Group #28505024 — subgroups

Subgroups — #28505024

Order 15104.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21432707	C₂	1	1	yes	maximal, characteristic
2	#21425710	C₂	1	1	yes	maximal, characteristic
3	#21422212	C₂	1	2	yes	maximal, complemented
4	#21422191	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1812	C₂	1	1	yes	maximal, characteristic
6	#16838169	—	2	1	no	—
7	C₂×#14273667	C₂²	1	1	yes	characteristic
8	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	#16837451	—	2	1	no	—
10	#16837451	C₄	1	4	yes	complemented
11	C₂×#14273663	—	2	2	no	—
12	C₂×#14273663	C₂²	1	2	yes	—
13	C₂²⋊(C₅₉⋊C₁₆)	—	2	2	no	—
14	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#427	C₂²	1	2	yes	—
16	C₅₉×#410	—	2	1	no	—
17	C₅₉×H(3,ℤ/4)	—	2	1	no	—
18	#14273734	—	4	4	no	—
19	C₂×#11745664	—	2	4	no	—
20	C₂×#11745664	—	2	2	no	—
21	#14273667	—	2	2	no	—
22	C₂×C₅₉⋊C₁₆	—	2	1	no	—
23	C₂×C₅₉⋊C₁₆	—	2	1	no	—
24	C₂×C₅₉⋊C₁₆	—	2	2	no	—
25	#14273663	—	2	4	no	—
26	#14273663	—	2	2	no	—
27	#14273663	C₂×C₄	1	4	yes	—
28	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
29	#131×C₅₉	—	2	2	no	—
30	#131×C₅₉	—	2	2	no	—
31	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	D₄	1	2	yes	—
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	#117×C₅₉	—	2	2	no	—
36	#107×C₅₉	—	2	2	no	—
37	#106×C₅₉	—	2	2	no	—
38	#99×C₅₉	—	4	2	no	—
39	#41×C₅₉	—	2	2	no	—
40	#41×C₅₉	—	2	2	no	—
41	C₂×Q₈×C₅₉	Q₁₆	1	1	yes	characteristic
42	C₂×D₄×C₅₉	SD₁₆	1	2	yes	—
43	C₂×D₄×C₅₉	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₅₉	—	2	2	no	—
47	M₁₆×C₅₉	—	2	4	no	—
48	C₂×C₈×C₅₉	—	2	2	no	—
49	C₂×C₈×C₅₉	—	2	2	no	—
50	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	#41	1	2	yes	—
54	C₂×C₈×C₅₉	#41	1	2	yes	—
55	C₄⋊C₄×C₅₉	—	4	2	no	—
56	C₂²⋊C₄×C₅₉	—	4	2	no	—
57	C₄²×C₅₉	—	4	2	no	—
58	#11745664	—	2	4	no	—
59	#11745664	—	4	4	no	—
60	#11745664	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	4	no	—
62	C₂×C₅₉⋊C₈	—	4	2	no	—
63	C₅₉⋊C₁₆	—	4	1	no	—
64	C₅₉⋊C₁₆	—	4	1	no	—
65	C₂³×C₅₉	#103	1	2	yes	—
66	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₅₉	—	2	2	no	—
68	D₄×C₅₉	—	2	4	no	—
69	D₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	8	2	no	—
73	C₂×C₄×C₅₉	#103	1	2	yes	—
74	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#123	1	2	yes	—
76	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
78	C₈×C₅₉	—	2	2	no	—
79	C₈×C₅₉	#117	1	2	yes	—
80	C₈×C₅₉	#123	1	1	yes	characteristic
81	C₈×C₅₉	#123	1	1	yes	characteristic
82	C₅₉⋊C₈	—	4	4	no	—
83	#12877	—	59	1	no	maximal, Sylow
84	C₂²×C₅₉	—	2	2	no	—
85	C₂²×C₅₉	—	4	1	no	—
86	C₂²×C₅₉	—	4	2	no	—
87	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
88	C₂²×C₅₉	#481	1	2	yes	—
89	C₄×C₅₉	—	2	2	no	—
90	C₄×C₅₉	—	16	2	no	—
