Group #28506722 — subgroups

Subgroups — #28506722

Order 15104.

326 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂×#16837230	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21424827	C₂	1	2	yes	maximal
4	#21422187	C₂	1	2	yes	maximal
5	C₂×C₅₉×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
9	#16837298	—	2	2	no	—
10	#16837230	C₄	1	4	yes	complemented
11	C₁₆×Dic₅₉	—	2	2	no	—
12	#16837164	—	2	2	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#367	—	2	2	no	—
16	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×Dic₅₉⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	C₂×C₅₉⋊C₁₆	—	2	1	no	—
22	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	—
23	#14273661	—	2	2	no	—
24	#14273661	C₂×C₄	1	4	yes	—
25	C₈×Dic₅₉	—	2	1	no	—
26	C₈×Dic₅₉	—	2	1	no	—
27	C₅₉⋊#95	—	4	2	no	—
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
33	#107×C₅₉	—	2	2	no	—
34	#107×C₅₉	—	2	2	no	—
35	C₂²×C₄×C₅₉	—	4	2	no	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₉	—	4	2	no	—
38	C₂²×Dic₅₉	D₈	1	2	yes	—
39	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
44	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₅₉	—	4	2	no	—
46	C₄⋊C₄×C₅₉	—	4	2	no	—
47	C₄⋊Dic₅₉	—	2	4	no	—
48	C₄⋊Dic₅₉	—	4	2	no	—
49	Dic₅₉⋊C₄	—	8	4	no	—
50	C₄×Dic₅₉	—	2	1	no	—
51	C₄×Dic₅₉	—	2	1	no	—
52	C₂×C₅₉⋊C₈	—	2	1	no	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
55	C₁₆×C₅₉	C₄⋊C₄	1	4	yes	complemented
56	C₅₉⋊C₁₆	—	2	2	no	—
57	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	8	2	no	—
59	C₂×C₄×C₅₉	—	8	2	no	—
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
67	C₂×Dic₅₉	—	2	2	no	—
68	C₂×Dic₅₉	—	2	2	no	—
69	C₂×Dic₅₉	—	4	4	no	—
70	C₂×Dic₅₉	—	8	2	no	—
71	C₂×Dic₅₉	—	8	2	no	—
72	C₂×Dic₅₉	SD₃₂	1	4	yes	—
73	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₅₉	#125	1	1	yes	characteristic
77	C₅₉⋊C₈	—	2	2	no	—
78	#13166	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
80	C₂²×C₅₉	#466	1	1	yes	characteristic
81	C₂²×C₅₉	#500	1	1	yes	characteristic
82	C₂²×C₅₉	C₂×SD₃₂	1	2	yes	—
83	C₂²×C₅₉	#508	1	2	yes	—
84	C₄×C₅₉	—	16	2	no	—
85	C₄×C₅₉	#389	1	1	yes	characteristic
86	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
87	C₄×C₅₉	#467	1	1	yes	characteristic
88	C₄×C₅₉	#499	1	1	yes	characteristic
89	Dic₅₉	—	2	4	no	—
90	Dic₅₉	—	8	4	no	—
91	C₂×#367	—	59	1	no	—
92	C₂×#367	D₅₉	1	1	yes	characteristic
93	C₂×C₄×C₁₆	—	59	1	no	—
94	#1942	—	59	2	no	—
95	#1380	—	59	2	no	—
96	C₂×C₅₉	#1942	1	1	yes	characteristic
97	C₂×C₅₉	C₄×SD₃₂	1	2	yes	—
98	C₂×C₅₉	#2217	1	2	yes	—
99	C₂×C₅₉	#2253	1	1	yes	characteristic
100	C₂×C₅₉	#2254	1	1	yes	characteristic
101	C₂²×C₁₆	—	59	1	no	—
