Group #28488157 — subgroups

Subgroups — #28488157

Order 15104.

582 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 159 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	#21424268	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422764	C₂	1	4	yes	maximal, complemented
3	#21422379	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₉×#493	C₂	1	1	yes	maximal, characteristic
5	#16837455	C₂²	1	2	yes	—
6	C₂×#14273663	—	2	2	no	—
7	#16837426	C₂²	1	2	yes	—
8	#16837146	—	2	8	no	—
9	#16837123	—	2	1	no	—
10	C₈×C₅₉⋊C₈	—	2	1	no	—
11	C₂²×D₈×C₅₉	—	2	2	no	—
12	C₂×#127×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#493	—	2	1	no	—
14	C₅₉×#493	—	2	1	no	—
15	C₅₉×#493	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745664	—	2	2	no	—
18	#14273663	—	4	4	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
20	C₄⋊(C₅₉⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄×C₅₉⋊C₈	—	2	2	no	—
23	C₂²×D₄×C₅₉	—	2	2	no	—
24	C₂²×D₄×C₅₉	—	2	4	no	—
25	C₂×D₈×C₅₉	—	2	4	no	—
26	C₂×D₈×C₅₉	—	2	4	no	—
27	C₂×D₈×C₅₉	—	4	4	no	—
28	C₂×D₈×C₅₉	—	4	4	no	—
29	C₂²×C₈×C₅₉	D₄	1	2	yes	—
30	#127×C₅₉	—	2	4	no	—
31	#127×C₅₉	—	2	2	no	—
32	#127×C₅₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
37	C₂⁴×C₅₉	—	4	2	no	—
38	C₂×D₄×C₅₉	—	2	4	no	—
39	C₂×D₄×C₅₉	—	2	8	no	—
40	C₂×D₄×C₅₉	—	4	4	no	—
41	C₂×D₄×C₅₉	—	4	2	no	—
42	C₂×D₄×C₅₉	—	4	4	no	—
43	C₂×D₄×C₅₉	—	4	4	no	—
44	C₂×D₄×C₅₉	—	8	8	no	—
45	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	#41	1	2	yes	—
47	D₈×C₅₉	—	4	8	no	—
48	D₈×C₅₉	—	4	8	no	—
49	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₉	D₈	1	4	yes	—
52	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
53	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₉	C₂×D₄	1	2	yes	—
56	#11745664	—	4	4	no	—
57	C₂×C₅₉⋊C₈	—	2	4	no	—
58	C₂×C₅₉⋊C₈	—	4	2	no	—
59	C₂³×C₅₉	—	4	4	no	—
60	C₂³×C₅₉	—	8	8	no	—
61	C₂³×C₅₉	—	8	4	no	—
62	C₂³×C₅₉	#124	1	1	yes	characteristic
63	D₄×C₅₉	—	4	8	no	—
64	D₄×C₅₉	—	8	8	no	—
65	D₄×C₅₉	—	8	8	no	—
66	D₄×C₅₉	—	8	2	no	—
67	D₄×C₅₉	—	8	2	no	—
68	C₂×C₄×C₅₉	—	2	4	no	—
69	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₅₉	#121	1	2	yes	—
72	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
75	C₈×C₅₉	#102	1	4	yes	—
76	C₈×C₅₉	#102	1	4	yes	—
77	C₅₉⋊C₈	—	4	4	no	—
78	C₅₉⋊C₈	—	8	2	no	—
79	#9773	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	—	8	4	no	—
82	C₂²×C₅₉	—	8	8	no	—
83	C₂²×C₅₉	—	16	8	no	—
84	C₂²×C₅₉	#390	1	1	yes	characteristic
85	C₂²×C₅₉	#486	1	2	yes	—
86	C₄×C₅₉	—	2	2	no	—
87	C₄×C₅₉	C₂×#102	1	2	yes	—
88	C₄×C₅₉	#472	1	4	yes	—
89	C₄×C₅₉	#492	1	1	yes	characteristic
90	C₄×C₅₉	#493	1	1	yes	characteristic
91	C₂×#493	D₅₉	1	1	yes	characteristic
