Group #28422794 — subgroups

Subgroups — C₅₉×#10685 #28422794

Order 15104.

442 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 178 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	C₅₉×#1990	C₂	1	2	yes	maximal
2	C₅₉×#1956	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1844	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1844	C₂	1	2	yes	maximal
5	C₅₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
7	C₂×#106×C₅₉	—	2	2	no	—
8	C₂×#106×C₅₉	—	2	4	no	—
9	C₂×#103×C₅₉	—	2	2	no	—
10	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#384	C₂²	1	2	yes	—
13	C₅₉×#384	C₂²	1	1	yes	characteristic
14	C₅₉×#340	—	2	4	no	—
15	C₂²×Q₈×C₅₉	—	2	1	no	—
16	C₂²×C₈×C₅₉	D₄	1	4	yes	—
17	C₂²×C₈×C₅₉	D₄	1	2	yes	—
18	C₄⋊Q₈×C₅₉	—	2	2	no	—
19	C₄⋊Q₈×C₅₉	—	2	1	no	—
20	C₄⋊Q₈×C₅₉	Q₈	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
24	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
27	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
28	#106×C₅₉	—	4	4	no	—
29	#106×C₅₉	—	4	8	no	—
30	#103×C₅₉	—	4	4	no	—
31	C₄×C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	—	2	2	no	—
33	C₄×C₈×C₅₉	—	2	2	no	—
34	C₂×Q₈×C₅₉	—	2	2	no	—
35	C₂×Q₈×C₅₉	—	4	1	no	—
36	C₂×Q₈×C₅₉	—	4	2	no	—
37	C₂×Q₈×C₅₉	—	4	2	no	—
38	C₂²×C₄×C₅₉	—	4	1	no	—
39	C₂²×C₄×C₅₉	—	4	1	no	—
40	C₂²×C₄×C₅₉	—	4	2	no	—
41	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	4	no	—
45	C₂×C₈×C₅₉	—	2	2	no	—
46	C₂×C₈×C₅₉	—	2	4	no	—
47	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
48	C₂×C₈×C₅₉	SD₁₆	1	8	yes	—
49	C₄⋊C₄×C₅₉	—	2	4	no	—
50	C₄⋊C₄×C₅₉	—	4	2	no	—
51	C₄⋊C₄×C₅₉	—	4	2	no	—
52	C₄⋊C₄×C₅₉	—	4	4	no	—
53	C₄⋊C₄×C₅₉	—	4	4	no	—
54	C₄⋊C₄×C₅₉	—	4	4	no	—
55	C₄⋊C₄×C₅₉	—	4	4	no	—
56	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₅₉	#41	1	2	yes	—
58	C₄²×C₅₉	#41	1	1	yes	characteristic
59	C₂³×C₅₉	#127	1	1	yes	characteristic
60	Q₈×C₅₉	—	4	4	no	—
61	Q₈×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	—	4	2	no	—
63	C₂×C₄×C₅₉	—	8	1	no	—
64	C₂×C₄×C₅₉	—	8	1	no	—
65	C₂×C₄×C₅₉	—	8	1	no	—
66	C₂×C₄×C₅₉	—	8	2	no	—
67	C₂×C₄×C₅₉	—	8	2	no	—
68	C₂×C₄×C₅₉	—	8	2	no	—
69	C₂×C₄×C₅₉	—	8	2	no	—
70	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#123	1	2	yes	—
74	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
75	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₅₉	#137	1	4	yes	—
78	C₈×C₅₉	—	2	4	no	—
79	C₈×C₅₉	—	2	8	no	—
80	#10685	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂²×C₅₉	#399	1	1	yes	characteristic
82	C₂²×C₅₉	#492	1	1	yes	characteristic