102	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
103	C₂×#107	—	59	2	no	—
104	C₂×#107	C₂×D₅₉	1	2	yes	—
105	C₂×C₄×C₈	—	59	1	no	—
106	#389	—	118	2	no	—
107	#367	—	2	2	no	—
108	#367	—	59	4	no	—
109	C₄×C₁₆	—	118	2	no	—
110	#340	—	118	2	no	—
111	C₅₉	#13166	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	—	59	1	no	—
113	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	118	2	no	—
116	C₂×C₄⋊C₄	—	236	2	no	—
117	C₂×C₄²	—	59	1	no	—
118	C₂×C₁₆	—	59	2	no	—
119	C₂×C₁₆	—	118	1	no	—
120	C₂×C₁₆	—	118	1	no	—
121	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
122	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	59	4	no	—
127	#107	—	118	2	no	—
128	C₄×C₈	—	118	1	no	—
129	C₄×C₈	—	118	1	no	—
130	#95	—	236	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	59	2	no	—
133	C₂²×C₄	—	236	2	no	—
134	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
135	C₂×C₈	—	59	2	no	—
136	C₂×C₈	—	118	1	no	—
137	C₂×C₈	—	118	1	no	—
138	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
140	C₂×C₈	#11745687	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
142	C₂×C₈	#11745689	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	118	4	no	—
147	C₄⋊C₄	—	472	4	no	—
148	C₄⋊C₄	—	236	2	no	—
149	C₄²	—	118	1	no	—
150	C₄²	—	118	1	no	—
151	C₁₆	—	118	2	no	—
152	C₁₆	Dic₅₉⋊C₄	1	4	yes	complemented
153	C₂³	D₈×D₅₉	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	59	4	no	—
158	C₂×C₄	—	118	2	no	—
159	C₂×C₄	—	118	2	no	—
160	C₂×C₄	—	472	2	no	—
161	C₂×C₄	—	472	2	no	—
162	C₂×C₄	—	236	4	no	—
163	C₂×C₄	#14273689	1	1	yes	characteristic
164	C₂×C₄	#14273698	1	1	yes	characteristic
165	C₂×C₄	#14273726	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
167	C₂×C₄	#14273746	1	1	yes	characteristic
168	C₂×C₄	#14273750	1	1	yes	characteristic
169	C₈	—	118	2	no	—
170	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
171	C₈	#14273692	1	1	yes	characteristic
172	C₈	#14273703	1	1	yes	characteristic
173	C₈	#14273718	1	1	yes	characteristic
174	C₂²	#16837484	1	1	yes	characteristic
175	C₂²	#16837635	1	1	yes	characteristic
176	C₂²	#16837806	1	1	yes	characteristic
177	C₂²	SD₃₂×D₅₉	1	2	yes	—
178	C₂²	#16837833	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	118	4	no	—
181	C₄	—	472	4	no	—
182	C₄	#16837298	1	1	yes	characteristic
183	C₄	#16837496	1	1	yes	characteristic
184	C₄	#16837640	1	1	yes	characteristic
185	C₄	#16837803	1	1	yes	characteristic
186	C₂	#21424827	1	1	yes	characteristic
187	C₂	#21425947	1	2	yes	—
188	C₂	#21426189	1	2	yes	—
189	C₂	#21426374	1	1	yes	characteristic
190	C₂	#21426377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837230

