Group #28492822 — subgroups

Subgroups — #28492822

Order 15104.

382 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21435441	C₂	1	1	yes	maximal, characteristic
2	#21425073	C₂	1	2	yes	maximal
3	#21422784	C₂	1	2	yes	maximal
4	#21421971	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1831	C₂	1	1	yes	maximal, characteristic
6	#16837959	C₂²	1	2	yes	—
7	#16837698	—	2	2	no	—
8	#16837643	—	2	2	no	—
9	#16837613	—	2	2	no	—
10	#16837426	C₂²	1	2	yes	—
11	#16837162	—	2	2	no	—
12	#16837162	—	2	2	no	—
13	#16837150	—	2	2	no	—
14	#16837149	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₄³×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#336	—	2	2	no	—
19	C₂×Dic₂₃₆	—	2	2	no	—
20	C₂×#11745664	—	2	2	no	—
21	#14273727	D₄	1	2	yes	—
22	#14273727	D₄	1	2	yes	—
23	#14273719	—	2	2	no	—
24	#14273706	—	2	2	no	—
25	#14273706	—	2	2	no	—
26	C₄×Dic₁₁₈	—	2	2	no	—
27	C₄×Dic₁₁₈	—	2	2	no	—
28	#14273686	—	2	2	no	—
29	#14273661	—	2	2	no	—
30	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
31	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
32	#14273652	—	2	4	no	—
33	C₂²×C₈×C₅₉	—	2	1	no	—
34	C₂²×C₈×C₅₉	D₄	1	2	yes	—
35	C₂×C₄²×C₅₉	—	2	1	no	—
36	C₂×C₄²×C₅₉	—	2	1	no	—
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₉	—	2	2	no	—
41	C₄⋊C₈×C₅₉	—	2	2	no	—
42	C₂²×C₄×C₅₉	—	2	1	no	—
43	C₂²×C₄×C₅₉	—	2	1	no	—
44	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂×Dic₁₁₈	—	2	2	no	—
48	C₂×Dic₁₁₈	—	2	2	no	—
49	C₂×C₈×C₅₉	—	2	2	no	—
50	C₂×C₈×C₅₉	—	4	2	no	—
51	C₂×C₈×C₅₉	—	4	2	no	—
52	C₂×C₈×C₅₉	—	4	1	no	—
53	C₄²×C₅₉	—	2	1	no	—
54	C₄²×C₅₉	—	2	2	no	—
55	C₄²×C₅₉	—	4	1	no	—
56	C₄²×C₅₉	—	4	1	no	—
57	C₄²×C₅₉	—	4	2	no	—
58	C₄²×C₅₉	Q₁₆	1	2	yes	complemented
59	C₄²×C₅₉	C₂×D₄	1	2	yes	—
60	C₄²×C₅₉	#41	1	2	yes	—
61	C₂²⋊Dic₅₉	—	4	2	no	—
62	C₄⋊Dic₅₉	—	2	2	no	—
63	C₄⋊Dic₅₉	—	2	2	no	—
64	C₄⋊Dic₅₉	—	2	2	no	—
65	C₄⋊Dic₅₉	—	2	2	no	—
66	Dic₅₉⋊C₄	—	4	2	no	—
67	Dic₅₉⋊C₄	—	4	2	no	—
68	C₄×Dic₅₉	—	4	2	no	—
69	#11745664	—	4	2	no	—
70	#11745664	—	4	2	no	—
71	C₂×C₅₉⋊C₈	—	4	2	no	—
72	Dic₂₃₆	—	8	2	no	—
73	C₂³×C₅₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₅₉	—	2	1	no	—
75	C₂×C₄×C₅₉	—	2	1	no	—
76	C₂×C₄×C₅₉	—	2	2	no	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	4	1	no	—
79	C₂×C₄×C₅₉	—	4	1	no	—
80	C₂×C₄×C₅₉	—	4	1	no	—
81	C₂×C₄×C₅₉	—	4	1	no	—
82	C₂×C₄×C₅₉	—	4	1	no	—
83	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
84	C₂×C₄×C₅₉	#121	1	2	yes	—
85	C₂×C₄×C₅₉	#123	1	2	yes	—
86	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
89	C₂×C₄×C₅₉	#137	1	2	yes	—
90	C₂×Dic₅₉	—	4	2	no	—
91	C₂×Dic₅₉	—	4	2	no	—
92	Dic₁₁₈	—	4	2	no	—
93	Dic₁₁₈	—	4	2	no	—
94	Dic₁₁₈	—	8	2	no	—
95	C₈×C₅₉	—	4	2	no	—
96	C₈×C₅₉	—	8	1	no	—
97	C₅₉⋊C₈	—	8	2	no	—
98	#10523	—	59	1	no	maximal, Sylow
99	C₂²×C₅₉	—	2	1	no	—
100	C₂²×C₅₉	—	2	1	no	—
101	C₂²×C₅₉	#394	1	1	yes	characteristic
102	C₂²×C₅₉	#467	1	1	yes	characteristic
103	C₂²×C₅₉	#470	1	1	yes	characteristic
104	C₄×C₅₉	—	2	2	no	—
105	C₄×C₅₉	—	2	1	no	—
106	C₄×C₅₉	—	4	2	no	—
107	C₄×C₅₉	—	4	2	no	—
108	C₄×C₅₉	#454	1	1	yes	characteristic
109	C₄×C₅₉	#455	1	1	yes	characteristic
110	C₄×C₅₉	#462	1	2	yes	—
111	C₄×C₅₉	#464	1	2	yes	—
112	Dic₅₉	—	8	2	no	—
113	Dic₅₉	—	8	2	no	—
114	#3249	—	59	1	no	—
115	#1994	—	59	2	no	—
116	#1831	D₅₉	1	1	yes	characteristic
117	#1550	—	59	2	no	—
118	#1292	—	59	1	no	—
119	C₂×C₅₉	—	2	1	no	—
120	C₂×C₅₉	—	2	1	no	—
121	C₂×C₅₉	#2040	1	1	yes	characteristic
122	C₂×C₅₉	#2050	1	1	yes	characteristic
123	C₂×C₅₉	#2056	1	1	yes	characteristic
124	#533	—	59	2	no	—