83	C₂²×C₅₉	#492	1	2	yes	—
84	C₂²×C₅₉	#496	1	1	yes	characteristic
85	C₂²×C₅₉	#497	1	2	yes	—
86	C₄×C₅₉	—	8	2	no	—
87	C₄×C₅₉	—	16	1	no	—
88	C₄×C₅₉	—	16	2	no	—
89	C₄×C₅₉	#475	1	4	yes	—
90	C₄×C₅₉	#476	1	2	yes	—
91	C₄×C₅₉	#481	1	2	yes	—
92	C₄×C₅₉	#483	1	4	yes	—
93	#1990	C₂×C₅₉	1	2	yes	—
94	#1956	C₂×C₅₉	1	1	yes	characteristic
95	#1844	C₂×C₅₉	1	2	yes	—
96	#1844	C₂×C₅₉	1	1	yes	characteristic
97	#1273	C₂×C₅₉	1	1	yes	characteristic
98	C₂×C₅₉	#1705	1	1	yes	characteristic
99	C₂×C₅₉	#1714	1	2	yes	—
100	C₂×C₅₉	#1716	1	1	yes	characteristic
101	C₂×C₅₉	#2094	1	1	yes	characteristic
102	C₂×C₅₉	#2096	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
104	C₂×#106	—	2	2	no	—
105	C₂×#106	—	2	4	no	—
106	C₂×#103	—	2	2	no	—
107	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
109	#384	C₂²×C₅₉	1	1	yes	characteristic
110	#384	C₂²×C₅₉	1	2	yes	—
111	#340	—	2	4	no	—
112	C₅₉	#10685	1	1	yes	characteristic, direct product factor, Sylow
113	C₂²×Q₈	—	2	1	no	—
114	C₂²×C₈	D₄×C₅₉	1	4	yes	—
115	C₂²×C₈	D₄×C₅₉	1	2	yes	—
116	C₄⋊Q₈	—	2	2	no	—
117	C₄⋊Q₈	—	2	1	no	—
118	C₄⋊Q₈	Q₈×C₅₉	1	2	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
126	#106	—	4	4	no	—
127	#106	—	4	8	no	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	SD₁₆×C₅₉	1	4	yes	—
146	C₂×C₈	SD₁₆×C₅₉	1	8	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄²	C₂×Q₈×C₅₉	1	1	yes	characteristic
155	C₄²	#41×C₅₉	1	2	yes	—
156	C₄²	#41×C₅₉	1	1	yes	characteristic
157	C₂³	#127×C₅₉	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	1	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₂²⋊Q₈×C₅₉	1	2	yes	—
169	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
170	C₂×C₄	#123×C₅₉	1	1	yes	characteristic
171	C₂×C₄	#123×C₅₉	1	2	yes	—
172	C₂×C₄	C₂×SD₁₆×C₅₉	1	4	yes	—
173	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
174	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
175	C₂×C₄	#137×C₅₉	1	4	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	C₅₉×#399	1	1	yes	characteristic
179	C₂²	C₅₉×#492	1	2	yes	—
180	C₂²	C₅₉×#492	1	1	yes	characteristic
181	C₂²	C₅₉×#496	1	1	yes	characteristic
182	C₂²	C₅₉×#497	1	2	yes	—
183	C₄	—	8	2	no	—
184	C₄	—	16	2	no	—
185	C₄	—	16	1	no	—
186	C₄	C₅₉×#475	1	4	yes	—
187	C₄	C₅₉×#476	1	2	yes	—
188	C₄	C₅₉×#481	1	2	yes	—
189	C₄	C₅₉×#483	1	4	yes	—
190	C₂	C₅₉×#1705	1	1	yes	characteristic
191	C₂	C₅₉×#1714	1	2	yes	—
192	C₂	C₅₉×#1716	1	1	yes	characteristic
193	C₂	C₅₉×#2094	1	1	yes	characteristic
194	C₂	C₅₉×#2096	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₉×#1990

