Group #28493752 — subgroups

Subgroups — #28493752

Order 15104.

410 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 145 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	#21424223	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×#14273652	—	2	2	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
9	#16837272	C₂²	1	1	yes	characteristic
10	#16837164	—	2	4	no	—
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	#14273652	—	4	4	no	—
24	C₄×C₅₉⋊C₈	—	2	2	no	—
25	C₄×C₅₉⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₅₉	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	#107×C₅₉	—	4	4	no	—
32	C₄×C₈×C₅₉	—	2	2	no	—
33	C₂²×C₄×C₅₉	—	4	2	no	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
36	C₂²×Dic₅₉	—	4	1	no	—
37	C₂²×Dic₅₉	—	4	1	no	—
38	C₂×Dic₁₁₈	—	2	2	no	—
39	C₂×Dic₁₁₈	—	4	2	no	—
40	C₂×Dic₁₁₈	—	4	2	no	—
41	C₂×Dic₁₁₈	—	4	1	no	—
42	C₂×C₈×C₅₉	—	2	2	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	D₈	1	4	yes	—
45	C₄⋊C₄×C₅₉	—	4	4	no	—
46	C₄⋊C₄×C₅₉	—	4	4	no	—
47	C₄⋊C₄×C₅₉	—	4	4	no	—
48	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₉	#41	1	2	yes	—
50	C₄²×C₅₉	#41	1	1	yes	characteristic
51	C₄⋊Dic₅₉	—	2	4	no	—
52	C₄⋊Dic₅₉	—	4	2	no	—
53	C₄⋊Dic₅₉	—	4	2	no	—
54	C₄⋊Dic₅₉	—	4	4	no	—
55	C₂×C₅₉⋊C₈	—	2	4	no	—
56	C₂×C₅₉⋊C₈	—	2	4	no	—
57	C₂×C₅₉⋊C₈	—	2	4	no	—
58	C₂³×C₅₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	8	2	no	—
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#123	1	2	yes	—
66	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₅₉	#135	1	4	yes	—
68	C₂×C₄×C₅₉	#137	1	2	yes	—
69	C₂×C₄×C₅₉	#137	1	4	yes	—
70	C₂×Dic₅₉	—	4	2	no	—
71	C₂×Dic₅₉	—	8	1	no	—
72	C₂×Dic₅₉	—	8	1	no	—
73	C₂×Dic₅₉	—	8	1	no	—
74	C₂×Dic₅₉	—	8	2	no	—
75	Dic₁₁₈	—	4	4	no	—
76	Dic₁₁₈	—	8	2	no	—
77	C₈×C₅₉	—	2	4	no	—
78	C₅₉⋊C₈	—	4	4	no	—
79	#10676	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	#399	1	1	yes	characteristic
81	C₂²×C₅₉	#493	1	1	yes	characteristic
82	C₂²×C₅₉	#495	1	2	yes	—
83	C₂²×C₅₉	#497	1	1	yes	characteristic
84	C₂²×C₅₉	#497	1	2	yes	—
85	C₄×C₅₉	—	16	2	no	—
86	C₄×C₅₉	#474	1	2	yes	—
87	C₄×C₅₉	#479	1	4	yes	—
88	C₄×C₅₉	#480	1	2	yes	—
89	C₄×C₅₉	#485	1	4	yes	—
90	Dic₅₉	—	8	2	no	—
91	Dic₅₉	—	16	1	no	—
92	#1990	—	59	2	no	—
93	#1956	—	59	1	no	—
94	#1845	D₅₉	1	1	yes	characteristic
95	#1834	—	59	2	no	—
96	#1273	—	59	1	no	—
97	C₂×C₅₉	#1709	1	1	yes	characteristic
98	C₂×C₅₉	#1718	1	2	yes	—
99	C₂×C₅₉	#1722	1	1	yes	characteristic
100	C₂×C₅₉	#2093	1	1	yes	characteristic
101	C₂×C₅₉	#2097	1	2	yes	—
102	C₂×C₄⋊Q₈	—	59	1	no	—
