Group #28493464 — subgroups

Subgroups — #28493464

Order 15104.

292 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 93 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	#21425077	C₂	1	1	yes	maximal, characteristic
2	#21424250	C₂	1	1	yes	maximal, characteristic
3	#21422751	C₂	1	2	yes	maximal
4	#21421992	C₂	1	2	yes	maximal
5	C₅₉×#1274	C₂	1	1	yes	maximal, characteristic
6	#16837959	C₂²	1	1	yes	characteristic
7	#16837426	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	#16837162	—	2	1	no	—
12	#16837162	—	2	1	no	—
13	#16837150	—	2	2	no	—
14	#16837149	—	2	2	no	—
15	C₅₉×#431	C₂²	1	2	yes	—
16	C₅₉×#376	C₂²	1	1	yes	characteristic
17	C₂×#11745664	—	2	1	no	—
18	C₂²×C₅₉⋊C₈	—	2	2	no	—
19	#14273727	D₄	1	2	yes	—
20	#14273719	—	2	1	no	—
21	#14273706	—	2	2	no	—
22	C₄×Dic₁₁₈	—	2	2	no	—
23	#14273686	—	2	1	no	—
24	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
28	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
29	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₉	—	2	2	no	—
33	C₂²⋊C₈×C₅₉	—	2	2	no	—
34	C₄×C₈×C₅₉	—	2	2	no	—
35	#95×C₅₉	—	2	2	no	—
36	C₂²×C₄×C₅₉	—	2	1	no	—
37	C₂²×C₄×C₅₉	—	2	1	no	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	#41	1	2	yes	—
40	C₂×Dic₁₁₈	—	2	2	no	—
41	C₂×C₈×C₅₉	—	4	2	no	—
42	C₄²×C₅₉	—	2	1	no	—
43	C₄²×C₅₉	—	2	1	no	—
44	C₄²×C₅₉	SD₁₆	1	2	yes	—
45	C₄²×C₅₉	Q₁₆	1	2	yes	—
46	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₉	#41	1	1	yes	characteristic
48	C₄²×C₅₉	#41	1	2	yes	—
49	C₂²⋊Dic₅₉	—	4	1	no	—
50	C₄⋊Dic₅₉	—	2	2	no	—
51	C₄⋊Dic₅₉	—	2	2	no	—
52	Dic₅₉⋊C₄	—	4	2	no	—
53	C₄×Dic₅₉	—	4	1	no	—
54	#11745664	—	4	1	no	—
55	#11745664	—	4	1	no	—
56	C₂×C₅₉⋊C₈	—	4	2	no	—
57	C₂×C₅₉⋊C₈	—	4	1	no	—
58	C₂×C₅₉⋊C₈	—	4	2	no	—
59	C₂×C₅₉⋊C₈	—	4	2	no	—
60	C₂³×C₅₉	#123	1	1	yes	characteristic
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	—	2	2	no	—
63	C₂×C₄×C₅₉	—	4	2	no	—
64	C₂×C₄×C₅₉	—	4	2	no	—
65	C₂×C₄×C₅₉	—	4	2	no	—
66	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#123	1	2	yes	—
68	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#126	1	2	yes	—
70	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
74	C₂×Dic₅₉	—	4	2	no	—
75	Dic₁₁₈	—	8	2	no	—
76	C₈×C₅₉	—	8	2	no	—
77	C₅₉⋊C₈	—	8	2	no	—
78	C₅₉⋊C₈	—	8	1	no	—
79	#10628	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	#397	1	1	yes	characteristic
82	C₂²×C₅₉	#483	1	1	yes	characteristic
83	C₂²×C₅₉	#484	1	1	yes	characteristic
84	C₄×C₅₉	—	2	2	no	—
85	C₄×C₅₉	—	2	2	no	—
86	C₄×C₅₉	—	4	2	no	—
87	C₄×C₅₉	—	4	2	no	—
88	C₄×C₅₉	#460	1	1	yes	characteristic
89	C₄×C₅₉	#462	1	1	yes	characteristic
90	C₄×C₅₉	#487	1	1	yes	characteristic
91	C₄×C₅₉	#489	1	1	yes	characteristic
