Group #28423279 — subgroups

Subgroups — C₅₉×#11170 #28423279

Order 15104.

962 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 222 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×#100×C₅₉	C₂	1	2	yes	maximal
2	C₅₉×#2309	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#344	C₂	1	4	yes	maximal, complemented
4	C₂³×D₄×C₅₉	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₅₉	C₂²	1	4	yes	—
6	C₂×#117×C₅₉	C₂²	1	2	yes	—
7	C₂×#100×C₅₉	—	2	4	no	—
8	C₂×#100×C₅₉	—	2	8	no	—
9	C₅₉×#390	—	2	4	no	—
10	C₅₉×#379	C₄	1	4	yes	—
11	C₅₉×#344	—	2	8	no	—
12	C₂⁵×C₅₉	C₂×C₄	1	2	yes	—
13	C₂²×D₄×C₅₉	—	2	4	no	—
14	C₂²×D₄×C₅₉	—	2	4	no	—
15	C₂²×D₄×C₅₉	—	2	4	no	—
16	C₂²×D₄×C₅₉	D₄	1	4	yes	—
17	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₅₉	—	2	4	no	—
19	C₂×M₁₆×C₅₉	—	2	4	no	—
20	C₂×M₁₆×C₅₉	—	2	4	no	—
21	C₂×M₁₆×C₅₉	—	2	4	no	—
22	C₂×M₁₆×C₅₉	—	2	4	no	—
23	C₂×M₁₆×C₅₉	D₄	1	8	yes	—
24	C₂²×C₈×C₅₉	—	2	4	no	—
25	#117×C₅₉	—	2	4	no	—
26	#117×C₅₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₅₉	—	2	8	no	—
29	C₂×C₂²⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₉	—	2	2	no	—
31	#100×C₅₉	—	2	16	no	—
32	C₂⁴×C₅₉	—	2	4	no	—
33	C₂⁴×C₅₉	—	2	8	no	—
34	C₂⁴×C₅₉	—	8	4	no	—
35	C₂⁴×C₅₉	C₂²⋊C₄	1	4	yes	—
36	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×D₄×C₅₉	—	2	4	no	—
38	C₂×D₄×C₅₉	—	2	8	no	—
39	C₂×D₄×C₅₉	—	2	8	no	—
40	C₂×D₄×C₅₉	—	2	4	no	—
41	C₂×D₄×C₅₉	—	4	4	no	—
42	C₂×D₄×C₅₉	—	4	4	no	—
43	C₂×D₄×C₅₉	—	4	4	no	—
44	C₂×D₄×C₅₉	—	4	4	no	—
45	C₂²×C₄×C₅₉	—	2	4	no	—
46	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₉	#41	1	4	yes	—
49	C₂²×C₄×C₅₉	#41	1	4	yes	—
50	M₁₆×C₅₉	—	2	8	no	—
51	M₁₆×C₅₉	—	2	8	no	—
52	M₁₆×C₅₉	—	4	4	no	—
53	M₁₆×C₅₉	—	4	4	no	—
54	C₂×C₈×C₅₉	—	2	8	no	—
55	C₂×C₈×C₅₉	—	4	4	no	—
56	C₂×C₈×C₅₉	—	4	4	no	—
57	C₄⋊C₄×C₅₉	—	4	2	no	—
58	C₄⋊C₄×C₅₉	—	4	2	no	—
59	C₂²⋊C₄×C₅₉	—	2	8	no	—
60	C₂²⋊C₄×C₅₉	—	8	8	no	—
61	C₄²×C₅₉	—	4	4	no	—
62	C₂³×C₅₉	—	2	4	no	—
63	C₂³×C₅₉	—	2	8	no	—
64	C₂³×C₅₉	—	4	2	no	—
65	C₂³×C₅₉	—	4	8	no	—
66	C₂³×C₅₉	—	4	8	no	—
67	C₂³×C₅₉	—	8	8	no	—
68	C₂³×C₅₉	—	8	16	no	—
69	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂³×C₅₉	#117	1	2	yes	—
71	C₂³×C₅₉	C₄×D₄	1	4	yes	—
72	D₄×C₅₉	—	4	8	no	—
73	D₄×C₅₉	—	4	8	no	—
74	C₂×C₄×C₅₉	—	4	8	no	—
75	C₂×C₄×C₅₉	—	4	4	no	—
76	C₂×C₄×C₅₉	#120	1	2	yes	—
77	C₂×C₄×C₅₉	#121	1	4	yes	—
78	C₂×C₄×C₅₉	#121	1	4	yes	—
79	C₂×C₄×C₅₉	#123	1	4	yes	—
80	C₂×C₄×C₅₉	#123	1	4	yes	—
81	C₂×C₄×C₅₉	#123	1	2	yes	—
82	C₂×C₄×C₅₉	#124	1	4	yes	—
83	C₂×C₄×C₅₉	#126	1	4	yes	—
84	C₈×C₅₉	—	4	8	no	—
85	#11170	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₅₉	—	2	2	no	—
87	C₂²×C₅₉	—	2	4	no	—
88	C₂²×C₅₉	—	4	4	no	—
89	C₂²×C₅₉	—	4	8	no	—
90	C₂²×C₅₉	—	4	4	no	—
91	C₂²×C₅₉	—	8	8	no	—
92	C₂²×C₅₉	—	8	16	no	—
93	C₂²×C₅₉	#381	1	1	yes	characteristic
94	C₂²×C₅₉	#386	1	2	yes	—
95	C₂²×C₅₉	#388	1	4	yes	—
96	C₄×C₅₉	—	8	4	no	—
97	C₄×C₅₉	#394	1	4	yes	—
