Group #28487782 — subgroups

Subgroups — #28487782

Order 15104.

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

normal subgroups only

This group has 538 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	#21432494	C₂	1	1	yes	maximal, characteristic
2	#21423949	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837147	C₂	1	4	yes	maximal
4	C₅₉×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273727	—	2	1	no	—
6	C₂×#14273727	—	2	1	no	—
7	C₂×#14273727	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14273638	—	2	1	no	—
10	#16837272	—	2	2	no	—
11	#16837272	C₄	1	4	yes	—
12	#16837157	—	2	1	no	—
13	#16837147	—	2	8	no	—
14	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
15	C₄³×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#336	—	2	2	no	—
17	C₂²×Dic₁₁₈	—	2	2	no	—
18	C₂²×C₅₉⋊C₈	—	2	2	no	—
19	#14273727	—	2	4	no	—
20	#14273727	—	2	4	no	—
21	#14273727	—	4	4	no	—
22	#14273727	—	4	2	no	—
23	#14273727	—	4	2	no	—
24	C₂×C₄⋊Dic₅₉	—	2	4	no	—
25	C₂×C₄⋊Dic₅₉	—	2	4	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
28	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
29	C₄⋊(C₅₉⋊C₈)	D₄	1	4	yes	—
30	#14273638	—	4	2	no	—
31	C₂²×C₈×C₅₉	—	2	2	no	—
32	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	D₄	1	2	yes	—
36	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₉	—	2	4	no	—
38	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₉	—	2	4	no	—
44	C₂×Dic₁₁₈	—	4	4	no	—
45	C₂×Dic₁₁₈	—	4	2	no	—
46	C₂×Dic₁₁₈	—	4	2	no	—
47	C₂×Dic₁₁₈	—	4	4	no	—
48	C₂×C₈×C₅₉	—	4	2	no	—
49	C₂×C₈×C₅₉	—	4	4	no	—
50	C₄²×C₅₉	—	2	4	no	—
51	C₄²×C₅₉	—	2	2	no	—
52	C₄²×C₅₉	—	2	4	no	—
53	C₄²×C₅₉	D₈	1	2	yes	—
54	C₄²×C₅₉	SD₁₆	1	2	yes	—
55	C₄²×C₅₉	C₂×D₄	1	2	yes	—
56	C₄²×C₅₉	#41	1	2	yes	—
57	C₄⋊Dic₅₉	—	2	8	no	—
58	C₄⋊Dic₅₉	—	4	8	no	—
59	C₄⋊Dic₅₉	—	4	8	no	—
60	C₄⋊Dic₅₉	—	4	2	no	—
61	C₄⋊Dic₅₉	—	4	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	—	2	2	no	—
67	C₂×C₄×C₅₉	—	2	2	no	—
68	C₂×C₄×C₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	#101	1	4	yes	—
72	C₂×C₄×C₅₉	#102	1	4	yes	—
73	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₅₉	#121	1	2	yes	—
75	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₅₉	#137	1	4	yes	—
81	C₂×Dic₅₉	—	4	4	no	—
82	C₂×Dic₅₉	—	4	4	no	—
83	C₂×Dic₅₉	—	4	4	no	—
84	Dic₁₁₈	—	8	4	no	—
85	Dic₁₁₈	—	8	4	no	—
86	C₈×C₅₉	—	8	2	no	—
87	C₅₉⋊C₈	—	4	4	no	—
88	#9705	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	#390	1	1	yes	characteristic
90	C₂²×C₅₉	C₂×#101	1	2	yes	—
91	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
92	C₂²×C₅₉	#418	1	1	yes	characteristic
93	C₂²×C₅₉	#419	1	2	yes	—
94	C₄×C₅₉	—	2	4	no	—
95	C₄×C₅₉	—	2	4	no	—
96	C₄×C₅₉	—	2	2	no	—
97	C₄×C₅₉	#459	1	2	yes	—
98	C₄×C₅₉	#460	1	2	yes	—
99	C₄×C₅₉	#491	1	2	yes	—
100	C₄×C₅₉	#497	1	2	yes	—
101	Dic₅₉	—	8	4	no	—
102	#2868	—	59	1	no	—
103	#1831	D₅₉	1	1	yes	characteristic
104	#1767	—	59	1	no	—
105	C₂×#332	—	59	4	no	—