103	C₂×#107	—	2	2	no	—
104	C₂×#103	—	118	2	no	—
105	C₂×C₄×C₈	—	59	2	no	—
106	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
107	#384	—	59	1	no	—
108	#384	C₂×D₅₉	1	2	yes	—
109	#340	—	118	4	no	—
110	#340	—	118	4	no	—
111	C₅₉	#10676	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	118	1	no	—
113	C₂²×C₈	—	59	4	no	—
114	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
115	C₄⋊Q₈	—	59	2	no	—
116	C₄⋊Q₈	—	118	2	no	—
117	C₄⋊Q₈	—	118	1	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	118	2	no	—
122	C₂×C₄⋊C₄	—	118	2	no	—
123	C₂×C₄⋊C₄	—	118	1	no	—
124	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
125	#107	—	4	4	no	—
126	#103	—	236	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	118	2	no	—
129	C₄×C₈	—	118	2	no	—
130	C₂×Q₈	—	118	2	no	—
131	C₂×Q₈	—	236	2	no	—
132	C₂×Q₈	—	236	2	no	—
133	C₂×Q₈	—	236	1	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	236	1	no	—
136	C₂²×C₄	—	236	1	no	—
137	C₂²×C₄	D₄×D₅₉	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	118	4	no	—
142	C₂×C₈	—	118	4	no	—
143	C₂×C₈	—	118	4	no	—
144	C₂×C₈	#11745669	1	4	yes	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	118	4	no	—
149	C₄⋊C₄	—	236	2	no	—
150	C₄⋊C₄	—	236	4	no	—
151	C₄⋊C₄	—	236	2	no	—
152	C₄²	#11745685	1	2	yes	—
153	C₄²	Q₈×D₅₉	1	1	yes	characteristic
154	C₄²	#11745689	1	1	yes	characteristic
155	C₂³	#14273723	1	1	yes	characteristic
156	Q₈	—	472	2	no	—
157	Q₈	—	236	4	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	472	1	no	—
162	C₂×C₄	—	472	2	no	—
163	C₂×C₄	—	472	1	no	—
164	C₂×C₄	—	472	1	no	—
165	C₂×C₄	—	236	2	no	—
166	C₂×C₄	#14273703	1	1	yes	characteristic
167	C₂×C₄	#14273705	1	2	yes	—
168	C₂×C₄	#14273710	1	1	yes	characteristic
169	C₂×C₄	#14273712	1	2	yes	—
170	C₂×C₄	C₂×#11745669	1	2	yes	—
171	C₂×C₄	#14273746	1	4	yes	—
172	C₂×C₄	#14273758	1	2	yes	—
173	C₂×C₄	#14273759	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	236	4	no	—
176	C₂²	#16837337	1	1	yes	characteristic
177	C₂²	#16837778	1	1	yes	characteristic
178	C₂²	#16837785	1	2	yes	—
179	C₂²	#16837794	1	1	yes	characteristic
180	C₂²	#16837795	1	2	yes	—
181	C₄	—	16	2	no	—
182	C₄	—	944	1	no	—
183	C₄	—	472	2	no	—
184	C₄	#16837679	1	2	yes	—
185	C₄	#16837708	1	4	yes	—
186	C₄	#16837715	1	2	yes	—
187	C₄	#16837745	1	4	yes	—
188	C₂	#21423701	1	1	yes	characteristic
189	C₂	#21423741	1	2	yes	—
190	C₂	#21423768	1	1	yes	characteristic
191	C₂	#21425625	1	1	yes	characteristic
192	C₂	#21425641	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425056