98	C₄×C₅₉	#397	1	4	yes	—
99	C₂²×#100	C₂×C₅₉	1	2	yes	—
100	#2309	C₂×C₅₉	1	1	yes	characteristic
101	C₂×#344	C₂×C₅₉	1	4	yes	complemented
102	C₂×C₅₉	—	2	2	no	—
103	C₂×C₅₉	—	2	4	no	—
104	C₂×C₅₉	—	8	4	no	—
105	C₂×C₅₉	#1440	1	1	yes	characteristic
106	C₂×C₅₉	#1829	1	2	yes	—
107	C₂³×D₄	C₂²×C₅₉	1	1	yes	characteristic
108	C₂²×M₁₆	C₂²×C₅₉	1	4	yes	—
109	C₂×#117	C₂²×C₅₉	1	2	yes	—
110	C₂×#100	—	2	4	no	—
111	C₂×#100	—	2	8	no	—
112	#390	—	2	4	no	—
113	#379	C₄×C₅₉	1	4	yes	—
114	#344	—	2	8	no	—
115	C₅₉	#11170	1	1	yes	characteristic, direct product factor, Sylow
116	C₂⁵	C₂×C₄×C₅₉	1	2	yes	—
117	C₂²×D₄	—	2	4	no	—
118	C₂²×D₄	—	2	4	no	—
119	C₂²×D₄	—	2	4	no	—
120	C₂²×D₄	D₄×C₅₉	1	4	yes	—
121	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	—	2	4	no	—
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	—	2	4	no	—
126	C₂×M₁₆	—	2	4	no	—
127	C₂×M₁₆	D₄×C₅₉	1	8	yes	—
128	C₂²×C₈	—	2	4	no	—
129	#117	—	2	4	no	—
130	#117	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₂²⋊C₄	—	2	8	no	—
133	C₂×C₂²⋊C₄	C₂×C₄×C₅₉	1	4	yes	—
134	C₂×C₄²	—	2	2	no	—
135	#100	—	2	16	no	—
136	C₂⁴	—	2	4	no	—
137	C₂⁴	—	2	8	no	—
138	C₂⁴	—	8	4	no	—
139	C₂⁴	C₂²⋊C₄×C₅₉	1	4	yes	—
140	C₂⁴	C₂²×C₄×C₅₉	1	1	yes	characteristic
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	—	2	8	no	—
143	C₂×D₄	—	2	8	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	—	4	4	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
151	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
152	C₂²×C₄	#41×C₅₉	1	4	yes	—
153	C₂²×C₄	#41×C₅₉	1	4	yes	—
154	M₁₆	—	2	8	no	—
155	M₁₆	—	2	8	no	—
156	M₁₆	—	4	4	no	—
157	M₁₆	—	4	4	no	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	4	4	no	—
160	C₂×C₈	—	4	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	2	8	no	—
164	C₂²⋊C₄	—	8	8	no	—
165	C₄²	—	4	4	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	4	2	no	—
169	C₂³	—	4	8	no	—
170	C₂³	—	4	8	no	—
171	C₂³	—	8	8	no	—
172	C₂³	—	8	16	no	—
173	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
174	C₂³	#117×C₅₉	1	2	yes	—
175	C₂³	C₄×D₄×C₅₉	1	4	yes	—
176	D₄	—	4	8	no	—
177	D₄	—	4	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	#120×C₅₉	1	2	yes	—
181	C₂×C₄	#121×C₅₉	1	4	yes	—
182	C₂×C₄	#121×C₅₉	1	4	yes	—
183	C₂×C₄	#123×C₅₉	1	4	yes	—
184	C₂×C₄	#123×C₅₉	1	4	yes	—
185	C₂×C₄	#123×C₅₉	1	2	yes	—
186	C₂×C₄	#124×C₅₉	1	4	yes	—
187	C₂×C₄	#126×C₅₉	1	4	yes	—
188	C₈	—	4	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	4	4	no	—
192	C₂²	—	4	8	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	8	8	no	—
195	C₂²	—	8	16	no	—
196	C₂²	C₅₉×#381	1	1	yes	characteristic
197	C₂²	C₅₉×#386	1	2	yes	—
198	C₂²	C₅₉×#388	1	4	yes	—
199	C₄	—	8	4	no	—
200	C₄	C₅₉×#394	1	4	yes	—
201	C₄	C₅₉×#397	1	4	yes	—
202	C₂	—	2	2	no	—
203	C₂	—	2	4	no	—
204	C₂	—	8	4	no	—
205	C₂	C₅₉×#1440	1	1	yes	characteristic
206	C₂	C₅₉×#1829	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#100×C₅₉