92	#1848	—	59	1	no	—
93	#1545	—	59	4	no	—
94	#1450	—	59	1	no	—
95	C₂×C₅₉	—	2	2	no	—
96	C₂×C₅₉	—	16	4	no	—
97	C₂×C₅₉	#1957	1	1	yes	characteristic
98	C₂×C₅₉	#1978	1	2	yes	—
99	C₂²×D₈	—	2	2	no	—
100	C₂×#127	C₂×D₅₉	1	2	yes	—
101	#493	—	2	1	no	—
102	#493	—	2	1	no	—
103	#493	Dic₅₉	1	4	yes	—
104	#431	—	59	2	no	—
105	C₂×#108	—	118	2	no	—
106	#423	—	59	2	no	—
107	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
108	#331	—	118	8	no	—
109	#322	—	118	1	no	—
110	C₈²	—	118	1	no	—
111	C₅₉	#9773	1	1	yes	characteristic, complemented, Sylow
112	C₂²×D₄	—	2	4	no	—
113	C₂²×D₄	—	2	2	no	—
114	C₂×D₈	—	2	4	no	—
115	C₂×D₈	—	2	4	no	—
116	C₂×D₈	—	4	4	no	—
117	C₂×D₈	—	4	4	no	—
118	C₂×M₁₆	—	118	2	no	—
119	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
120	#127	—	2	4	no	—
121	#127	—	2	2	no	—
122	#127	C₂×Dic₅₉	1	4	yes	—
123	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
124	#108	—	236	4	no	—
125	C₄⋊C₈	—	59	4	no	—
126	C₄⋊C₈	—	118	4	no	—
127	C₂²⋊C₈	—	118	2	no	—
128	C₄×C₈	—	118	2	no	—
129	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
130	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
132	C₂⁴	—	4	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	4	no	—
138	C₂×D₄	—	4	4	no	—
139	C₂×D₄	—	8	8	no	—
140	C₂²×C₄	#11745687	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
142	D₈	—	4	8	no	—
143	D₈	—	4	8	no	—
144	M₁₆	—	236	4	no	—
145	C₂×C₈	—	118	4	no	—
146	C₂×C₈	—	236	2	no	—
147	C₂×C₈	#11745669	1	4	yes	—
148	C₂×C₈	#11745670	1	4	yes	—
149	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
150	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
151	C₄²	D₄×D₅₉	1	2	yes	—
152	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
154	C₂³	—	4	4	no	—
155	C₂³	—	8	8	no	—
156	C₂³	—	8	4	no	—
157	C₂³	#14273714	1	1	yes	characteristic
158	D₄	—	4	8	no	—
159	D₄	—	8	8	no	—
160	D₄	—	8	8	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
166	C₂×C₄	#14273698	1	2	yes	—
167	C₂×C₄	#14273723	1	1	yes	characteristic
168	C₂×C₄	C₂×#11745669	1	2	yes	—
169	C₂×C₄	C₂×#11745670	1	2	yes	—
170	C₈	—	472	2	no	—
171	C₈	—	236	4	no	—
172	C₈	#14273648	1	4	yes	—
173	C₈	#14273648	1	4	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	8	4	no	—
176	C₂²	—	8	8	no	—
177	C₂²	—	16	8	no	—
178	C₂²	#16837305	1	1	yes	characteristic
179	C₂²	#16837748	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	C₂×#14273648	1	2	yes	—
182	C₄	#16837669	1	4	yes	—
183	C₄	#16837774	1	1	yes	characteristic
184	C₄	#16837778	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	—	16	4	no	—
187	C₂	#21424903	1	1	yes	characteristic
188	C₂	#21425007	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424268

