Group #28506926 — subgroups

Subgroups — #28506926

Order 15104.

486 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 167 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×#16837206	C₂	1	4	yes	maximal
2	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
3	#21424260	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273661	C₂²	1	2	yes	—
7	C₂×#14273659	—	2	1	no	—
8	#16837268	—	2	1	no	—
9	#16837206	—	2	8	no	—
10	#16837178	—	2	2	no	—
11	C₂²×Q₁₆×C₅₉	—	2	1	no	—
12	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
14	C₅₉×#494	—	2	2	no	—
15	C₅₉×#494	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	—
21	#14273661	—	2	4	no	—
22	#14273659	—	4	2	no	—
23	C₅₉⋊#95	—	4	2	no	—
24	C₂²×Q₈×C₅₉	—	2	2	no	—
25	C₂×Q₁₆×C₅₉	—	2	4	no	—
26	C₂×Q₁₆×C₅₉	—	2	4	no	—
27	C₂×Q₁₆×C₅₉	—	2	2	no	—
28	C₂×Q₁₆×C₅₉	—	2	4	no	—
29	C₂×Q₁₆×C₅₉	—	4	2	no	—
30	C₂×Q₁₆×C₅₉	—	4	2	no	—
31	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₉	—	2	4	no	—
35	C₄⋊Q₈×C₅₉	—	4	2	no	—
36	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₉	D₄	1	2	yes	—
40	C₂×Q₈×C₅₉	—	2	4	no	—
41	C₂×Q₈×C₅₉	—	4	2	no	—
42	C₂×Q₈×C₅₉	—	4	2	no	—
43	C₂×Q₈×C₅₉	—	4	2	no	—
44	C₂×Q₈×C₅₉	—	4	2	no	—
45	C₂×Q₈×C₅₉	—	4	2	no	—
46	C₂²×C₄×C₅₉	—	4	2	no	—
47	C₂²×C₄×C₅₉	D₈	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₉	—	4	2	no	—
50	Q₁₆×C₅₉	—	2	8	no	—
51	Q₁₆×C₅₉	—	4	8	no	—
52	Q₁₆×C₅₉	—	4	4	no	—
53	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₉	D₈	1	2	yes	—
55	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	#41	1	2	yes	—
60	C₄⋊C₄×C₅₉	—	4	8	no	—
61	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₉	SD₁₆	1	2	yes	—
63	C₄⋊Dic₅₉	—	4	2	no	—
64	C₄⋊Dic₅₉	—	4	2	no	—
65	C₅₉⋊C₁₆	—	2	4	no	—
66	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₅₉	—	4	4	no	—
68	Q₈×C₅₉	—	4	4	no	—
69	Q₈×C₅₉	—	8	2	no	—
70	Q₈×C₅₉	—	8	2	no	—
71	C₂×C₄×C₅₉	—	2	4	no	—
72	C₂×C₄×C₅₉	—	4	4	no	—
73	C₂×C₄×C₅₉	—	8	2	no	—
74	C₂×C₄×C₅₉	—	8	2	no	—
75	C₂×C₄×C₅₉	#102	1	4	yes	—
76	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×Dic₅₉	—	8	2	no	—
83	C₂×Dic₅₉	—	8	4	no	—
84	C₈×C₅₉	#102	1	4	yes	—
85	C₈×C₅₉	C₄×D₄	1	2	yes	—
86	C₈×C₅₉	#121	1	2	yes	—
87	#13207	—	59	1	no	maximal, Sylow
88	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
89	C₂²×C₅₉	#486	1	1	yes	characteristic
90	C₂²×C₅₉	#493	1	1	yes	characteristic
91	C₂²×C₅₉	#510	1	4	yes	—
92	C₄×C₅₉	—	2	4	no	—
93	C₄×C₅₉	—	8	4	no	—
94	C₄×C₅₉	#390	1	1	yes	characteristic
95	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
96	C₄×C₅₉	#486	1	1	yes	characteristic
97	C₄×C₅₉	#492	1	1	yes	characteristic
98	Dic₅₉	—	16	2	no	—
99	C₂×#494	D₅₉	1	1	yes	characteristic
100	C₂×#358	—	59	4	no	—
101	C₂×#346	—	59	1	no	—
102	#1846	—	59	1	no	—
103	C₂×C₅₉	#1957	1	1	yes	characteristic
104	C₂×C₅₉	#2141	1	2	yes	—
105	C₂×C₅₉	#2245	1	2	yes	—
106	C₂×C₅₉	#2251	1	2	yes	—
107	C₂²×Q₁₆	—	2	1	no	—
