Group #28493742 — subgroups

Subgroups — #28493742

Order 15104.

442 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 165 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	#21425057	C₂	1	2	yes	maximal
2	#21424898	C₂	1	1	yes	maximal, characteristic
3	#21424256	C₂	1	1	yes	maximal, characteristic
4	#21424256	C₂	1	2	yes	maximal
5	C₅₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	C₂²	1	1	yes	characteristic
7	C₂×#14273661	—	2	2	no	—
8	C₂×#14273661	—	2	4	no	—
9	C₂×#14273652	—	2	2	no	—
10	#16837272	C₂²	1	1	yes	characteristic
11	#16837272	C₂²	1	2	yes	—
12	#16837165	—	2	4	no	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₂²×Dic₁₁₈	—	2	1	no	—
16	#14273727	—	2	2	no	—
17	#14273727	—	2	1	no	—
18	#14273727	Q₈	1	2	yes	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	1	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₂×C₄⋊Dic₅₉	—	2	2	no	—
24	C₂×C₄⋊Dic₅₉	—	2	2	no	—
25	#14273661	—	4	8	no	—
26	#14273661	—	4	4	no	—
27	#14273652	—	4	4	no	—
28	C₂²×C₈×C₅₉	D₄	1	2	yes	—
29	C₂²×C₈×C₅₉	D₄	1	4	yes	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	—	2	2	no	—
33	C₄×C₈×C₅₉	—	2	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	C₂²×Dic₅₉	—	4	2	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	4	no	—
45	C₂×C₈×C₅₉	—	2	4	no	—
46	C₂×C₈×C₅₉	—	2	2	no	—
47	C₂×C₈×C₅₉	D₈	1	4	yes	—
48	C₂×C₈×C₅₉	Q₁₆	1	8	yes	—
49	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅₉	#41	1	1	yes	characteristic
51	C₄²×C₅₉	#41	1	2	yes	—
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	C₄⋊Dic₅₉	—	4	4	no	—
57	C₄⋊Dic₅₉	—	4	4	no	—
58	C₄⋊Dic₅₉	—	4	4	no	—
59	C₂³×C₅₉	#127	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#123	1	2	yes	—
64	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
66	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
67	C₂×C₄×C₅₉	#137	1	2	yes	—
68	C₂×Dic₅₉	—	4	2	no	—
69	C₂×Dic₅₉	—	8	1	no	—
70	C₂×Dic₅₉	—	8	1	no	—
71	C₂×Dic₅₉	—	8	1	no	—
72	C₂×Dic₅₉	—	8	2	no	—
73	C₂×Dic₅₉	—	8	2	no	—
74	C₂×Dic₅₉	—	8	2	no	—
75	C₂×Dic₅₉	—	8	2	no	—
76	Dic₁₁₈	—	4	4	no	—
77	Dic₁₁₈	—	8	2	no	—
78	C₈×C₅₉	—	2	4	no	—
79	C₈×C₅₉	—	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	#474	1	2	yes	—
87	C₄×C₅₉	#477	1	4	yes	—
88	C₄×C₅₉	#480	1	2	yes	—
89	C₄×C₅₉	#484	1	4	yes	—
90	Dic₅₉	—	8	2	no	—
91	Dic₅₉	—	16	1	no	—
92	Dic₅₉	—	16	2	no	—
93	#1990	—	59	2	no	—
94	#1956	—	59	1	no	—
95	#1845	—	59	2	no	—
96	#1845	—	59	1	no	—
97	#1273	D₅₉	1	1	yes	characteristic
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	—	118	4	no	—
105	C₂×#107	—	118	2	no	—
106	C₂×#103	—	118	2	no	—
107	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
109	#384	—	59	1	no	—
110	#384	—	59	2	no	—
111	#340	—	118	4	no	—
112	C₅₉	#10674	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	118	1	no	—
114	C₂²×C₈	D₂₃₆	1	2	yes	—
115	C₂²×C₈	D₂₃₆	1	4	yes	—
116	C₄⋊Q₈	—	59	2	no	—
117	C₄⋊Q₈	—	118	2	no	—
118	C₄⋊Q₈	—	118	1	no	—
119	C₂×C₄⋊C₄	—	118	2	no	—
120	C₂×C₄⋊C₄	—	118	2	no	—
121	C₂×C₄⋊C₄	—	118	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	—	236	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	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	—	236	2	no	—
137	C₂²×C₄	—	236	1	no	—
138	C₂²×C₄	—	236	1	no	—
139	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	D₄₇₂	1	4	yes	—
146	C₂×C₈	Dic₂₃₆	1	8	yes	—
147	C₄⋊C₄	—	118	4	no	—
148	C₄⋊C₄	—	236	2	no	—
149	C₄⋊C₄	—	236	4	no	—
150	C₄⋊C₄	—	236	4	no	—
151	C₄⋊C₄	—	236	4	no	—
152	C₄⋊C₄	—	236	4	no	—
153	C₄⋊C₄	—	236	2	no	—
154	C₄²	#11745687	1	1	yes	characteristic
155	C₄²	#11745687	1	2	yes	—
156	C₄²	Q₈×D₅₉	1	1	yes	characteristic
157	C₂³	#14273724	1	1	yes	characteristic
158	Q₈	—	472	2	no	—
159	Q₈	—	236	4	no	—
160	C₂×C₄	—	472	1	no	—
161	C₂×C₄	—	472	2	no	—
162	C₂×C₄	—	472	2	no	—
163	C₂×C₄	—	472	2	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	#14273707	1	1	yes	characteristic
169	C₂×C₄	#14273707	1	2	yes	—
170	C₂×C₄	#14273711	1	1	yes	characteristic
171	C₂×C₄	#14273711	1	2	yes	—
172	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
173	C₂×C₄	C₂×Dic₂₃₆	1	4	yes	—
174	C₂×C₄	#14273755	1	4	yes	—
175	C₂×C₄	#14273760	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	#16837339	1	1	yes	characteristic
179	C₂²	#16837779	1	1	yes	characteristic
180	C₂²	#16837782	1	2	yes	—
181	C₂²	#16837791	1	2	yes	—
182	C₂²	#16837796	1	1	yes	characteristic
183	C₄	—	944	1	no	—
184	C₄	—	944	2	no	—
185	C₄	—	472	2	no	—
186	C₄	#16837683	1	2	yes	—
187	C₄	#16837698	1	4	yes	—
188	C₄	#16837716	1	2	yes	—
189	C₄	#16837738	1	4	yes	—
190	C₂	#21423714	1	1	yes	characteristic
191	C₂	#21423762	1	1	yes	characteristic
192	C₂	#21423773	1	2	yes	—
193	C₂	#21425623	1	1	yes	characteristic
194	C₂	#21425633	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425057