C₂

yes

maximal

C₅₉×#1956

C₂

yes

maximal, characteristic

C₅₉×#1844

C₂

yes

maximal, characteristic

C₅₉×#1844

C₂

yes

maximal

C₅₉×#1273

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₂×#106×C₅₉

—

C₂×#106×C₅₉

—

C₂×#103×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#384

C₂²

yes

—

C₅₉×#384

C₂²

yes

characteristic

C₅₉×#340

—

C₂²×Q₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

Q₈

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#106×C₅₉

—

#106×C₅₉

—

#103×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄²×C₅₉

yes

C₂³×C₅₉

yes

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

#10685

C₅₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#496

yes

characteristic

C₂²×C₅₉

#497

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#475

yes

—

C₄×C₅₉

#476

yes

—

C₄×C₅₉

#481

yes

—

C₄×C₅₉

#483

yes

—

#1990

C₂×C₅₉

yes

—

#1956

C₂×C₅₉

yes

characteristic

#1844

C₂×C₅₉

yes

—

#1844

C₂×C₅₉

yes

characteristic

#1273

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

100

C₂×C₅₉

#1716

yes

characteristic

101

C₂×C₅₉

#2094

yes

characteristic

102

C₂×C₅₉

#2096

yes

—

103

C₂×C₄⋊Q₈

C₂²×C₅₉

yes

characteristic

104

C₂×#106

—

105

C₂×#106

—

106

C₂×#103

—

107

C₂×C₄×C₈

C₂²×C₅₉

yes

characteristic

108

C₂×C₄×C₈

C₂²×C₅₉

yes

—

109

#384

C₂²×C₅₉

yes

characteristic

110

#384

C₂²×C₅₉

yes

—

111

#340

—

112

C₅₉

#10685

yes

characteristic, direct product factor, Sylow

113

C₂²×Q₈

—

114

C₂²×C₈

D₄×C₅₉

yes

—

115

C₂²×C₈

D₄×C₅₉

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

Q₈×C₅₉

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂³×C₅₉

yes

characteristic

126

#106

—

127

#106

—

128

#103

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

C₂×D₄×C₅₉

yes

characteristic

140

C₂²×C₄

C₂×D₄×C₅₉

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

SD₁₆×C₅₉

yes

—

146

C₂×C₈

SD₁₆×C₅₉

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄²

C₂×Q₈×C₅₉

yes

characteristic

155

C₄²

#41×C₅₉

yes

—

156

C₄²

#41×C₅₉

yes

characteristic

157

C₂³

#127×C₅₉

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

C₂²⋊Q₈×C₅₉

yes

—

169

C₂×C₄

C₂²⋊Q₈×C₅₉

yes

characteristic

170

C₂×C₄

#123×C₅₉

yes

characteristic

171

C₂×C₄

#123×C₅₉

yes

—

172

C₂×C₄

C₂×SD₁₆×C₅₉

yes

—

173

C₂×C₄

C₂×SD₁₆×C₅₉

yes

—

174

C₂×C₄

AGL(1,ℤ/8)×C₅₉

yes

—

175

C₂×C₄

#137×C₅₉

yes

—

176

C₈

—

177

C₈

—

178

C₂²

C₅₉×#399

yes

characteristic

179

C₂²

C₅₉×#492

yes

—

180

C₂²

C₅₉×#492

yes

characteristic

181

C₂²

C₅₉×#496

yes

characteristic

182

C₂²

C₅₉×#497

yes

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

C₅₉×#475

yes

—

187

C₄

C₅₉×#476

yes

—

188

C₄

C₅₉×#481

yes

—

189

C₄

C₅₉×#483

yes

—

190

C₂

C₅₉×#1705

yes

characteristic

191

C₂

C₅₉×#1714

yes

—

192

C₂

C₅₉×#1716

yes

characteristic

193

C₂

C₅₉×#2094

yes

characteristic

194

C₂

C₅₉×#2096

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅₉

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₉

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

N	H ≅ G/N	type
80. #10685	C₅₉	direct (N × H)
112. C₅₉	#10685	direct (N × H)

H ≅ G/N

type

80. #10685

C₅₉

direct (N × H)

112. C₅₉

#10685

direct (N × H)

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

subgroup	index
C₅₉×#1990	2
C₅₉×#1956	2
C₅₉×#1844	2
C₅₉×#1273	2
#10685	59

subgroup

index

C₅₉×#1990

C₅₉×#1956

C₅₉×#1844

C₅₉×#1273

#10685

Subgroups — C₅₉×#10685 #28422794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C59×#10685 #28422794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#10685 #28422794