Group #28508230 — subgroups

Subgroups — #28508230

Order 15104.

418 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂²×D₉₄₄	C₂	1	1	yes	maximal, characteristic
2	#21425854	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837214	C₂	1	2	yes	maximal, complemented
4	#21425004	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₇₂	C₂²	1	2	yes	—
7	C₂×D₉₄₄	—	2	4	no	—
8	C₂×D₉₄₄	—	2	2	no	—
9	C₂×#14273667	C₂²	1	1	yes	characteristic
10	#16837408	—	2	2	no	—
11	C₂×#14273643	—	2	2	no	—
12	#16837357	C₂²	1	1	yes	characteristic
13	Dic₅₉⋊C₁₆	—	2	2	no	—
14	#16837214	—	2	4	no	—
15	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₉	C₂²	1	2	yes	—
17	C₂²×D₂₃₆	—	2	2	no	—
18	C₂×D₄₇₂	—	2	2	no	—
19	C₂×D₄₇₂	—	2	2	no	—
20	C₂×D₄₇₂	—	2	2	no	—
21	C₂×D₄₇₂	—	2	2	no	—
22	C₂×D₄₇₂	—	2	2	no	—
23	C₂×D₄₇₂	D₄	1	4	yes	—
24	C₂×#11745664	D₄	1	1	yes	characteristic
25	#14273686	D₄	1	1	yes	characteristic
26	D₉₄₄	—	2	8	no	—
27	#14273667	—	2	2	no	—
28	C₂×C₅₉⋊C₁₆	—	2	1	no	—
29	#14273649	—	4	4	no	—
30	#14273643	—	4	4	no	—
31	#14273637	—	2	1	no	—
32	C₈×Dic₅₉	—	2	1	no	—
33	C₂×M₁₆×C₅₉	—	2	2	no	—
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₁₆×C₅₉	—	2	2	no	—
36	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
37	#108×C₅₉	—	2	2	no	—
38	#108×C₅₉	—	2	2	no	—
39	C₂³×D₅₉	—	4	2	no	—
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×D₂₃₆	—	2	4	no	—
42	C₂×D₂₃₆	—	4	2	no	—
43	C₂×D₂₃₆	—	4	2	no	—
44	C₂×D₂₃₆	—	4	2	no	—
45	C₂×D₂₃₆	—	4	2	no	—
46	C₂×D₂₃₆	—	4	2	no	—
47	M₁₆×C₅₉	—	4	2	no	—
48	M₁₆×C₅₉	—	4	2	no	—
49	C₂×C₈×C₅₉	—	4	2	no	—
50	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₅₉	#41	1	2	yes	—
54	C₂²⋊Dic₅₉	—	2	1	no	—
55	C₄⋊Dic₅₉	D₈	1	2	yes	—
56	C₄×Dic₅₉	—	2	1	no	—
57	#11745664	—	2	2	no	—
58	C₂×C₅₉⋊C₈	—	2	1	no	—
59	D₄₇₂	—	2	4	no	—
60	D₄₇₂	—	2	4	no	—
61	D₄₇₂	—	4	2	no	—
62	D₄₇₂	—	4	2	no	—
63	C₁₆×C₅₉	—	2	2	no	—
64	C₅₉⋊C₁₆	—	4	1	no	—
65	C₂³×C₅₉	#135	1	1	yes	characteristic
66	C₂²×D₅₉	—	4	4	no	—
67	C₂²×D₅₉	—	8	2	no	—
68	C₂²×D₅₉	—	8	2	no	—
69	C₂²×D₅₉	—	16	4	no	—
70	C₂×C₄×C₅₉	#121	1	2	yes	—
71	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×Dic₅₉	—	2	2	no	—
75	D₂₃₆	—	4	4	no	—
76	D₂₃₆	—	4	4	no	—
77	D₂₃₆	—	8	2	no	—
78	D₂₃₆	—	8	2	no	—
79	C₈×C₅₉	—	8	2	no	—
80	C₈×C₅₉	#120	1	1	yes	characteristic
81	C₈×C₅₉	#123	1	2	yes	—
82	C₈×C₅₉	#124	1	1	yes	characteristic
83	C₅₉⋊C₈	—	4	1	no	—
84	#13439	—	59	1	no	maximal, Sylow
85	C₂²×C₅₉	—	2	2	no	—
86	C₂²×C₅₉	#463	1	2	yes	—
87	C₂²×C₅₉	#466	1	1	yes	characteristic
88	C₂×D₅₉	—	8	4	no	—
89	C₂×D₅₉	—	16	4	no	—
90	C₂×D₅₉	—	16	4	no	—
91	C₂×D₅₉	—	16	4	no	—
92	C₄×C₅₉	#394	1	1	yes	characteristic
93	C₄×C₅₉	#459	1	2	yes	—
94	C₄×C₅₉	#469	1	1	yes	characteristic
95	Dic₅₉	—	4	2	no	—
96	C₂²×D₁₆	—	59	1	no	—
97	#2151	—	59	1	no	—
98	C₂×#361	—	59	2	no	—
99	#1978	—	59	2	no	—
100	#1381	D₅₉	1	1	yes	characteristic, complemented