108	C₂²×Q₁₆	C₂×D₅₉	1	2	yes	—
109	C₂×C₄⋊Q₈	—	2	1	no	—
110	C₂²×C₁₆	—	59	2	no	—
111	#494	—	2	2	no	—
112	#494	Dic₅₉	1	4	yes	complemented
113	C₂×#107	—	59	2	no	—
114	C₂×#106	—	118	1	no	—
115	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
116	#382	—	118	1	no	—
117	#358	—	118	8	no	—
118	#346	—	118	2	no	—
119	C₅₉	#13207	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	2	2	no	—
121	C₂×Q₁₆	—	2	4	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	4	no	—
124	C₂×Q₁₆	—	2	2	no	—
125	C₂×Q₁₆	—	4	2	no	—
126	C₂×Q₁₆	—	4	2	no	—
127	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
128	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	4	2	no	—
132	C₂×C₄⋊C₄	—	4	2	no	—
133	C₂×C₄⋊C₄	—	118	2	no	—
134	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
135	C₂×C₁₆	—	59	4	no	—
136	C₂×C₁₆	—	118	2	no	—
137	C₂×C₁₆	—	118	2	no	—
138	#107	—	118	4	no	—
139	#106	—	236	2	no	—
140	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
141	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
142	#95	—	236	2	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	236	2	no	—
151	C₂²×C₄	#11745669	1	2	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
153	Q₁₆	—	2	8	no	—
154	Q₁₆	—	4	8	no	—
155	Q₁₆	—	4	4	no	—
156	C₂×C₈	#11745669	1	2	yes	—
157	C₂×C₈	#11745670	1	2	yes	—
158	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
159	C₂×C₈	D₄×D₅₉	1	2	yes	—
160	C₂×C₈	#11745687	1	2	yes	—
161	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
163	C₄⋊C₄	—	4	8	no	—
164	C₄⋊C₄	—	236	2	no	—
165	C₄⋊C₄	—	236	2	no	—
166	C₄²	#11745670	1	2	yes	—
167	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
168	C₁₆	—	118	4	no	—
169	C₂³	C₂×#11745669	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	4	no	—
172	Q₈	—	8	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	472	2	no	—
179	C₂×C₄	—	472	4	no	—
180	C₂×C₄	#14273648	1	4	yes	—
181	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
182	C₂×C₄	#14273714	1	1	yes	characteristic
183	C₂×C₄	#14273723	1	1	yes	characteristic
184	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
185	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
186	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
187	C₈	#14273648	1	4	yes	—
188	C₈	D₄×Dic₅₉	1	2	yes	—
189	C₈	#14273698	1	2	yes	—
190	C₂²	C₂×#14273648	1	1	yes	characteristic
191	C₂²	#16837748	1	1	yes	characteristic
192	C₂²	#16837778	1	1	yes	characteristic
193	C₂²	#16837843	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	—	944	2	no	—
197	C₄	#16837305	1	1	yes	characteristic
198	C₄	C₂×#14273648	1	1	yes	characteristic
199	C₄	#16837748	1	1	yes	characteristic
200	C₄	#16837774	1	1	yes	characteristic
201	C₂	#21424903	1	1	yes	characteristic
202	C₂	#21425816	1	2	yes	—
203	C₂	#21426341	1	2	yes	—
204	C₂	#21426365	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837206

