Group #28493145 — subgroups

Subgroups — #28493145

Order 15104.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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	#21424854	C₂	1	2	yes	maximal
2	#21424252	C₂	1	1	yes	maximal, characteristic
3	#21421979	C₂	1	1	yes	maximal, characteristic
4	#21421937	C₂	1	2	yes	maximal
5	C₅₉×#2571	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273660	—	2	2	no	—
7	C₂×#14273658	—	2	2	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#16837272	C₂²	1	1	yes	characteristic
11	#16837164	—	2	2	no	—
12	C₅₉×#400	—	2	2	no	—
13	C₅₉×#389	—	2	2	no	—
14	C₅₉×#384	C₂²	1	1	yes	characteristic
15	C₅₉×#382	—	2	2	no	—
16	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
17	C₂²×C₅₉⋊C₈	—	2	1	no	—
18	C₂²×C₅₉⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	1	no	—
21	#14273660	—	4	4	no	—
22	#14273658	—	4	4	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	Q₈	1	4	yes	—
25	C₄×C₅₉⋊C₈	—	2	1	no	—
26	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
32	C₂×C₄²×C₅₉	—	2	2	no	—
33	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	#95×C₅₉	—	2	2	no	—
35	#95×C₅₉	—	2	4	no	—
36	#95×C₅₉	—	2	2	no	—
37	C₂²×C₄×C₅₉	—	2	2	no	—
38	C₂²×C₄×C₅₉	—	2	2	no	—
39	C₂²×C₄×C₅₉	—	2	2	no	—
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
42	C₂²×Dic₅₉	—	4	1	no	—
43	C₄⋊C₄×C₅₉	—	2	2	no	—
44	C₄⋊C₄×C₅₉	—	2	2	no	—
45	C₄⋊C₄×C₅₉	—	2	2	no	—
46	C₄⋊C₄×C₅₉	—	2	2	no	—
47	C₄⋊C₄×C₅₉	—	4	2	no	—
48	C₄⋊C₄×C₅₉	—	4	2	no	—
49	C₄⋊C₄×C₅₉	—	4	2	no	—
50	C₄⋊C₄×C₅₉	—	8	2	no	—
51	C₄²×C₅₉	—	4	2	no	—
52	C₄²×C₅₉	—	4	2	no	—
53	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
55	C₄²×C₅₉	#41	1	1	yes	characteristic
56	C₄⋊Dic₅₉	—	4	2	no	—
57	C₄⋊Dic₅₉	—	4	2	no	—
58	C₄⋊Dic₅₉	—	4	2	no	—
59	C₂×C₅₉⋊C₈	—	2	2	no	—
60	C₂×C₅₉⋊C₈	—	2	2	no	—
61	C₂×C₅₉⋊C₈	—	2	2	no	—
62	C₂×C₅₉⋊C₈	—	2	4	no	—
63	C₂×C₅₉⋊C₈	—	4	2	no	—
64	C₂×C₅₉⋊C₈	—	4	2	no	—
65	C₂³×C₅₉	#120	1	1	yes	characteristic
66	C₂×C₄×C₅₉	—	2	1	no	—
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	—	4	2	no	—
69	C₂×C₄×C₅₉	—	4	2	no	—
70	C₂×C₄×C₅₉	—	4	2	no	—
71	C₂×C₄×C₅₉	—	4	2	no	—
72	C₂×C₄×C₅₉	—	4	4	no	—
73	C₂×C₄×C₅₉	—	4	2	no	—
74	C₂×C₄×C₅₉	—	4	2	no	—
75	C₂×C₄×C₅₉	—	4	2	no	—
76	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₅₉	#135	1	2	yes	—
81	C₂×C₄×C₅₉	#135	1	2	yes	—
82	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₅₉	#137	1	2	yes	—
84	C₂×Dic₅₉	—	8	2	no	—
85	C₂×Dic₅₉	—	8	1	no	—
86	C₅₉⋊C₈	—	2	4	no	—
87	C₅₉⋊C₈	—	4	4	no	—
88	#10577	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	#395	1	1	yes	characteristic
90	C₂²×C₅₉	#449	1	2	yes	—
91	C₂²×C₅₉	#452	1	2	yes	—
92	C₂²×C₅₉	#453	1	1	yes	characteristic
93	C₂²×C₅₉	#456	1	1	yes	characteristic
94	C₄×C₅₉	—	2	1	no	—
95	C₄×C₅₉	—	2	1	no	—
96	C₄×C₅₉	—	8	2	no	—
