₆	—	8	1	no	—
63	C₁₆×C₅₉	—	4	1	no	—
64	C₅₉⋊C₁₆	—	2	1	no	—
65	C₅₉⋊C₁₆	—	2	1	no	—
66	Q₈×C₅₉	—	8	1	no	—
67	Q₈×C₅₉	—	8	1	no	—
68	C₂×C₄×C₅₉	—	8	1	no	—
69	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
72	C₂×Dic₅₉	—	2	1	no	—
73	C₂×Dic₅₉	—	2	1	no	—
74	C₂×Dic₅₉	—	8	1	no	—
75	Dic₁₁₈	—	2	1	no	—
76	Dic₁₁₈	—	8	1	no	—
77	Dic₁₁₈	—	8	1	no	—
78	Dic₁₁₈	SD₃₂	1	2	yes	—
79	C₈×C₅₉	—	2	1	no	—
80	C₈×C₅₉	—	8	1	no	—
81	C₈×C₅₉	#135	1	1	yes	characteristic
82	C₈×C₅₉	#137	1	1	yes	characteristic
83	C₅₉⋊C₈	—	4	1	no	—
84	C₅₉⋊C₈	—	8	1	no	—
85	#12478	—	59	1	no	maximal, Sylow
86	C₂²×C₅₉	#466	1	1	yes	characteristic
87	C₄×C₅₉	—	2	1	no	—
88	C₄×C₅₉	—	16	1	no	—
89	C₄×C₅₉	#464	1	1	yes	characteristic
90	C₄×C₅₉	#471	1	1	yes	characteristic
91	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
92	C₄×C₅₉	#508	1	1	yes	characteristic
93	Dic₅₉	—	2	2	no	—
94	Dic₅₉	—	4	1	no	—
95	Dic₅₉	—	16	1	no	—
96	#2241	—	59	1	no	—
97	Q₈×C₁₆	—	59	1	no	—
98	#1674	—	59	1	no	—
99	#1670	—	59	1	no	—
100	#1372	—	59	1	no	—
101	#1364	D₅₉	1	1	yes	characteristic
102	#1362	—	59	1	no	—
103	C₂×C₅₉	#1682	1	1	yes	characteristic
104	C₂×C₅₉	#2217	1	1	yes	characteristic
105	C₂×C₅₉	#2222	1	1	yes	characteristic
106	#500	C₂×D₅₉	1	1	yes	characteristic
107	#494	—	59	1	no	—
108	#475	—	118	1	no	—
109	#462	—	118	1	no	—
110	C₈×Q₈	—	59	1	no	—
111	C₄⋊C₁₆	—	118	1	no	—
112	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
113	#358	—	118	1	no	—
114	#358	—	118	1	no	—
115	C₄×C₁₆	—	59	1	no	—
116	C₄×C₁₆	—	118	1	no	—
117	#335	—	59	1	no	—
118	#335	C₂×D₅₉	1	1	yes	characteristic
119	#332	—	118	1	no	—
120	#332	—	118	1	no	—
121	C₅₉	#12478	1	1	yes	characteristic, complemented, Sylow
122	C₂×Q₁₆	—	118	1	no	—
123	C₂×Q₁₆	—	118	1	no	—
124	C₂×Q₁₆	—	236	1	no	—
125	C₄⋊Q₈	—	2	1	no	—
126	C₄⋊Q₈	—	118	1	no	—
127	C₄×Q₈	—	59	1	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	59	1	no	—
130	C₂×C₁₆	—	59	1	no	—
131	#107	—	2	1	no	—
132	#107	—	2	1	no	—
133	#106	—	236	1	no	—
134	C₄⋊C₈	—	2	1	no	—
135	C₄⋊C₈	—	59	1	no	—
136	C₄⋊C₈	—	118	1	no	—
137	C₄⋊C₈	—	118	1	no	—
138	#103	—	236	1	no	—
139	#103	—	236	1	no	—
140	C₄×C₈	—	118	1	no	—
141	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	59	1	no	—
144	C₂×Q₈	—	236	1	no	—
145	Q₁₆	—	472	1	no	—
146	Q₁₆	—	472	2	no	—
147	Q₁₆	—	236	1	no	—
148	Q₁₆	—	236	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	118	1	no	—
151	C₂×C₈	—	236	1	no	—
152	C₂×C₈	#11745685	1	1	yes	characteristic
153	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	—	59	1	no	—
157	C₄⋊C₄	—	118	1	no	—
158	C₄⋊C₄	—	236	1	no	—
159	C₄⋊C₄	—	236	1	no	—
160	C₄²	—	118	1	no	—
161	C₄²	D₄×D₅₉	1	1	yes	characteristic
162	C₁₆	—	4	1	no	—
163	C₁₆	—	118	1	no	—
164	C₁₆	—	118	1	no	—
165	Q₈	—	8	1	no	—
166	Q₈	—	8	1	no	—
167	Q₈	—	59	2	no	—
168	Q₈	—	118	1	no	—
169	Q₈	—	472	1	no	—
170	Q₈	—	472	1	no	—
171	C₂×C₄	—	8	1	no	—
172	C₂×C₄	—	118	1	no	—
173	C₂×C₄	—	118	1	no	—
174	C₂×C₄	—	472	1	no	—
175	C₂×C₄	#14273700	1	1	yes	characteristic
176	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
177	C₂×C₄	#14273748	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	472	1	no	—
181	C₈	—	236	1	no	—
182	C₈	#14273748	1	1	yes	characteristic
183	C₈	#14273759	1	1	yes	characteristic
184	C₂²	#16837637	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	16	1	no	—
187	C₄	—	118	2	no	—
188	C₄	—	944	1	no	—
189	C₄	—	236	1	no	—
190	C₄	#16837627	1	1	yes	characteristic
191	C₄	#16837668	1	1	yes	characteristic
192	C₄	SD₃₂×D₅₉	1	1	yes	characteristic
193	C₄	#16837831	1	1	yes	characteristic
194	C₂	#21423543	1	1	yes	characteristic
195	C₂	#21426191	1	1	yes	characteristic
196	C₂	#21426220	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426325

