Group #28488551 — subgroups

Subgroups — #28488551

Order 15104.

510 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 199 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	#21425058	C₂	1	2	yes	maximal
2	#21424899	C₂	1	2	yes	maximal
3	#21424256	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837123	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	2	no	—
7	C₂×#14273651	—	2	4	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
9	#16837272	C₂²	1	1	yes	characteristic
10	#16837272	C₂²	1	1	yes	characteristic
11	#16837164	—	2	4	no	—
12	#16837123	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
14	C₅₉×#498	—	2	2	no	—
15	C₅₉×#498	C₄	1	4	yes	complemented
16	C₂×#106×C₅₉	—	2	2	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²×C₅₉⋊C₈	D₄	1	4	yes	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	1	no	—
22	C₂×C₄⋊Dic₅₉	—	2	1	no	—
23	#14273661	—	4	4	no	—
24	#14273651	—	4	8	no	—
25	C₄×C₅₉⋊C₈	—	2	2	no	—
26	C₄×C₅₉⋊C₈	—	2	2	no	—
27	C₂²×Q₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	D₄	1	2	yes	—
29	C₄⋊Q₈×C₅₉	—	2	2	no	—
30	C₄⋊Q₈×C₅₉	—	2	2	no	—
31	C₄⋊Q₈×C₅₉	—	2	2	no	—
32	C₄⋊Q₈×C₅₉	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
35	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
36	#106×C₅₉	—	2	8	no	—
37	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₉	D₄	1	2	yes	—
39	C₂×Q₈×C₅₉	—	2	4	no	—
40	C₂×Q₈×C₅₉	—	4	2	no	—
41	C₂×Q₈×C₅₉	—	4	4	no	—
42	C₂×Q₈×C₅₉	—	4	4	no	—
43	C₂²×C₄×C₅₉	—	4	2	no	—
44	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
46	C₂²×Dic₅₉	—	4	1	no	—
47	C₂²×Dic₅₉	—	4	1	no	—
48	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
49	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
50	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
51	C₄⋊C₄×C₅₉	—	2	4	no	—
52	C₄⋊C₄×C₅₉	—	2	4	no	—
53	C₄⋊C₄×C₅₉	—	4	2	no	—
54	C₄⋊C₄×C₅₉	—	4	2	no	—
55	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₉	#41	1	2	yes	—
58	C₄⋊Dic₅₉	—	4	2	no	—
59	C₄⋊Dic₅₉	—	4	2	no	—
60	C₄⋊Dic₅₉	—	4	4	no	—
61	C₄⋊Dic₅₉	—	4	4	no	—
62	C₂×C₅₉⋊C₈	—	2	4	no	—
63	C₂×C₅₉⋊C₈	—	2	4	no	—
64	C₂×C₅₉⋊C₈	—	2	4	no	—
65	C₂³×C₅₉	#127	1	1	yes	characteristic
66	Q₈×C₅₉	—	4	8	no	—
67	Q₈×C₅₉	—	8	4	no	—
68	C₂×C₄×C₅₉	—	4	4	no	—
69	C₂×C₄×C₅₉	—	8	2	no	—
70	C₂×C₄×C₅₉	—	8	2	no	—
71	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₅₉	#121	1	2	yes	—
74	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×C₄×C₅₉	#137	1	4	yes	—
79	C₂×Dic₅₉	—	8	1	no	—
80	C₂×Dic₅₉	—	8	1	no	—
81	C₂×Dic₅₉	—	8	2	no	—
82	C₂×Dic₅₉	—	8	2	no	—
83	C₈×C₅₉	#103	1	8	yes	complemented
84	C₅₉⋊C₈	—	4	4	no	—
85	#9841	—	59	1	no	maximal, Sylow
86	C₂²×C₅₉	#390	1	1	yes	characteristic
87	C₂²×C₅₉	#492	1	1	yes	characteristic
88	C₂²×C₅₉	#494	1	1	yes	characteristic
89	C₂²×C₅₉	#496	1	2	yes	—
90	C₂²×C₅₉	#497	1	2	yes	—
91	C₄×C₅₉	—	8	4	no	—
92	C₄×C₅₉	C₂×#103	1	2	yes	—
93	C₄×C₅₉	#440	1	4	yes	—
94	C₄×C₅₉	#468	1	4	yes	—