97	C₄×C₅₉	—	8	2	no	—
98	C₄×C₅₉	—	8	2	no	—
99	C₄×C₅₉	#478	1	2	yes	—
100	C₄×C₅₉	#479	1	2	yes	—
101	C₄×C₅₉	#499	1	2	yes	—
102	C₄×C₅₉	#501	1	2	yes	—
103	Dic₅₉	—	16	1	no	—
104	#2571	D₅₉	1	1	yes	characteristic
105	#1947	—	59	2	no	—
106	#1844	—	59	1	no	—
107	#1296	—	59	1	no	—
108	#1279	—	59	2	no	—
109	C₂×C₅₉	#1624	1	1	yes	characteristic
110	C₂×C₅₉	#1630	1	1	yes	characteristic
111	C₂×C₅₉	#1652	1	2	yes	—
112	C₂×C₅₉	#2059	1	2	yes	—
113	C₂×C₅₉	#2060	1	1	yes	characteristic
114	C₂×#107	—	118	2	no	—
115	C₂×#106	—	118	2	no	—
116	C₂×C₄⋊C₈	—	59	2	no	—
117	C₂×C₄×C₈	—	59	1	no	—
118	#400	—	2	2	no	—
119	#389	—	2	2	no	—
120	#384	—	59	1	no	—
121	#384	C₂×D₅₉	1	1	yes	characteristic
122	#382	—	2	2	no	—
123	C₄×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
124	#340	—	118	2	no	—
125	C₅₉	#10577	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	118	1	no	—
127	C₂²×C₈	—	118	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	118	2	no	—
131	C₂×C₄⋊C₄	—	118	1	no	—
132	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
137	#107	—	236	4	no	—
138	#106	—	236	4	no	—
139	C₄⋊C₈	—	59	4	no	—
140	C₄⋊C₈	—	118	2	no	—
141	C₄×C₈	—	59	2	no	—
142	C₄×C₈	—	118	1	no	—
143	#95	—	2	2	no	—
144	#95	—	2	4	no	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	236	1	no	—
150	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
152	C₂×C₈	—	118	2	no	—
153	C₂×C₈	—	118	2	no	—
154	C₂×C₈	—	118	2	no	—
155	C₂×C₈	—	118	4	no	—
156	C₂×C₈	—	236	2	no	—
157	C₂×C₈	—	236	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	8	2	no	—
166	C₄⋊C₄	—	236	2	no	—
167	C₄⋊C₄	—	236	2	no	—
168	C₄⋊C₄	—	236	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	Q₈×D₅₉	1	1	yes	characteristic
172	C₄²	Q₈×D₅₉	1	2	yes	—
173	C₄²	#11745689	1	1	yes	characteristic
174	C₂³	#14273696	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	472	2	no	—
186	C₂×C₄	—	472	1	no	—
187	C₂×C₄	#14273703	1	1	yes	characteristic
188	C₂×C₄	#14273703	1	2	yes	—
189	C₂×C₄	#14273725	1	1	yes	characteristic
190	C₂×C₄	#14273725	1	2	yes	—
191	C₂×C₄	#14273747	1	2	yes	—
192	C₂×C₄	#14273747	1	2	yes	—
193	C₂×C₄	#14273752	1	2	yes	—
194	C₂×C₄	#14273757	1	2	yes	—
195	C₈	—	118	4	no	—
196	C₈	—	236	4	no	—
197	C₂²	#16837321	1	1	yes	characteristic
198	C₂²	#16837547	1	2	yes	—
199	C₂²	#16837564	1	2	yes	—
200	C₂²	#16837570	1	1	yes	characteristic
201	C₂²	#16837585	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	8	2	no	—
206	C₄	—	8	2	no	—
207	C₄	—	944	1	no	—
208	C₄	#16837699	1	2	yes	—
209	C₄	#16837706	1	2	yes	—
210	C₄	#16837801	1	2	yes	—
211	C₄	#16837812	1	2	yes	—
212	C₂	#21423179	1	1	yes	characteristic
213	C₂	#21423217	1	1	yes	characteristic
214	C₂	#21423366	1	2	yes	—
215	C₂	#21425431	1	2	yes	—
216	C₂	#21425438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424854