C₂

yes

maximal, characteristic, complemented

#21422764

C₂

yes

maximal, complemented

#21422379

C₂

yes

maximal, characteristic, complemented

C₂×C₅₉×#493

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×#14273663

—

#16837426

C₂²

yes

—

#16837146

—

#16837123

—

C₈×C₅₉⋊C₈

—

C₂²×D₈×C₅₉

—

C₂×#127×C₅₉

C₂²

yes

—

C₅₉×#493

—

C₅₉×#493

—

C₅₉×#493

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂²×D₄×C₅₉

—

C₂²×D₄×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

#127×C₅₉

—

#127×C₅₉

—

#127×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂⁴×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

#124

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₈×C₅₉

#102

yes

—

C₈×C₅₉

#102

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9773

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#486

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#102

yes

—

C₄×C₅₉

#472

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1957

yes

characteristic

C₂×C₅₉

#1978

yes

—

C₂²×D₈

—

100

C₂×#127

C₂×D₅₉

yes

—

101

#493

—

102

#493

—

103

#493

Dic₅₉

yes

—

104

#431

—

105

C₂×#108

—

118

—

106

#423

—

107

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

108

#331

—

118

—

109

#322

—

118

—

110

C₈²

—

118

—

111

C₅₉

#9773

yes

characteristic, complemented, Sylow

112

C₂²×D₄

—

113

C₂²×D₄

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂×M₁₆

—

118

—

119

C₂²×C₈

C₂²⋊D₅₉

yes

—

120

#127

—

121

#127

—

122

#127

C₂×Dic₅₉

yes

—

123

C₂×C₄²

C₂²×D₅₉

yes

characteristic

124

#108

—

236

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

118

—

127

C₂²⋊C₈

—

118

—

128

C₄×C₈

—

118

—

129

C₄×C₈

C₂×Dic₅₉

yes

—

130

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

131

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

132

C₂⁴

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

#11745687

yes

—

141

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

142

D₈

—

143

D₈

—

144

M₁₆

—

236

—

145

C₂×C₈

—

118

—

146

C₂×C₈

—

236

—

147

C₂×C₈

#11745669

yes

—

148

C₂×C₈

#11745670

yes

—

149

C₂×C₈

C₂²⋊Dic₅₉

yes

—

150

C₂×C₈

C₂²⋊Dic₅₉

yes

—

151

C₄²

D₄×D₅₉

yes

—

152

C₄²

C₂²×Dic₅₉

yes

characteristic

153

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

#14273714

yes

characteristic

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

165

C₂×C₄

D₄×Dic₅₉

yes

—

166

C₂×C₄

#14273698

yes

—

167

C₂×C₄

#14273723

yes

characteristic

168

C₂×C₄

C₂×#11745669

yes

—

169

C₂×C₄

C₂×#11745670

yes

—

170

C₈

—

472

—

171

C₈

—

236

—

172

C₈

#14273648

yes

—

173

C₈

#14273648

yes

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#16837305

yes

characteristic

179

C₂²

#16837748

yes

—

180

C₄

—

181

C₄

C₂×#14273648

yes

—

182

C₄

#16837669

yes

—

183

C₄

#16837774

yes

characteristic

184

C₄

#16837778

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#21424903

yes

characteristic

188

C₂

#21425007

yes

—

	subgroup	G/H
center	C₂²	#16837305
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting, prime radical	C₂×C₅₉×#493	C₂
socle	C₂²×C₅₉	#390
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#493	D₅₉
hypocenter	C₅₉	#9773
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂²

#16837305

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting, prime radical

C₂×C₅₉×#493

C₂

socle

C₂²×C₅₉

#390

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #21424268	C₂	semidirect
2. #21422764	C₂	semidirect
3. #21422379	C₂	semidirect
111. C₅₉	#9773	semidirect

H ≅ G/N

type

1. #21424268

C₂

semidirect

2. #21422764

C₂

semidirect

3. #21422379

C₂

semidirect

111. C₅₉

#9773

semidirect

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

subgroup	index
#21424268	2
#21422764	2
#21422379	2
C₂×C₅₉×#493	2
#9773	59

subgroup

index

C₂×C₅₉×#493

Subgroups — #28488157

Conjugacy classes of subgroups

Complements

Maximal subgroups