95	C₄×C₅₉	#487	1	2	yes	—
96	Dic₅₉	—	16	1	no	—
97	Dic₅₉	—	16	1	no	—
98	C₂×#498	D₅₉	1	1	yes	characteristic
99	#1990	—	59	2	no	—
100	#1956	—	59	2	no	—
101	#1845	—	59	1	no	—
102	C₂×#322	—	59	1	no	—
103	C₂×C₅₉	#1715	1	1	yes	characteristic
104	C₂×C₅₉	#1716	1	1	yes	characteristic
105	C₂×C₅₉	#1723	1	2	yes	—
106	C₂×C₅₉	#1956	1	1	yes	characteristic
107	C₂×C₅₉	#1973	1	2	yes	—
108	C₂×C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
109	#498	—	2	2	no	—
110	#498	Dic₅₉	1	4	yes	complemented
111	C₂×#107	—	118	2	no	—
112	C₂×#106	—	2	2	no	—
113	C₂×#103	—	118	4	no	—
114	C₂×C₄×C₈	—	59	2	no	—
115	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
116	#384	—	59	1	no	—
117	#384	—	59	1	no	—
118	#340	—	118	4	no	—
119	#322	—	118	2	no	—
120	C₅₉	#9841	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×C₈	—	59	4	no	—
123	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×Dic₅₉	1	4	yes	—
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	2	no	—
132	C₂×C₄⋊C₄	—	118	1	no	—
133	C₂×C₄⋊C₄	—	118	1	no	—
134	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
135	#107	—	236	4	no	—
136	#106	—	2	8	no	—
137	#103	—	236	8	no	—
138	C₄×C₈	—	118	2	no	—
139	C₄×C₈	—	118	2	no	—
140	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
141	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	236	1	no	—
148	C₂²×C₄	—	236	1	no	—
149	C₂²×C₄	D₄×D₅₉	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
151	C₂×C₈	—	118	4	no	—
152	C₂×C₈	—	118	4	no	—
153	C₂×C₈	—	118	4	no	—
154	C₂×C₈	#11745671	1	4	yes	—
155	C₂×C₈	#11745672	1	4	yes	—
156	C₂×C₈	C₂²⋊Dic₅₉	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	236	2	no	—
162	C₄⋊C₄	—	236	2	no	—
163	C₄⋊C₄	—	236	4	no	—
164	C₄⋊C₄	—	236	4	no	—
165	C₄²	#11745687	1	2	yes	—
166	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
168	C₂³	#14273723	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	472	1	no	—
175	C₂×C₄	—	472	1	no	—
176	C₂×C₄	—	472	2	no	—
177	C₂×C₄	—	472	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
180	C₂×C₄	#14273698	1	2	yes	—
181	C₂×C₄	#14273714	1	1	yes	characteristic
182	C₂×C₄	C₂×#11745671	1	2	yes	—
183	C₂×C₄	C₂×#11745672	1	2	yes	—
184	C₂×C₄	#14273754	1	4	yes	—
185	C₂×C₄	#14273756	1	4	yes	—
186	C₈	—	236	4	no	—
187	C₈	#14273651	1	8	yes	complemented
188	C₂²	#16837305	1	1	yes	characteristic
189	C₂²	#16837776	1	1	yes	characteristic
190	C₂²	#16837781	1	1	yes	characteristic
191	C₂²	#16837788	1	2	yes	—
192	C₂²	#16837793	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	944	1	no	—
195	C₄	—	944	1	no	—
196	C₄	C₂×#14273651	1	2	yes	—
197	C₄	#16837503	1	4	yes	—
198	C₄	#16837645	1	4	yes	—
199	C₄	#16837752	1	2	yes	—
200	C₂	#21423725	1	1	yes	characteristic
201	C₂	#21423730	1	1	yes	characteristic
202	C₂	#21423770	1	2	yes	—
203	C₂	#21424899	1	1	yes	characteristic
204	C₂	#21424979	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425058