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₅₉

C₂

yes

maximal, characteristic

#21424827

C₂

yes

maximal

#21422187

C₂

yes

maximal

C₂×C₅₉×#367

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273661

C₂²

yes

—

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

#16837298

—

#16837230

C₄

yes

complemented

C₁₆×Dic₅₉

—

#16837164

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#107×C₅₉

C₂²

yes

—

C₅₉×#367

—

C₂²×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₅₉

—

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

Q₈

yes

—

#14273661

—

#14273661

C₂×C₄

yes

—

C₈×Dic₅₉

—

C₈×Dic₅₉

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

Q₈

yes

—

#107×C₅₉

—

#107×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₁₆

yes

—

C₁₆×C₅₉

C₄⋊C₄

yes

complemented

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

SD₃₂

yes

—

C₈×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

characteristic

C₈×C₅₉

C₂²⋊Q₈

yes

characteristic

C₈×C₅₉

#125

yes

characteristic

C₅₉⋊C₈

—

#13166

—

maximal, Sylow

C₂²×C₅₉

C₄×D₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×SD₃₂

yes

—

C₂²×C₅₉

#508

yes

—

C₄×C₅₉

—

C₄×C₅₉

#389

yes

characteristic

C₄×C₅₉

C₄×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

C₂×#367

—

C₂×#367

D₅₉

yes

characteristic

C₂×C₄×C₁₆

—

#1942

—

#1380

—

C₂×C₅₉

#1942

yes

characteristic

C₂×C₅₉

C₄×SD₃₂

yes

—

C₂×C₅₉

#2217

yes

—

C₂×C₅₉

#2253

yes

characteristic

100

C₂×C₅₉

#2254

yes

characteristic

101

C₂²×C₁₆

—

102

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

103

C₂×#107

—

104

C₂×#107

C₂×D₅₉

yes

—

105

C₂×C₄×C₈

—

106

#389

—

118

—

107

#367

—

108

#367

—

109

C₄×C₁₆

—

118

—

110

#340

—

118

—

111

C₅₉

#13166

yes

characteristic, complemented, Sylow

112

C₂²×C₈

—

113

C₂²×C₈

C₂²×D₅₉

yes

characteristic

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

118

—

116

C₂×C₄⋊C₄

—

236

—

117

C₂×C₄²

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

118

—

120

C₂×C₁₆

—

118

—

121

C₂×C₁₆

Dic₁₁₈

yes

—

122

C₂×C₁₆

C₄×D₅₉

yes

—

123

C₂×C₁₆

C₂²⋊D₅₉

yes

—

124

#107

—

125

#107

—

126

#107

—

127

#107

—

118

—

128

C₄×C₈

—

118

—

129

C₄×C₈

—

118

—

130

#95

—

236

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

236

—

134

C₂²×C₄

D₄×D₅₉

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

118

—

137

C₂×C₈

—

118

—

138

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

139

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

140

C₂×C₈

#11745687

yes

characteristic

141

C₂×C₈

Q₈×D₅₉

yes

characteristic

142

C₂×C₈

#11745689

yes

characteristic

143

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

118

—

147

C₄⋊C₄

—

472

—

148

C₄⋊C₄

—

236

—

149

C₄²

—

118

—

150

C₄²

—

118

—

151

C₁₆

—

118

—

152

C₁₆

Dic₅₉⋊C₄

yes

complemented

153

C₂³

D₈×D₅₉

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

118

—

159

C₂×C₄

—

118

—

160

C₂×C₄

—

472

—

161

C₂×C₄

—

472

—

162

C₂×C₄

—

236

—

163

C₂×C₄

#14273689

yes

characteristic

164

C₂×C₄

#14273698

yes

characteristic

165

C₂×C₄

#14273726

yes

characteristic

166

C₂×C₄

Q₁₆×D₅₉

yes

characteristic

167

C₂×C₄

#14273746

yes

characteristic

168

C₂×C₄

#14273750

yes

characteristic

169

C₈

—

118

—

170

C₈

C₂×Dic₅₉⋊C₄

yes

characteristic

171

C₈

#14273692

yes

characteristic

172

C₈

#14273703

yes

characteristic

173

C₈

#14273718

yes

characteristic

174

C₂²

#16837484

yes

characteristic

175

C₂²

#16837635

yes

characteristic

176

C₂²

#16837806

yes

characteristic

177

C₂²

SD₃₂×D₅₉

yes

—

178

C₂²

#16837833

yes

—

179

C₄

—

180

C₄

—

118

—

181

C₄

—

472

—

182

C₄

#16837298

yes

characteristic

183

C₄

#16837496

yes

characteristic

184

C₄

#16837640

yes

characteristic

185

C₄

#16837803

yes

characteristic

186

C₂

#21424827

yes

characteristic

187

C₂

#21425947

yes

—

188

C₂

#21426189

yes

—

189

C₂

#21426374

yes

characteristic

190

C₂

#21426377

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#367	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#367	D₅₉
hypocenter	C₅₉	#13166
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#367

C₂

socle, prime radical

C₂³×C₅₉

C₂×D₈

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #16837230	C₄	semidirect
55. C₁₆×C₅₉	C₄⋊C₄	semidirect
111. C₅₉	#13166	semidirect
152. C₁₆	Dic₅₉⋊C₄	semidirect

H ≅ G/N

type

10. #16837230

C₄

semidirect

55. C₁₆×C₅₉

C₄⋊C₄

semidirect

111. C₅₉

#13166

semidirect

152. C₁₆

Dic₅₉⋊C₄

semidirect

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

subgroup	index
C₂×#16837230	2
C₂×C₁₆×Dic₅₉	2
#21424827	2
#21422187	2
C₂×C₅₉×#367	2
#13166	59

subgroup

index

C₂×C₁₆×Dic₅₉

C₂×C₅₉×#367

Subgroups — #28506722

Conjugacy classes of subgroups

Complements

Maximal subgroups