C₂

yes

maximal

#21424898

C₂

yes

maximal, characteristic

#21424223

C₂

yes

maximal

C₅₉⋊#1273

C₂

yes

maximal, characteristic

C₅₉×#1845

C₂

yes

maximal, characteristic

C₂×#14273727

C₂²

yes

characteristic

C₂×#14273652

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837272

C₂²

yes

characteristic

#16837164

—

#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

—

#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

—

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

#135

yes

—

C₂×C₄×C₅₉

#137

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

#497

yes

characteristic

C₂²×C₅₉

#497

yes

—

C₄×C₅₉

—

C₄×C₅₉

#474

yes

—

C₄×C₅₉

#479

yes

—

C₄×C₅₉

#480

yes

—

C₄×C₅₉

#485

yes

—

Dic₅₉

—

Dic₅₉

—

#1990

—

#1956

—

#1845

D₅₉

yes

characteristic

#1834

—

#1273

—

C₂×C₅₉

#1709

yes

characteristic

C₂×C₅₉

#1718

yes

—

C₂×C₅₉

#1722

yes

characteristic

100

C₂×C₅₉

#2093

yes

characteristic

101

C₂×C₅₉

#2097

yes

—

102

C₂×C₄⋊Q₈

—

103

C₂×#107

—

104

C₂×#103

—

118

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

107

#384

—

108

#384

C₂×D₅₉

yes

—

109

#340

—

118

—

110

#340

—

118

—

111

C₅₉

#10676

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

118

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₅₉

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

118

—

117

C₄⋊Q₈

—

118

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

118

—

122

C₂×C₄⋊C₄

—

118

—

123

C₂×C₄⋊C₄

—

118

—

124

C₂×C₄²

C₂²×D₅₉

yes

characteristic

125

#107

—

126

#103

—

236

—

127

C₄×C₈

—

128

C₄×C₈

—

118

—

129

C₄×C₈

—

118

—

130

C₂×Q₈

—

118

—

131

C₂×Q₈

—

236

—

132

C₂×Q₈

—

236

—

133

C₂×Q₈

—

236

—

134

C₂²×C₄

—

135

C₂²×C₄

—

236

—

136

C₂²×C₄

—

236

—

137

C₂²×C₄

D₄×D₅₉

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

118

—

142

C₂×C₈

—

118

—

143

C₂×C₈

—

118

—

144

C₂×C₈

#11745669

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

118

—

149

C₄⋊C₄

—

236

—

150

C₄⋊C₄

—

236

—

151

C₄⋊C₄

—

236

—

152

C₄²

#11745685

yes

—

153

C₄²

Q₈×D₅₉

yes

characteristic

154

C₄²

#11745689

yes

characteristic

155

C₂³

#14273723

yes

characteristic

156

Q₈

—

472

—

157

Q₈

—

236

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

472

—

162

C₂×C₄

—

472

—

163

C₂×C₄

—

472

—

164

C₂×C₄

—

472

—

165

C₂×C₄

—

236

—

166

C₂×C₄

#14273703

yes

characteristic

167

C₂×C₄

#14273705

yes

—

168

C₂×C₄

#14273710

yes

characteristic

169

C₂×C₄

#14273712

yes

—

170

C₂×C₄

C₂×#11745669

yes

—

171

C₂×C₄

#14273746

yes

—

172

C₂×C₄

#14273758

yes

—

173

C₂×C₄

#14273759

yes

—

174

C₈

—

175

C₈

—

236

—

176

C₂²

#16837337

yes

characteristic

177

C₂²

#16837778

yes

characteristic

178

C₂²

#16837785

yes

—

179

C₂²

#16837794

yes

characteristic

180

C₂²

#16837795

yes

—

181

C₄

—

182

C₄

—

944

—

183

C₄

—

472

—

184

C₄

#16837679

yes

—

185

C₄

#16837708

yes

—

186

C₄

#16837715

yes

—

187

C₄

#16837745

yes

—

188

C₂

#21423701

yes

characteristic

189

C₂

#21423741

yes

—

190

C₂

#21423768

yes

characteristic

191

C₂

#21425625

yes

characteristic

192

C₂

#21425641

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	#10676
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
111. C₅₉	#10676	semidirect

H ≅ G/N

type

111. C₅₉

#10676

semidirect

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

subgroup	index
#21425056	2
#21424898	2
#21424223	2
C₅₉⋊#1273	2
C₅₉×#1845	2
#10676	59

subgroup

index

#21425056

#21424898

#21424223

C₅₉⋊#1273

C₅₉×#1845

#10676

Subgroups — #28493752

Conjugacy classes of subgroups

Complements

Maximal subgroups