C₂

yes

maximal, characteristic

#21425998

C₂

yes

maximal, characteristic

#21423487

C₂

yes

maximal, characteristic

#21423467

C₂

yes

maximal, characteristic

#21422169

C₂

yes

maximal, characteristic

#21422146

C₂

yes

maximal, characteristic

C₅₉×#1364

C₂

yes

maximal, characteristic

#16837782

C₂²

yes

characteristic

#16837688

—

#16837609

—

C₈×Dic₁₁₈

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

#16837205

—

#16837205

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₁₆×Dic₅₉

—

#16837155

C₂²

yes

characteristic

#16837148

—

#16837147

—

C₅₉×#500

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

C₂²

yes

characteristic

C₅₉×#335

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×#11745672

—

#14273727

—

C₄×Dic₁₁₈

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

D₄

yes

characteristic

#14273659

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

characteristic

Dic₅₉⋊C₈

—

#14273652

—

#14273651

—

C₈×Dic₅₉

—

C₄⋊Q₈×C₅₉

—

C₂×C₁₆×C₅₉

—

#107×C₅₉

—

#107×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

D₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₈

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

SD₃₂

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×SD₃₂

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2241

—

Q₈×C₁₆

—

#1674

—

#1670

—

100

#1372

—

101

#1364

D₅₉

yes

characteristic

102

#1362

—

103

C₂×C₅₉

#1682

yes

characteristic

104

C₂×C₅₉

#2217

yes

characteristic

105

C₂×C₅₉

#2222

yes

characteristic

106

#500

C₂×D₅₉

yes

characteristic

107

#494

—

108

#475

—

118

—

109

#462

—

118

—

110

C₈×Q₈

—

111

C₄⋊C₁₆

—

118

—

112

C₄⋊C₁₆

C₂×D₅₉

yes

characteristic

113

#358

—

118

—

114

#358

—

118

—

115

C₄×C₁₆

—

116

C₄×C₁₆

—

118

—

117

#335

—

118

#335

C₂×D₅₉

yes

characteristic

119

#332

—

118

—

120

#332

—

118

—

121

C₅₉

#12478

yes

characteristic, complemented, Sylow

122

C₂×Q₁₆

—

118

—

123

C₂×Q₁₆

—

118

—

124

C₂×Q₁₆

—

236

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

118

—

127

C₄×Q₈

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#107

—

132

#107

—

133

#106

—

236

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

118

—

137

C₄⋊C₈

—

118

—

138

#103

—

236

—

139

#103

—

236

—

140

C₄×C₈

—

118

—

141

C₄×C₈

C₂²×D₅₉

yes

characteristic

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

236

—

145

Q₁₆

—

472

—

146

Q₁₆

—

472

—

147

Q₁₆

—

236

—

148

Q₁₆

—

236

—

149

C₂×C₈

—

150

C₂×C₈

—

118

—

151

C₂×C₈

—

236

—

152

C₂×C₈

#11745685

yes

characteristic

153

C₂×C₈

D₄×D₅₉

yes

characteristic

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

118

—

158

C₄⋊C₄

—

236

—

159

C₄⋊C₄

—

236

—

160

C₄²

—

118

—

161

C₄²

D₄×D₅₉

yes

characteristic

162

C₁₆

—

163

C₁₆

—

118

—

164

C₁₆

—

118

—

165

Q₈

—

166

Q₈

—

167

Q₈

—

168

Q₈

—

118

—

169

Q₈

—

472

—

170

Q₈

—

472

—

171

C₂×C₄

—

172

C₂×C₄

—

118

—

173

C₂×C₄

—

118

—

174

C₂×C₄

—

472

—

175

C₂×C₄

#14273700

yes

characteristic

176

C₂×C₄

D₈×D₅₉

yes

characteristic

177

C₂×C₄

#14273748

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

472

—

181

C₈

—

236

—

182

C₈

#14273748

yes

characteristic

183

C₈

#14273759

yes

characteristic

184

C₂²

#16837637

yes

characteristic

185

C₄

—

186

C₄

—

187

C₄

—

118

—

188

C₄

—

944

—

189

C₄

—

236

—

190

C₄

#16837627

yes

characteristic

191

C₄

#16837668

yes

characteristic

192

C₄

SD₃₂×D₅₉

yes

characteristic

193

C₄

#16837831

yes

characteristic

194

C₂

#21423543

yes

characteristic

195

C₂

#21426191

yes

characteristic

196

C₂

#21426220

yes

characteristic

	subgroup	G/H
center	C₂²	#16837637
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1364	C₂
socle, prime radical	C₂²×C₅₉	#466
hypercenter	#1364	D₅₉
hypocenter	C₅₉	#12478
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1364

subgroup

G/H

center

C₂²

#16837637

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1364

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1364

N	H ≅ G/N	type
121. C₅₉	#12478	semidirect

H ≅ G/N

type

121. C₅₉

#12478

semidirect

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

subgroup	index
#21426325	2
#21425998	2
#21423487	2
#21423467	2
#21422169	2
#21422146	2
C₅₉×#1364	2
#12478	59

subgroup

index

C₅₉×#1364

Subgroups — #28503186

Conjugacy classes of subgroups

Complements

Maximal subgroups