C₂

yes

maximal

#21424252

C₂

yes

maximal, characteristic

#21421979

C₂

yes

maximal, characteristic

#21421937

C₂

yes

maximal

C₅₉×#2571

C₂

yes

maximal, characteristic

C₂×#14273660

—

C₂×#14273658

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

characteristic

#16837272

C₂²

yes

characteristic

#16837164

—

C₅₉×#400

—

C₅₉×#389

—

C₅₉×#384

C₂²

yes

characteristic

C₅₉×#382

—

C₄×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273660

—

#14273658

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

Q₈

yes

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

Q₈

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#95×C₅₉

—

#95×C₅₉

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

—

C₄²×C₅₉

#41

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#120

yes

characteristic

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

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#452

yes

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#478

yes

—

100

C₄×C₅₉

#479

yes

—

101

C₄×C₅₉

#499

yes

—

102

C₄×C₅₉

#501

yes

—

103

Dic₅₉

—

104

#2571

D₅₉

yes

characteristic

105

#1947

—

106

#1844

—

107

#1296

—

108

#1279

—

109

C₂×C₅₉

#1624

yes

characteristic

110

C₂×C₅₉

#1630

yes

characteristic

111

C₂×C₅₉

#1652

yes

—

112

C₂×C₅₉

#2059

yes

—

113

C₂×C₅₉

#2060

yes

characteristic

114

C₂×#107

—

118

—

115

C₂×#106

—

118

—

116

C₂×C₄⋊C₈

—

117

C₂×C₄×C₈

—

118

#400

—

119

#389

—

120

#384

—

121

#384

C₂×D₅₉

yes

characteristic

122

#382

—

123

C₄×C₄⋊C₄

C₂×D₅₉

yes

—

124

#340

—

118

—

125

C₅₉

#10577

yes

characteristic, complemented, Sylow

126

C₂²×C₈

—

118

—

127

C₂²×C₈

—

118

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

118

—

131

C₂×C₄⋊C₄

—

118

—

132

C₂×C₄⋊C₄

C₂²⋊D₅₉

yes

—

133

C₂×C₄⋊C₄

C₂²⋊D₅₉

yes

—

134

C₂×C₄⋊C₄

C₂²⋊D₅₉

yes

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²×D₅₉

yes

characteristic

137

#107

—

236

—

138

#106

—

236

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

118

—

141

C₄×C₈

—

142

C₄×C₈

—

118

—

143

#95

—

144

#95

—

145

#95

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

236

—

150

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

151

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

—

152

C₂×C₈

—

118

—

153

C₂×C₈

—

118

—

154

C₂×C₈

—

118

—

155

C₂×C₈

—

118

—

156

C₂×C₈

—

236

—

157

C₂×C₈

—

236

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

236

—

167

C₄⋊C₄

—

236

—

168

C₄⋊C₄

—

236

—

169

C₄²

—

170

C₄²

—

171

C₄²

Q₈×D₅₉

yes

characteristic

172

C₄²

Q₈×D₅₉

yes

—

173

C₄²

#11745689

yes

characteristic

174

C₂³

#14273696

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

472

—

186

C₂×C₄

—

472

—

187

C₂×C₄

#14273703

yes

characteristic

188

C₂×C₄

#14273703

yes

—

189

C₂×C₄

#14273725

yes

characteristic

190

C₂×C₄

#14273725

yes

—

191

C₂×C₄

#14273747

yes

—

192

C₂×C₄

#14273747

yes

—

193

C₂×C₄

#14273752

yes

—

194

C₂×C₄

#14273757

yes

—

195

C₈

—

118

—

196

C₈

—

236

—

197

C₂²

#16837321

yes

characteristic

198

C₂²

#16837547

yes

—

199

C₂²

#16837564

yes

—

200

C₂²

#16837570

yes

characteristic

201

C₂²

#16837585

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

944

—

208

C₄

#16837699

yes

—

209

C₄

#16837706

yes

—

210

C₄

#16837801

yes

—

211

C₄

#16837812

yes

—

212

C₂

#21423179

yes

characteristic

213

C₂

#21423217

yes

characteristic

214

C₂

#21423366

yes

—

215

C₂

#21425431

yes

—

216

C₂

#21425438

yes

characteristic

	subgroup	G/H
center	C₂³	#14273696
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2571	C₂
socle, prime radical	C₂³×C₅₉	#120
hypercenter	#2571	D₅₉
hypocenter	C₅₉	#10577
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2571

subgroup

G/H

center

C₂³

#14273696

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#2571

C₂

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2571

N	H ≅ G/N	type
125. C₅₉	#10577	semidirect

H ≅ G/N

type

125. C₅₉

#10577

semidirect

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

subgroup	index
#21424854	2
#21424252	2
#21421979	2
#21421937	2
C₅₉×#2571	2
#10577	59

subgroup

index

C₅₉×#2571

Subgroups — #28493145

Conjugacy classes of subgroups

Complements

Maximal subgroups