C₂

yes

maximal

C₅₉×#2309

C₂

yes

maximal, characteristic

C₂×C₅₉×#344

C₂

yes

maximal, complemented

C₂³×D₄×C₅₉

C₂²

yes

characteristic

C₂²×M₁₆×C₅₉

C₂²

yes

—

C₂×#117×C₅₉

C₂²

yes

—

C₂×#100×C₅₉

—

C₂×#100×C₅₉

—

C₅₉×#390

—

C₅₉×#379

C₄

yes

—

C₅₉×#344

—

C₂⁵×C₅₉

C₂×C₄

yes

—

C₂²×D₄×C₅₉

—

C₂²×D₄×C₅₉

—

C₂²×D₄×C₅₉

—

C₂²×D₄×C₅₉

D₄

yes

—

C₂³×C₄×C₅₉

C₂³

yes

characteristic

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

—

#117×C₅₉

—

#117×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₂²⋊C₄×C₅₉

—

C₂×C₂²⋊C₄×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

—

#100×C₅₉

—

C₂⁴×C₅₉

—

C₂⁴×C₅₉

—

C₂⁴×C₅₉

—

C₂⁴×C₅₉

C₂²⋊C₄

yes

—

C₂⁴×C₅₉

C₂²×C₄

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×C₄×C₅₉

#41

yes

—

C₂²×C₄×C₅₉

#41

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₅₉

#117

yes

—

C₂³×C₅₉

C₄×D₄

yes

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#120

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

#126

yes

—

C₈×C₅₉

—

#11170

C₅₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#381

yes

characteristic

C₂²×C₅₉

#386

yes

—

C₂²×C₅₉

#388

yes

—

C₄×C₅₉

—

C₄×C₅₉

#394

yes

—

C₄×C₅₉

#397

yes

—

C₂²×#100

C₂×C₅₉

yes

—

100

#2309

C₂×C₅₉

yes

characteristic

101

C₂×#344

C₂×C₅₉

yes

complemented

102

C₂×C₅₉

—

103

C₂×C₅₉

—

104

C₂×C₅₉

—

105

C₂×C₅₉

#1440

yes

characteristic

106

C₂×C₅₉

#1829

yes

—

107

C₂³×D₄

C₂²×C₅₉

yes

characteristic

108

C₂²×M₁₆

C₂²×C₅₉

yes

—

109

C₂×#117

C₂²×C₅₉

yes

—

110

C₂×#100

—

111

C₂×#100

—

112

#390

—

113

#379

C₄×C₅₉

yes

—

114

#344

—

115

C₅₉

#11170

yes

characteristic, direct product factor, Sylow

116

C₂⁵

C₂×C₄×C₅₉

yes

—

117

C₂²×D₄

—

118

C₂²×D₄

—

119

C₂²×D₄

—

120

C₂²×D₄

D₄×C₅₉

yes

—

121

C₂³×C₄

C₂³×C₅₉

yes

characteristic

122

C₂×M₁₆

—

123

C₂×M₁₆

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂×M₁₆

D₄×C₅₉

yes

—

128

C₂²×C₈

—

129

#117

—

130

#117

—

131

C₂×C₄⋊C₄

—

132

C₂×C₂²⋊C₄

—

133

C₂×C₂²⋊C₄