91	C₄×C₅₉	#381	1	1	yes	characteristic
92	C₄×C₅₉	#418	1	1	yes	characteristic
93	C₄×C₅₉	#484	1	2	yes	—
94	#2916	—	59	1	no	—
95	#2113	—	59	1	no	—
96	#1812	D₅₉	1	1	yes	characteristic
97	#1389	—	59	2	no	—
98	#1382	—	59	2	no	—
99	C₂×C₅₉	—	2	2	no	—
100	C₂×C₅₉	—	2	1	no	—
101	C₂×C₅₉	—	4	2	no	—
102	C₂×C₅₉	#1824	1	1	yes	characteristic
103	#568	—	118	1	no	—
104	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
105	C₂×M₃₂	—	59	1	no	—
106	C₂²×C₁₆	—	59	1	no	—
107	#430	—	59	4	no	—
108	#430	—	118	1	no	—
109	C₂×#108	—	59	2	no	—
110	C₂×#108	—	118	2	no	—
111	#427	C₂×D₅₉	1	2	yes	—
112	#410	—	2	1	no	—
113	C₂²⋊C₁₆	—	118	2	no	—
114	H(3,ℤ/4)	—	2	1	no	—
115	C₅₉	#12877	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	C₂²⋊D₅₉	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	236	4	no	—
120	C₂×M₁₆	—	118	4	no	—
121	C₂×M₁₆	—	118	2	no	—
122	C₂×M₁₆	C₄×D₅₉	1	2	yes	—
123	C₂×M₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
124	C₂²×C₈	D₂₃₆	1	2	yes	—
125	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
126	#117	—	2	2	no	—
127	M₃₂	—	118	2	no	—
128	C₂×C₁₆	—	118	1	no	—
129	C₂×C₁₆	—	118	1	no	—
130	C₂×C₁₆	—	118	2	no	—
131	#108	—	59	4	no	—
132	#108	—	118	4	no	—
133	#108	—	118	2	no	—
134	#107	—	2	2	no	—
135	#106	—	2	2	no	—
136	#99	—	4	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	#11745672	1	1	yes	characteristic
140	C₂×D₄	#11745670	1	2	yes	—
141	C₂×D₄	#11745672	1	1	yes	characteristic
142	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	118	4	no	—
147	M₁₆	—	236	4	no	—
148	M₁₆	—	236	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	236	4	no	—
152	C₂×C₈	—	236	2	no	—
153	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
155	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
156	C₂×C₈	#11745687	1	2	yes	—
157	C₂×C₈	#11745687	1	2	yes	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₁₆	—	236	1	no	—
162	C₁₆	—	236	1	no	—
163	C₂³	#14273650	1	2	yes	—
164	C₂³	C₂×#11745670	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	#14273650	1	2	yes	—
172	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#14273708	1	2	yes	—
174	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
175	C₂×C₄	#14273752	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	236	4	no	—
178	C₈	#14273683	1	2	yes	—
179	C₈	#14273711	1	1	yes	characteristic
180	C₈	#14273711	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#14273650	1	1	yes	characteristic
185	C₂²	#16837718	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	16	2	no	—
188	C₄	#16837263	1	1	yes	characteristic
189	C₄	#16837406	1	1	yes	characteristic
190	C₄	#16837735	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#21424184	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21432707