101	C₂×C₅₉	—	2	2	no	—
102	C₂×C₅₉	#2054	1	1	yes	characteristic
103	C₂×C₅₉	#2221	1	2	yes	—
104	D₅₉	—	16	4	no	—
105	C₂²×D₈	—	59	2	no	—
106	C₂×D₁₆	—	118	4	no	—
107	C₂×D₁₆	—	118	2	no	—
108	C₂×M₃₂	—	59	1	no	—
109	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
110	C₂×#108	C₂×D₅₉	1	2	yes	—
111	#418	—	118	2	no	—
112	C₂×#100	—	118	2	no	—
113	#405	—	59	1	no	—
114	C₄⋊C₁₆	—	118	2	no	—
115	#361	—	118	4	no	—
116	C₅₉	#13439	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₄	—	118	2	no	—
118	C₂×D₈	—	59	4	no	—
119	C₂×D₈	—	118	2	no	—
120	C₂×D₈	—	118	2	no	—
121	C₂×D₈	—	118	2	no	—
122	C₂×D₈	—	118	2	no	—
123	C₂×D₈	—	118	2	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	—	59	1	no	—
126	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
127	#117	—	59	1	no	—
128	D₁₆	—	118	8	no	—
129	M₃₂	—	118	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	118	1	no	—
132	C₂×C₁₆	D₂₃₆	1	2	yes	—
133	#108	—	2	2	no	—
134	#108	—	2	2	no	—
135	#102	—	236	4	no	—
136	#100	—	236	4	no	—
137	#97	—	118	1	no	—
138	C₄×C₈	—	118	1	no	—
139	C₂⁴	—	236	2	no	—
140	C₂×D₄	—	118	4	no	—
141	C₂×D₄	—	236	2	no	—
142	C₂×D₄	—	236	2	no	—
143	C₂×D₄	—	236	2	no	—
144	C₂×D₄	—	236	2	no	—
145	C₂×D₄	—	236	2	no	—
146	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
147	D₈	—	118	4	no	—
148	D₈	—	118	4	no	—
149	D₈	—	236	2	no	—
150	D₈	—	236	2	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	118	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	118	1	no	—
156	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
157	C₂×C₈	#11745685	1	2	yes	—
158	C₂×C₈	D₄×D₅₉	1	2	yes	—
159	C₂×C₈	#11745689	1	1	yes	characteristic
160	C₄⋊C₄	—	59	2	no	—
161	C₂²⋊C₄	—	118	1	no	—
162	C₄²	—	118	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	236	1	no	—
165	C₂³	—	944	4	no	—
166	C₂³	—	472	2	no	—
167	C₂³	—	472	2	no	—
168	C₂³	—	236	4	no	—
169	C₂³	#14273750	1	1	yes	characteristic
170	D₄	—	472	2	no	—
171	D₄	—	472	2	no	—
172	D₄	—	236	4	no	—
173	D₄	—	236	4	no	—
174	C₂×C₄	—	118	2	no	—
175	C₂×C₄	#14273700	1	2	yes	—
176	C₂×C₄	#14273702	1	1	yes	characteristic
177	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
178	C₂×C₄	#14273754	1	2	yes	—
179	C₈	—	8	2	no	—
180	C₈	—	236	1	no	—
181	C₈	#14273697	1	1	yes	characteristic
182	C₈	#14273712	1	2	yes	—
183	C₈	#14273715	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	944	4	no	—
186	C₂²	—	944	4	no	—
187	C₂²	—	944	4	no	—
188	C₂²	—	472	4	no	—
189	C₂²	#16837620	1	2	yes	—
190	C₂²	#16837639	1	1	yes	characteristic
191	C₄	—	236	2	no	—
192	C₄	#16837320	1	1	yes	characteristic
193	C₄	#16837596	1	2	yes	—
194	C₄	#16837656	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	944	4	no	—
197	C₂	#21425408	1	1	yes	characteristic
198	C₂	#21426217	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₉₄₄