C₂

yes

maximal

#21424899

C₂

yes

maximal

#21424256

C₂

yes

maximal, characteristic

C₂×#16837123

C₂

yes

maximal, characteristic

C₂×C₅₉×#498

C₂

yes

maximal, characteristic

C₂×#14273661

—

C₂×#14273651

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837272

C₂²

yes

characteristic

#16837272

C₂²

yes

characteristic

#16837164

—

#16837123

—

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

—

C₅₉×#498

—

C₅₉×#498

C₄

yes

complemented

C₂×#106×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273651

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#106×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#127

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

#103

yes

complemented

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#497

yes

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#103

yes

—

C₄×C₅₉

#440

yes

—

C₄×C₅₉

#468

yes

—

C₄×C₅₉

#487

yes

—

Dic₅₉

—

Dic₅₉

—

C₂×#498

D₅₉

yes

characteristic

#1990

—

100

#1956

—

101

#1845

—

102

C₂×#322

—

103

C₂×C₅₉

#1715

yes

characteristic

104

C₂×C₅₉

#1716

yes

characteristic

105

C₂×C₅₉

#1723

yes

—

106

C₂×C₅₉

#1956

yes

characteristic

107

C₂×C₅₉

#1973

yes

—

108

C₂×C₄⋊Q₈

C₂×D₅₉

yes

—

109

#498

—

110

#498

Dic₅₉

yes

complemented

111

C₂×#107

—

118

—

112

C₂×#106

—

113

C₂×#103

—

118

—

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

116

#384

—

117

#384

—

118

#340

—

118

—

119

#322

—

118

—

120

C₅₉

#9841

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₅₉

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

C₂×Dic₅₉

yes

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

118

—

131

C₂×C₄⋊C₄

—

118

—

132

C₂×C₄⋊C₄

—

118

—

133

C₂×C₄⋊C₄

—

118

—

134

C₂×C₄²

C₂²×D₅₉

yes

characteristic

135

#107

—

236

—

136

#106

—

137

#103

—

236

—

138

C₄×C₈

—

118

—

139

C₄×C₈

—

118

—

140

C₄×C₈

C₂×Dic₅₉

yes

—

141

C₄×C₈

C₂²⋊D₅₉

yes

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

236

—

148

C₂²×C₄

—

236

—

149

C₂²×C₄

D₄×D₅₉

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

151

C₂×C₈

—

118

—

152

C₂×C₈

—

118

—

153

C₂×C₈

—

118

—

154

C₂×C₈

#11745671

yes

—

155

C₂×C₈

#11745672

yes

—

156

C₂×C₈

C₂²⋊Dic₅₉

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

236

—

162

C₄⋊C₄

—

236

—

163

C₄⋊C₄

—

236

—

164

C₄⋊C₄

—

236

—

165

C₄²

#11745687

yes

—

166

C₄²

C₂²×Dic₅₉

yes

characteristic

167

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

168

C₂³

#14273723

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

472

—

175

C₂×C₄

—

472

—

176

C₂×C₄

—

472

—

177

C₂×C₄

—

472

—

178

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

179

C₂×C₄

D₄×Dic₅₉

yes

—

180

C₂×C₄

#14273698

yes

—

181

C₂×C₄

#14273714

yes

characteristic

182

C₂×C₄

C₂×#11745671

yes

—

183

C₂×C₄

C₂×#11745672

yes

—

184

C₂×C₄

#14273754

yes

—

185

C₂×C₄

#14273756

yes

—

186

C₈

—

236

—

187

C₈

#14273651

yes

complemented

188

C₂²

#16837305

yes

characteristic

189

C₂²

#16837776

yes

characteristic

190

C₂²

#16837781

yes

characteristic

191

C₂²

#16837788

yes

—

192

C₂²

#16837793

yes

—

193

C₄

—

194

C₄

—

944

—

195

C₄

—

944

—

196

C₄

C₂×#14273651

yes

—

197

C₄

#16837503

yes

—

198

C₄

#16837645

yes

—

199

C₄

#16837752

yes

—

200

C₂

#21423725

yes

characteristic

201

C₂

#21423730

yes

characteristic

202

C₂

#21423770

yes

—

203

C₂

#21424899

yes

characteristic

204

C₂

#21424979

yes

—

	subgroup	G/H
center	C₂³	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#498	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#498	D₅₉
hypocenter	C₅₉	#9841
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#14273723

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#498

C₂

socle, prime radical

C₂³×C₅₉

#127

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
15. C₅₉×#498	C₄	semidirect
83. C₈×C₅₉	#103	semidirect
110. #498	Dic₅₉	semidirect
120. C₅₉	#9841	semidirect
187. C₈	#14273651	semidirect

H ≅ G/N

type

15. C₅₉×#498

C₄

semidirect

83. C₈×C₅₉

#103

semidirect

110. #498

Dic₅₉

semidirect

120. C₅₉

#9841

semidirect

187. C₈

#14273651

semidirect

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

subgroup	index
#21425058	2
#21424899	2
#21424256	2
C₂×#16837123	2
C₂×C₅₉×#498	2
#9841	59

subgroup

index

C₂×C₅₉×#498

Subgroups — #28488551

Conjugacy classes of subgroups

Complements

Maximal subgroups