106	C₂×C₅₉	#1546	1	2	yes	—
107	C₂×C₅₉	#1556	1	2	yes	—
108	C₂×C₅₉	#1960	1	1	yes	characteristic
109	C₂×C₅₉	#1966	1	1	yes	characteristic
110	C₂×C₅₉	#1982	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	59	2	no	—
112	C₂×C₄⋊Q₈	—	118	1	no	—
113	C₂×C₄⋊Q₈	—	118	1	no	—
114	C₂×C₄⋊C₈	—	59	2	no	—
115	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
116	C₂×#97	—	118	1	no	—
117	#384	—	59	4	no	—
118	#384	—	118	2	no	—
119	C₄³	C₂×D₅₉	1	1	yes	characteristic
120	#336	—	2	2	no	—
121	#336	—	118	1	no	—
122	#332	—	118	8	no	—
123	C₅₉	#9705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	118	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	118	2	no	—
127	C₄⋊Q₈	—	118	4	no	—
128	C₄⋊Q₈	—	118	4	no	—
129	C₄⋊Q₈	—	236	4	no	—
130	C₄⋊Q₈	—	236	2	no	—
131	C₄⋊Q₈	—	236	2	no	—
132	C₂×C₄⋊C₄	—	59	4	no	—
133	C₂×C₄⋊C₄	—	118	4	no	—
134	C₂×C₄⋊C₄	—	118	4	no	—
135	C₂×C₄⋊C₄	—	118	2	no	—
136	C₂×C₄²	C₄×D₅₉	1	2	yes	—
137	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
138	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
140	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	59	4	no	—
143	C₄⋊C₈	—	118	2	no	—
144	#97	—	236	2	no	—
145	C₂×Q₈	—	236	4	no	—
146	C₂×Q₈	—	236	2	no	—
147	C₂×Q₈	—	236	2	no	—
148	C₂×Q₈	—	236	4	no	—
149	C₂²×C₄	—	118	4	no	—
150	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
151	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	—	118	4	no	—
158	C₂×C₈	—	236	2	no	—
159	C₂×C₈	—	236	2	no	—
160	C₄⋊C₄	—	118	8	no	—
161	C₄⋊C₄	—	236	8	no	—
162	C₄⋊C₄	—	236	8	no	—
163	C₄⋊C₄	—	236	2	no	—
164	C₄⋊C₄	—	236	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	#11745660	1	2	yes	—
169	C₄²	D₄₇₂	1	2	yes	—
170	C₄²	D₄×D₅₉	1	2	yes	—
171	C₄²	#11745689	1	2	yes	—
172	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
173	Q₈	—	472	4	no	—
174	Q₈	—	472	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	236	4	no	—
181	C₂×C₄	—	236	4	no	—
182	C₂×C₄	—	236	4	no	—
183	C₂×C₄	#14273645	1	4	yes	—
184	C₂×C₄	#14273649	1	4	yes	—
185	C₂×C₄	#14273689	1	2	yes	—
186	C₂×C₄	#14273702	1	2	yes	—
187	C₂×C₄	#14273716	1	1	yes	characteristic
188	C₂×C₄	#14273723	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
190	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
191	C₂×C₄	#14273755	1	2	yes	—
192	C₂×C₄	#14273757	1	4	yes	—
193	C₈	—	8	2	no	—
194	C₈	—	236	4	no	—
195	C₂²	#16837303	1	1	yes	characteristic
196	C₂²	C₂×#14273645	1	2	yes	—
197	C₂²	C₂×#14273649	1	1	yes	characteristic
198	C₂²	#16837408	1	1	yes	characteristic
199	C₂²	#16837410	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	472	4	no	—
204	C₄	#16837598	1	2	yes	—
205	C₄	#16837603	1	2	yes	—
206	C₄	#16837772	1	2	yes	—
207	C₄	#16837797	1	2	yes	—
208	C₂	#21422767	1	2	yes	—
209	C₂	#21422812	1	2	yes	—
210	C₂	#21424915	1	1	yes	characteristic
211	C₂	#21424939	1	1	yes	characteristic
212	C₂	#21425022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21432494