C₂

yes

maximal, characteristic

#21425710

C₂

yes

maximal, characteristic

#21422212

C₂

yes

maximal, complemented

#21422191

C₂

yes

maximal, complemented

C₅₉×#1812

C₂

yes

maximal, characteristic

#16838169

—

C₂×#14273667

C₂²

yes

characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

#16837451

—

#16837451

C₄

yes

complemented

C₂×#14273663

—

C₂×#14273663

C₂²

yes

—

C₂²⋊(C₅₉⋊C₁₆)

—

C₂×#131×C₅₉

C₂²

yes

characteristic

C₅₉×#427

C₂²

yes

—

C₅₉×#410

—

C₅₉×H(3,ℤ/4)

—

—

—

—

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

#14273663

—

#14273663

—

#14273663

C₂×C₄

yes

—

C₂×#41×C₅₉

D₄

yes

characteristic

#131×C₅₉

—

#131×C₅₉

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

—

C₂×M₁₆×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

—

#107×C₅₉

—

#106×C₅₉

—

#99×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

Q₁₆

yes

characteristic

C₂×D₄×C₅₉

SD₁₆

yes

—

C₂×D₄×C₅₉

Q₁₆

yes

characteristic

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

#11745664

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#103

yes

—

C₂³×C₅₉

C₂×SD₁₆

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#103

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₈×C₅₉

#117

yes

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

#12877

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#103

yes

characteristic

C₂²×C₅₉

#481

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#2916

—

#2113

—

#1812

D₅₉

yes

characteristic

#1389

—

#1382

—

C₂×C₅₉

—

100

C₂×C₅₉

—

101

C₂×C₅₉

—

102

C₂×C₅₉

#1824

yes

characteristic

103

#568

—

118

—

104

C₂×#131

C₂×D₅₉

yes

characteristic

105

C₂×M₃₂

—

106

C₂²×C₁₆

—

107

#430

—

108

#430

—

118

—

109

C₂×#108

—

110

C₂×#108

—

118

—

111

#427

C₂×D₅₉

yes

—

112

#410

—

113

C₂²⋊C₁₆

—

118

—

114

H(3,ℤ/4)

—

115

C₅₉

#12877

yes

characteristic, complemented, Sylow

116

C₂×#41

C₂²⋊D₅₉

yes

characteristic

117

#131

—

118

#131

—

119

#131

—

236

—

120

C₂×M₁₆

—

118

—

121

C₂×M₁₆

—

118

—

122

C₂×M₁₆

C₄×D₅₉

yes

—

123

C₂×M₁₆

C₂²⋊D₅₉

yes

characteristic

124

C₂²×C₈

D₂₃₆

yes

—

125

C₂²×C₈

C₂²×D₅₉

yes

characteristic

126

#117

—

127

M₃₂

—

118

—

128

C₂×C₁₆

—

118

—

129

C₂×C₁₆

—

118

—

130

C₂×C₁₆

—

118

—

131

#108

—

132

#108

—

118

—

133

#108

—

118

—

134

#107

—

135

#106

—

136

#99

—

137

#41

—

138

#41

—

139

C₂×Q₈

#11745672

yes

characteristic

140

C₂×D₄

#11745670

yes

—

141

C₂×D₄

#11745672

yes

characteristic

142

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

118

—

147

M₁₆

—

236

—

148

M₁₆

—

236

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

236

—

152

C₂×C₈

—

236

—

153

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

154

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

155

C₂×C₈

C₂×D₂₃₆

yes

characteristic

156

C₂×C₈

#11745687

yes

—

157

C₂×C₈

#11745687

yes

—

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₁₆

—

236

—

162

C₁₆

—

236

—

163

C₂³

#14273650

yes

—

164

C₂³

C₂×#11745670

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#14273650

yes

—

172

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

173

C₂×C₄

#14273708

yes

—

174

C₂×C₄

C₂×#11745672

yes

characteristic

175

C₂×C₄

#14273752

yes

—

176

C₈

—

177

C₈

—

236

—

178

C₈

#14273683

yes

—

179

C₈

#14273711

yes

characteristic

180

C₈

#14273711

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#14273650

yes

characteristic

185

C₂²

#16837718

yes

—

186

C₄

—

187

C₄

—

188

C₄

#16837263

yes

characteristic

189

C₄

#16837406

yes

characteristic

190

C₄

#16837735

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#21424184

yes

characteristic

	subgroup	G/H
center	C₄	#16837406
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1812	C₂
socle	C₂×C₅₉	#1824
prime radical	C₂×#41×C₅₉	D₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1812	D₅₉
hypocenter	C₅₉	#12877
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #1812

subgroup

G/H

center

C₄

#16837406

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1812

C₂

socle

C₂×C₅₉

#1824

prime radical

C₂×#41×C₅₉

D₄

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #1812

N	H ≅ G/N	type
3. #21422212	C₂	semidirect
4. #21422191	C₂	semidirect
10. #16837451	C₄	semidirect
115. C₅₉	#12877	semidirect

H ≅ G/N

type

3. #21422212

C₂

semidirect

4. #21422191

C₂

semidirect

10. #16837451

C₄

semidirect

115. C₅₉

#12877

semidirect

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

subgroup	index
#21432707	2
#21425710	2
#21422212	2
#21422191	2
C₅₉×#1812	2
#12877	59

subgroup

index

C₅₉×#1812

Subgroups — #28505024

Conjugacy classes of subgroups

Complements

Maximal subgroups