92	Dic₅₉	—	8	2	no	—
93	#1995	—	59	1	no	—
94	#1843	—	59	1	no	—
95	#1542	—	59	2	no	—
96	#1301	—	59	2	no	—
97	#1274	D₅₉	1	1	yes	characteristic
98	C₂×C₅₉	—	2	1	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	#2072	1	1	yes	characteristic
101	C₂×C₅₉	#2080	1	2	yes	—
102	#533	—	59	1	no	—
103	#431	C₂×D₅₉	1	2	yes	—
104	#423	—	59	1	no	—
105	C₂×C₄⋊C₈	—	59	2	no	—
106	C₂×C₄⋊C₈	—	118	1	no	—
107	C₂×C₄⋊C₈	—	118	1	no	—
108	#376	C₂×D₅₉	1	1	yes	characteristic
109	#339	—	118	1	no	—
110	#339	—	118	1	no	—
111	#333	—	118	2	no	—
112	#332	—	118	2	no	—
113	C₅₉	#10628	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	118	1	no	—
115	C₂²×C₈	—	118	2	no	—
116	C₄⋊Q₈	—	59	2	no	—
117	#125	—	118	1	no	—
118	C₂²⋊Q₈	—	118	2	no	—
119	C₄×Q₈	—	118	2	no	—
120	#117	—	118	1	no	—
121	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
122	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	118	2	no	—
126	C₄⋊C₈	—	118	2	no	—
127	C₄⋊C₈	—	118	2	no	—
128	C₄⋊C₈	—	236	2	no	—
129	C₄⋊C₈	—	236	2	no	—
130	C₂²⋊C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	118	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
137	C₂²×C₄	#11745687	1	2	yes	—
138	M₁₆	—	236	1	no	—
139	M₁₆	—	236	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	236	2	no	—
142	C₂×C₈	—	236	1	no	—
143	C₂×C₈	—	236	2	no	—
144	C₂×C₈	—	236	2	no	—
145	C₄⋊C₄	—	118	2	no	—
146	C₄⋊C₄	—	118	2	no	—
147	C₄⋊C₄	—	236	2	no	—
148	C₂²⋊C₄	—	236	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	236	1	no	—
152	C₄²	#11745660	1	2	yes	—
153	C₄²	Dic₂₃₆	1	2	yes	—
154	C₄²	#11745685	1	2	yes	—
155	C₄²	D₄×D₅₉	1	1	yes	characteristic
156	C₄²	#11745689	1	1	yes	characteristic
157	C₂³	#14273711	1	1	yes	characteristic
158	Q₈	—	472	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	236	2	no	—
165	C₂×C₄	#14273702	1	1	yes	characteristic
166	C₂×C₄	#14273709	1	2	yes	—
167	C₂×C₄	#14273715	1	1	yes	characteristic
168	C₂×C₄	#14273722	1	2	yes	—
169	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
171	C₂×C₄	#14273755	1	1	yes	characteristic
172	C₂×C₄	#14273760	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	472	1	no	—
175	C₈	—	472	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#16837329	1	1	yes	characteristic
178	C₂²	#16837733	1	1	yes	characteristic
179	C₂²	#16837738	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	472	2	no	—
185	C₄	#16837603	1	1	yes	characteristic
186	C₄	#16837613	1	1	yes	characteristic
187	C₄	#16837753	1	1	yes	characteristic
188	C₄	#16837762	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#21425506	1	1	yes	characteristic
192	C₂	#21425548	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425077