C₂×C₄×C₅₉

yes

—

134

C₂×C₄²

—

135

#100

—

136

C₂⁴

—

137

C₂⁴

—

138

C₂⁴

—

139

C₂⁴

C₂²⋊C₄×C₅₉

yes

—

140

C₂⁴

C₂²×C₄×C₅₉

yes

characteristic

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂×D₄×C₅₉

yes

—

151

C₂²×C₄

C₂×D₄×C₅₉

yes

—

152

C₂²×C₄

#41×C₅₉

yes

—

153

C₂²×C₄

#41×C₅₉

yes

—

154

M₁₆

—

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

C₂×C₂²⋊C₄×C₅₉

yes

characteristic

174

C₂³

#117×C₅₉

yes

—

175

C₂³

C₄×D₄×C₅₉

yes

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#120×C₅₉

yes

—

181

C₂×C₄

#121×C₅₉

yes

—

182

C₂×C₄

#121×C₅₉

yes

—

183

C₂×C₄

#123×C₅₉

yes

—

184

C₂×C₄

#123×C₅₉

yes

—

185

C₂×C₄

#123×C₅₉

yes

—

186

C₂×C₄

#124×C₅₉

yes

—

187

C₂×C₄

#126×C₅₉

yes

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₅₉×#381

yes

characteristic

197

C₂²

C₅₉×#386

yes

—

198

C₂²

C₅₉×#388

yes

—

199

C₄

—

200

C₄

C₅₉×#394

yes

—

201

C₄

C₅₉×#397

yes

—

202

C₂

—

203

C₂

—

204

C₂

—

205

C₂

C₅₉×#1440

yes

characteristic

206

C₂

C₅₉×#1829

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₉	#381
commutator	C₂⁴	C₂²×C₄×C₅₉
Frattini, cosocle	C₂³×C₄	C₂³×C₅₉
prime radical	C₂³×D₄×C₅₉	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₉

#381

commutator

C₂⁴

C₂²×C₄×C₅₉

Frattini, cosocle

C₂³×C₄

C₂³×C₅₉

prime radical

C₂³×D₄×C₅₉

C₂²

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ G

N	H ≅ G/N	type
3. C₂×C₅₉×#344	C₂	semidirect
85. #11170	C₅₉	direct (N × H)
101. C₂×#344	C₂×C₅₉	semidirect
115. C₅₉	#11170	direct (N × H)

H ≅ G/N

type

3. C₂×C₅₉×#344

C₂

semidirect

85. #11170

C₅₉

direct (N × H)

101. C₂×#344

C₂×C₅₉

semidirect

115. C₅₉

#11170

direct (N × H)

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

subgroup	index
C₂²×#100×C₅₉	2
C₅₉×#2309	2
C₂×C₅₉×#344	2
#11170	59

subgroup

index

C₂²×#100×C₅₉

C₅₉×#2309

C₂×C₅₉×#344

#11170

Subgroups — C₅₉×#11170 #28423279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C59×#11170 #28423279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#11170 #28423279