C₂

yes

maximal

C₂×#16837178

C₂

yes

maximal, characteristic

#21424260

C₂

yes

maximal, characteristic

C₂×C₅₉×#494

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

C₂×#14273661

C₂²

yes

—

—

—

—

—

C₂²×Q₁₆×C₅₉

—

C₂²×Q₁₆×C₅₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₉

—

C₅₉×#494

—

C₅₉×#494

C₄

yes

complemented

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

#14273661

—

#14273659

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

SD₁₆

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

#102

yes

—

C₈×C₅₉

C₄×D₄

yes

—

C₈×C₅₉

#121

yes

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

C₂×#494

D₅₉

yes

characteristic

100

C₂×#358

—

101

C₂×#346

—

102

#1846

—

103

C₂×C₅₉

#1957

yes

characteristic

104

C₂×C₅₉

#2141

yes

—

105

C₂×C₅₉

#2245

yes

—

106

C₂×C₅₉

#2251

yes

—

107

C₂²×Q₁₆

—

108

C₂²×Q₁₆

C₂×D₅₉

yes

—

109

C₂×C₄⋊Q₈

—

110

C₂²×C₁₆

—

111

#494

—

112

#494

Dic₅₉

yes

complemented

113

C₂×#107

—

114

C₂×#106

—

118

—

115

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

116

#382

—

118

—

117

#358

—

118

—

118

#346

—

118

—

119

C₅₉

#13207

yes

characteristic, complemented, Sylow

120

C₂²×Q₈

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

C₂×Dic₅₉

yes

—

128

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

129

C₂²×C₈

C₂²×D₅₉

yes

characteristic

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

118

—

134

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

135

C₂×C₁₆

—

136

C₂×C₁₆

—

118

—

137

C₂×C₁₆

—

118

—

138

#107

—

118

—

139

#106

—

236

—

140

C₄×C₈

C₂×Dic₅₉

yes

—

141

C₄×C₈

C₂²⋊D₅₉

yes

—

142

#95

—

236

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

—

236

—

151

C₂²×C₄

#11745669

yes

—

152

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

153

Q₁₆

—

154

Q₁₆

—

155

Q₁₆

—

156

C₂×C₈

#11745669

yes

—

157

C₂×C₈

#11745670

yes

—

158

C₂×C₈

C₂²⋊Dic₅₉

yes

—

159

C₂×C₈

D₄×D₅₉

yes

—

160

C₂×C₈

#11745687

yes

—

161

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

162

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

163

C₄⋊C₄

—

164

C₄⋊C₄

—

236

—

165

C₄⋊C₄

—

236

—

166

C₄²

#11745670

yes

—

167

C₄²

C₂²⋊Dic₅₉

yes

—

168

C₁₆

—

118

—

169

C₂³

C₂×#11745669

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

472

—

179

C₂×C₄

—

472

—

180

C₂×C₄

#14273648

yes

—

181

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

182

C₂×C₄

#14273714

yes

characteristic

183

C₂×C₄

#14273723

yes

characteristic

184

C₂×C₄

C₂×#11745669

yes

characteristic

185

C₂×C₄

C₂×#11745670

yes

characteristic

186

C₂×C₄

C₂×#11745670

yes

characteristic

187

C₈

#14273648

yes

—

188

C₈

D₄×Dic₅₉

yes

—

189

C₈

#14273698

yes

—

190

C₂²

C₂×#14273648

yes

characteristic

191

C₂²

#16837748

yes

characteristic

192

C₂²

#16837778

yes

characteristic

193

C₂²

#16837843

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

944

—

197

C₄

#16837305

yes

characteristic

198

C₄

C₂×#14273648

yes

characteristic

199

C₄

#16837748

yes

characteristic

200

C₄

#16837774

yes

characteristic

201

C₂

#21424903

yes

characteristic

202

C₂

#21425816

yes

—

203

C₂

#21426341

yes

—

204

C₂

#21426365

yes

—

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	#13207
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#494

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 ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
15. C₅₉×#494	C₄	semidirect
112. #494	Dic₅₉	semidirect
119. C₅₉	#13207	semidirect

H ≅ G/N

type

15. C₅₉×#494

C₄

semidirect

112. #494

Dic₅₉

semidirect

119. C₅₉

#13207

semidirect

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

subgroup	index
C₂×#16837206	2
C₂×#16837178	2
#21424260	2
C₂×C₅₉×#494	2
#13207	59

subgroup

index

C₂×C₅₉×#494

Subgroups — #28506926

Conjugacy classes of subgroups

Complements

Maximal subgroups