C₂

yes

maximal, characteristic

#21424250

C₂

yes

maximal, characteristic

#21422751

C₂

yes

maximal

#21421992

C₂

yes

maximal

C₅₉×#1274

C₂

yes

maximal, characteristic

#16837959

C₂²

yes

characteristic

#16837426

C₂²

yes

characteristic

C₂×C₄⋊(C₅₉⋊C₈)

—

C₂×C₄⋊(C₅₉⋊C₈)

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

—

—

—

—

C₅₉×#431

C₂²

yes

—

C₅₉×#376

C₂²

yes

characteristic

C₂×#11745664

—

C₂²×C₅₉⋊C₈

—

#14273727

D₄

yes

—

#14273719

—

#14273706

—

C₄×Dic₁₁₈

—

#14273686

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

Q₁₆

yes

—

C₄²×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#123

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

#126

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10628

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

—

100

C₂×C₅₉

#2072

yes

characteristic

101

C₂×C₅₉

#2080

yes

—

102

#533

—

103

#431

C₂×D₅₉

yes

—

104

#423

—

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

118

—

107

C₂×C₄⋊C₈

—

118

—

108

#376

C₂×D₅₉

yes

characteristic

109

#339

—

118

—

110

#339

—

118

—

111

#333

—

118

—

112

#332

—

118

—

113

C₅₉

#10628

yes

characteristic, complemented, Sylow

114

C₂×M₁₆

—

118

—

115

C₂²×C₈

—

118

—

116

C₄⋊Q₈

—

117

#125

—

118

—

118

C₂²⋊Q₈

—

118

—

119

C₄×Q₈

—

118

—

120

#117

—

118

—

121

C₂×C₄²

D₂₃₆

yes

characteristic

122

C₂×C₄²

D₂₃₆

yes

characteristic

123

C₂×C₄²

C₂²×D₅₉

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

118

—

126

C₄⋊C₈

—

118

—

127

C₄⋊C₈

—

118

—

128

C₄⋊C₈

—

236

—

129

C₄⋊C₈

—

236

—

130

C₂²⋊C₈

—

131

C₄×C₈

—

132

#95

—

133

C₂×Q₈

—

118

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

137

C₂²×C₄

#11745687

yes

—

138

M₁₆

—

236

—

139

M₁₆

—

236

—

140

C₂×C₈

—

141

C₂×C₈

—

236

—

142

C₂×C₈

—

236

—

143

C₂×C₈

—

236

—

144

C₂×C₈

—

236

—

145

C₄⋊C₄

—

118

—

146

C₄⋊C₄

—

118

—

147

C₄⋊C₄

—

236

—

148

C₂²⋊C₄

—

236

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

236

—

152

C₄²

#11745660

yes

—

153

C₄²

Dic₂₃₆

yes

—

154

C₄²

#11745685

yes

—

155

C₄²

D₄×D₅₉

yes

characteristic

156

C₄²

#11745689

yes

characteristic

157

C₂³

#14273711

yes

characteristic

158

Q₈

—

472

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

236

—

165

C₂×C₄

#14273702

yes

characteristic

166

C₂×C₄

#14273709

yes

—

167

C₂×C₄

#14273715

yes

characteristic

168

C₂×C₄

#14273722

yes

—

169

C₂×C₄

C₂×#11745660

yes

characteristic

170

C₂×C₄

C₂×Dic₂₃₆

yes

characteristic

171

C₂×C₄

#14273755

yes

characteristic

172

C₂×C₄

#14273760

yes

characteristic

173

C₈

—

174

C₈

—

472

—

175

C₈

—

472

—

176

C₂²

—

177

C₂²

#16837329

yes

characteristic

178

C₂²

#16837733

yes

characteristic

179

C₂²

#16837738

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

472

—

185

C₄

#16837603

yes

characteristic

186

C₄

#16837613

yes

characteristic

187

C₄

#16837753

yes

characteristic

188

C₄

#16837762

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#21425506

yes

characteristic

192

C₂

#21425548

yes

—

	subgroup	G/H
center	C₂×C₄	#14273715
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1274	C₂
socle	C₂²×C₅₉	#397
prime radical	C₂³×C₅₉	#123
hypercenter	#1274	D₅₉
hypocenter	C₅₉	#10628
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#14273715

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1274

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
113. C₅₉	#10628	semidirect

H ≅ G/N

type

113. C₅₉

#10628

semidirect

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

subgroup	index
#21425077	2
#21424250	2
#21422751	2
#21421992	2
C₅₉×#1274	2
#10628	59

subgroup

index

C₅₉×#1274

Subgroups — #28493464

Conjugacy classes of subgroups

Complements

Maximal subgroups