C₂

yes

maximal, characteristic

#21425854

C₂

yes

maximal, characteristic, complemented

C₂×#16837214

C₂

yes

maximal, complemented

#21425004

C₂

yes

maximal, complemented

C₅₉×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₇₂

C₂²

yes

—

C₂×D₉₄₄

—

C₂×D₉₄₄

—

C₂×#14273667

C₂²

yes

—

—

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

#16837214

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₂²×D₂₃₆

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

D₄

yes

—

C₂×#11745664

D₄

yes

characteristic

#14273686

D₄

yes

characteristic

D₉₄₄

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

#14273649

—

#14273643

—

#14273637

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

—

#108×C₅₉

—

#108×C₅₉

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

D₈

yes

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#135

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

#120

yes

characteristic

C₈×C₅₉

#123

yes

—

C₈×C₅₉

#124

yes

characteristic

C₅₉⋊C₈

—

#13439

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#463

yes

—

C₂²×C₅₉

#466

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

#394

yes

characteristic

C₄×C₅₉

#459

yes

—

C₄×C₅₉

#469

yes

characteristic

Dic₅₉

—

C₂²×D₁₆

—

#2151

—

C₂×#361

—

#1978

—

100

#1381

D₅₉

yes

characteristic, complemented

101

C₂×C₅₉

—

102

C₂×C₅₉

#2054

yes

characteristic

103

C₂×C₅₉

#2221

yes

—

104

D₅₉

—

105

C₂²×D₈

—

106

C₂×D₁₆

—

118

—

107

C₂×D₁₆

—

118

—

108

C₂×M₃₂

—

109

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

110

C₂×#108

C₂×D₅₉

yes

—

111

#418

—

118

—

112

C₂×#100

—

118

—

113

#405

—

114

C₄⋊C₁₆

—

118

—

115

#361

—

118

—

116

C₅₉

#13439

yes

characteristic, complemented, Sylow

117

C₂²×D₄

—

118

—

118

C₂×D₈

—

119

C₂×D₈

—

118

—

120

C₂×D₈

—

118

—

121

C₂×D₈

—

118

—

122

C₂×D₈

—

118

—

123

C₂×D₈

—

118

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂²×C₈

C₂²×D₅₉

yes

characteristic

127

#117

—

128

D₁₆

—

118

—

129

M₃₂

—

118

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

118

—

132

C₂×C₁₆

D₂₃₆

yes

—

133

#108

—

134

#108

—

135

#102

—

236

—

136

#100

—

236

—

137

#97

—

118

—

138

C₄×C₈

—

118

—

139

C₂⁴

—

236

—

140

C₂×D₄

—

118

—

141

C₂×D₄

—

236

—

142

C₂×D₄

—

236

—

143

C₂×D₄

—

236

—

144

C₂×D₄

—

236

—

145

C₂×D₄

—

236

—

146

C₂²×C₄

D₄×D₅₉

yes

characteristic

147

D₈

—

118

—

148

D₈

—

118

—

149

D₈

—

236

—

150

D₈

—

236

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

118

—

154

C₂×C₈

—

155

C₂×C₈

—

118

—

156

C₂×C₈

C₂×D₂₃₆

yes

characteristic

157

C₂×C₈

#11745685

yes

—

158

C₂×C₈

D₄×D₅₉

yes

—

159

C₂×C₈

#11745689

yes

characteristic

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

118

—

162

C₄²

—

118

—

163

C₁₆

—

164

C₁₆

—

236

—

165

C₂³

—

944

—

166

C₂³

—

472

—

167

C₂³

—

472

—

168

C₂³

—

236

—

169

C₂³

#14273750

yes

characteristic

170

D₄

—

472

—

171

D₄

—

472

—

172

D₄

—

236

—

173

D₄

—

236

—

174

C₂×C₄

—

118

—

175

C₂×C₄

#14273700

yes

—

176

C₂×C₄

#14273702

yes

characteristic

177

C₂×C₄

D₈×D₅₉

yes

characteristic

178

C₂×C₄

#14273754

yes

—

179

C₈

—

180

C₈

—

236

—

181

C₈

#14273697

yes

characteristic

182

C₈

#14273712

yes

—

183

C₈

#14273715

yes

characteristic

184

C₂²

—

185

C₂²

—

944

—

186

C₂²

—

944

—

187

C₂²

—

944

—

188

C₂²

—

472

—

189

C₂²

#16837620

yes

—

190

C₂²

#16837639

yes

characteristic

191

C₄

—

236

—

192

C₄

#16837320

yes

characteristic

193

C₄

#16837596

yes

—

194

C₄

#16837656

yes

characteristic

195

C₂

—

196

C₂

—

944

—

197

C₂

#21425408

yes

characteristic

198

C₂

#21426217

yes

—

	subgroup	G/H
center	C₂²	#16837639
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1381	C₂
socle	C₂²×C₅₉	#466
prime radical	C₂²×D₉₄₄	C₂
hypercenter	#1381	D₅₉
hypocenter	C₅₉	#13439
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#16837639

commutator, cosocle

C₂²×C₈×C₅₉

C₂³

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1381

C₂

socle

C₂²×C₅₉

#466

prime radical

C₂²×D₉₄₄

C₂

D₅₉

C₅₉

G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. #21425854	C₂	semidirect
3. C₂×#16837214	C₂	semidirect
4. #21425004	C₂	semidirect
5. C₅₉×#1381	C₂	semidirect
100. #1381	D₅₉	semidirect
116. C₅₉	#13439	semidirect

H ≅ G/N

type

2. #21425854

C₂

semidirect

3. C₂×#16837214

C₂

semidirect

4. #21425004

C₂

semidirect

5. C₅₉×#1381

C₂

semidirect

100. #1381

D₅₉

semidirect

116. C₅₉

#13439

semidirect

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

subgroup	index
C₂²×D₉₄₄	2
#21425854	2
C₂×#16837214	2
#21425004	2
C₅₉×#1381	2
#13439	59

subgroup

index

C₂²×D₉₄₄

C₅₉×#1381

Subgroups — #28508230

Conjugacy classes of subgroups

Complements

Maximal subgroups