C₂

yes

maximal, characteristic

#21423949

C₂

yes

maximal, characteristic

C₂×#16837147

C₂

yes

maximal

C₅₉×#1831

C₂

yes

maximal, characteristic

C₂×#14273727

—

C₂×#14273727

—

C₂×#14273727

C₂²

yes

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂×#14273638

—

#16837272

—

#16837272

C₄

yes

—

#16837157

—

#16837147

—

C₂×C₄⋊C₈×C₅₉

C₂²

yes

—

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

—

C₂²×Dic₁₁₈

—

C₂²×C₅₉⋊C₈

—

—

—

—

—

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

#14273638

—

C₂²×C₈×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

D₈

yes

—

C₄²×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#101

yes

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

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

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#459

yes

—

C₄×C₅₉

#460

yes

—

C₄×C₅₉

#491

yes

—

100

C₄×C₅₉

#497

yes

—

101

Dic₅₉

—

102

#2868

—

103

#1831

D₅₉

yes

characteristic

104

#1767

—

105

C₂×#332

—

106

C₂×C₅₉

#1546

yes

—

107

C₂×C₅₉

#1556

yes

—

108

C₂×C₅₉

#1960

yes

characteristic

109

C₂×C₅₉

#1966

yes

characteristic

110

C₂×C₅₉

#1982

yes

characteristic

111

C₂×C₄⋊Q₈

—

112

C₂×C₄⋊Q₈

—

118

—

113

C₂×C₄⋊Q₈

—

118

—

114

C₂×C₄⋊C₈

—

115

C₂×C₄⋊C₈

C₂×D₅₉

yes

—

116

C₂×#97

—

118

—

117

#384

—

118

#384

—

118

—

119

C₄³

C₂×D₅₉

yes

characteristic

120

#336

—

121

#336

—

118

—

122

#332

—

118

—

123

C₅₉

#9705

yes

characteristic, complemented, Sylow

124

C₂²×Q₈

—

118

—

125

C₂²×C₈

—

126

C₂²×C₈

—

118

—

127

C₄⋊Q₈

—

118

—

128

C₄⋊Q₈

—

118

—

129

C₄⋊Q₈

—

236

—

130

C₄⋊Q₈

—

236

—

131

C₄⋊Q₈

—

236

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

118

—

134

C₂×C₄⋊C₄

—

118

—

135

C₂×C₄⋊C₄

—

118

—

136

C₂×C₄²

C₄×D₅₉

yes

—

137

C₂×C₄²

D₂₃₆

yes

characteristic

138

C₂×C₄²

D₂₃₆

yes

characteristic

139

C₂×C₄²

C₂²⋊D₅₉

yes

—

140

C₂×C₄²

C₂²×D₅₉

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

118

—

144

#97

—

236

—

145

C₂×Q₈

—

236

—

146

C₂×Q₈

—

236

—

147

C₂×Q₈

—

236

—

148

C₂×Q₈

—

236

—

149

C₂²×C₄

—

118

—

150

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

151

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

152

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

153

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

118

—

158

C₂×C₈

—

236

—

159

C₂×C₈

—

236

—

160

C₄⋊C₄

—

118

—

161

C₄⋊C₄

—

236

—

162

C₄⋊C₄

—

236

—

163

C₄⋊C₄

—

236

—

164

C₄⋊C₄

—

236

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

#11745660

yes

—

169

C₄²

D₄₇₂

yes

—

170

C₄²

D₄×D₅₉

yes

—

171

C₄²

#11745689

yes

—

172

C₂³

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

173

Q₈

—

472

—

174

Q₈

—

472

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

236

—

181

C₂×C₄

—

236

—

182

C₂×C₄

—

236

—

183

C₂×C₄

#14273645

yes

—

184

C₂×C₄

#14273649

yes

—

185

C₂×C₄

#14273689

yes

—

186

C₂×C₄

#14273702

yes

—

187

C₂×C₄

#14273716

yes

characteristic

188

C₂×C₄

#14273723

yes

characteristic

189

C₂×C₄

C₂×D₄₇₂

yes

characteristic

190

C₂×C₄

C₂×#11745660

yes

characteristic

191

C₂×C₄

#14273755

yes

—

192

C₂×C₄

#14273757

yes

—

193

C₈

—

194

C₈

—

236

—

195

C₂²

#16837303

yes

characteristic

196

C₂²

C₂×#14273645

yes

—

197

C₂²

C₂×#14273649

yes

characteristic

198

C₂²

#16837408

yes

characteristic

199

C₂²

#16837410

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

472

—

204

C₄

#16837598

yes

—

205

C₄

#16837603

yes

—

206

C₄

#16837772

yes

—

207

C₄

#16837797

yes

—

208

C₂

#21422767

yes

—

209

C₂

#21422812

yes

—

210

C₂

#21424915

yes

characteristic

211

C₂

#21424939

yes

characteristic

212

C₂

#21425022

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₅₉)⋊C₄
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1831	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1831	D₅₉
hypocenter	C₅₉	#9705
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×(C₂×D₅₉)⋊C₄

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1831

C₂

socle, prime radical

C₂³×C₅₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
123. C₅₉	#9705	semidirect

H ≅ G/N

type

123. C₅₉

#9705

semidirect

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

subgroup	index
#21432494	2
#21423949	2
C₂×#16837147	2
C₅₉×#1831	2
#9705	59

subgroup

index

C₅₉×#1831

Subgroups — #28487782

Conjugacy classes of subgroups

Complements

Maximal subgroups