C₂

yes

maximal

#21424898

C₂

yes

maximal, characteristic

#21424256

C₂

yes

maximal, characteristic

#21424256

C₂

yes

maximal

C₅₉×#1273

C₂

yes

maximal, characteristic

C₂×#14273727

C₂²

yes

—

—

—

C₂²

yes

characteristic

#16837272

C₂²

yes

—

#16837165

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂²×Dic₁₁₈

—

#14273727

—

#14273727

—

#14273727

Q₈

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273661

—

#14273652

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄²×C₅₉

C₂×Q₈

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

#127

yes

characteristic

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

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#496

yes

—

C₂²×C₅₉

#497

yes

characteristic

C₄×C₅₉

#474

yes

—

C₄×C₅₉

#477

yes

—

C₄×C₅₉

#480

yes

—

C₄×C₅₉

#484

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

—

—

—

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

—

104

C₂×#107

—

118

—

105

C₂×#107

—

118

—

106

C₂×#103

—

118

—

107

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

108

C₂×C₄×C₈

C₂×D₅₉

yes

—

109

#384

—

110

#384

—

111

#340

—

118

—

112

C₅₉

#10674

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

118

—

114

C₂²×C₈

D₂₃₆

yes

—

115

C₂²×C₈

D₂₃₆

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

—

118

C₄⋊Q₈

—

118

—

119

C₂×C₄⋊C₄

—

118

—

120

C₂×C₄⋊C₄

—

118

—

121

C₂×C₄⋊C₄

—

118

—

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

—

236

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

118

—

133

C₂×Q₈

—

236

—

134

C₂×Q₈

—

236

—

135

C₂×Q₈

—

236

—

136

C₂²×C₄

—

236

—

137

C₂²×C₄

—

236

—

138

C₂²×C₄

—

236

—

139

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

140

C₂²×C₄

C₂×D₂₃₆

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

D₄₇₂

yes

—

146

C₂×C₈

Dic₂₃₆

yes

—

147

C₄⋊C₄

—

118

—

148

C₄⋊C₄

—

236

—

149

C₄⋊C₄

—

236

—

150

C₄⋊C₄

—

236

—

151

C₄⋊C₄

—

236

—

152

C₄⋊C₄

—

236

—

153

C₄⋊C₄

—

236

—

154

C₄²

#11745687

yes

characteristic

155

C₄²

#11745687

yes

—

156

C₄²

Q₈×D₅₉

yes

characteristic

157

C₂³

#14273724

yes

characteristic

158

Q₈

—

472

—

159

Q₈

—

236

—

160

C₂×C₄

—

472

—

161

C₂×C₄

—

472

—

162

C₂×C₄

—

472

—

163

C₂×C₄

—

472

—

164

C₂×C₄

—

472

—

165

C₂×C₄

—

472

—

166

C₂×C₄

—

472

—

167

C₂×C₄

—

236

—

168

C₂×C₄

#14273707

yes

characteristic

169

C₂×C₄

#14273707

yes

—

170

C₂×C₄

#14273711

yes

characteristic

171

C₂×C₄

#14273711

yes

—

172

C₂×C₄

C₂×D₄₇₂

yes

—

173

C₂×C₄

C₂×Dic₂₃₆

yes

—

174

C₂×C₄

#14273755

yes

—

175

C₂×C₄

#14273760

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#16837339

yes

characteristic

179

C₂²

#16837779

yes

characteristic

180

C₂²

#16837782

yes

—

181

C₂²

#16837791

yes

—

182

C₂²

#16837796

yes

characteristic

183

C₄

—

944

—

184

C₄

—

944

—

185

C₄

—

472

—

186

C₄

#16837683

yes

—

187

C₄

#16837698

yes

—

188

C₄

#16837716

yes

—

189

C₄

#16837738

yes

—

190

C₂

#21423714

yes

characteristic

191

C₂

#21423762

yes

characteristic

192

C₂

#21423773

yes

—

193

C₂

#21425623

yes

characteristic

194

C₂

#21425633

yes

—

	subgroup	G/H
center	C₂³	#14273724
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1273	C₂
socle, prime radical	C₂³×C₅₉	#127
hypercenter	#1273	D₅₉
hypocenter	C₅₉	#10674
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14273724

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1273

C₂

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

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
#21425057	2
#21424898	2
#21424256	2
C₅₉×#1273	2
#10674	59

subgroup

index

C₅₉×#1273

Subgroups — #28493742

Conjugacy classes of subgroups

Complements

Maximal subgroups