125	#477	—	118	2	no	—
126	#467	—	118	2	no	—
127	#462	—	118	2	no	—
128	#423	—	59	2	no	—
129	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
130	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
131	C₄³	C₂×D₅₉	1	1	yes	characteristic
132	#339	—	118	2	no	—
133	#339	—	118	2	no	—
134	#336	—	2	2	no	—
135	#333	—	118	2	no	—
136	#332	—	118	2	no	—
137	C₅₉	#10523	1	1	yes	characteristic, complemented, Sylow
138	C₂×Q₁₆	—	118	2	no	—
139	C₂×M₁₆	—	118	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
142	C₄⋊Q₈	—	59	2	no	—
143	C₄⋊Q₈	—	59	2	no	—
144	#125	—	118	2	no	—
145	C₂²⋊Q₈	—	118	2	no	—
146	C₂²⋊Q₈	—	118	2	no	—
147	C₄×Q₈	—	118	2	no	—
148	C₄×Q₈	—	118	2	no	—
149	#117	—	118	2	no	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
153	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
155	#107	—	118	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	118	2	no	—
159	C₄⋊C₈	—	118	2	no	—
160	#103	—	118	4	no	—
161	C₂×Q₈	—	118	2	no	—
162	C₂×Q₈	—	118	2	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
166	C₂²×C₄	#11745685	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
168	Q₁₆	—	472	2	no	—
169	M₁₆	—	236	2	no	—
170	M₁₆	—	236	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	236	2	no	—
176	C₄⋊C₄	—	118	2	no	—
177	C₄⋊C₄	—	118	2	no	—
178	C₄⋊C₄	—	118	2	no	—
179	C₄⋊C₄	—	118	2	no	—
180	C₄⋊C₄	—	236	2	no	—
181	C₄⋊C₄	—	236	2	no	—
182	C₂²⋊C₄	—	236	2	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	1	no	—
186	C₄²	—	4	1	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	236	2	no	—
189	C₄²	Dic₂₃₆	1	2	yes	complemented
190	C₄²	D₄×D₅₉	1	2	yes	—
191	C₄²	#11745689	1	2	yes	—
192	C₂³	#14273701	1	1	yes	characteristic
193	Q₈	—	472	2	no	—
194	Q₈	—	236	2	no	—
195	Q₈	—	236	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	236	2	no	—
206	C₂×C₄	—	236	2	no	—
207	C₂×C₄	#14273695	1	1	yes	characteristic
208	C₂×C₄	#14273702	1	2	yes	—
209	C₂×C₄	#14273712	1	2	yes	—
210	C₂×C₄	#14273716	1	1	yes	characteristic
211	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
212	C₂×C₄	#14273749	1	1	yes	characteristic
213	C₂×C₄	#14273760	1	2	yes	—
214	C₈	—	4	2	no	—
215	C₈	—	8	1	no	—
216	C₈	—	472	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#16837319	1	1	yes	characteristic
220	C₂²	#16837643	1	1	yes	characteristic
221	C₂²	#16837660	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	472	2	no	—
227	C₄	—	472	2	no	—
228	C₄	#16837574	1	1	yes	characteristic
229	C₄	#16837579	1	1	yes	characteristic
230	C₄	#16837613	1	2	yes	—
231	C₄	#16837625	1	2	yes	—
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	#21425325	1	1	yes	characteristic
235	C₂	#21425383	1	1	yes	characteristic
236	C₂	#21425419	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21435441

C₂

yes

maximal, characteristic

#21425073

C₂

yes

maximal

#21422784

C₂

yes

maximal

#21421971

C₂

yes

maximal, characteristic

C₅₉×#1831

C₂

yes

maximal, characteristic

#16837959

C₂²

yes

—

—

—

—

C₂²

yes

—

—

—

—

—

C₂×C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

—

C₂×Dic₂₃₆

—

C₂×#11745664

—

#14273727

D₄

yes

—

#14273727

D₄

yes

—

#14273719

—

#14273706

—

#14273706

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

#14273686

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

#14273652

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

Q₁₆

yes

complemented

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

#41

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

C₂³×C₅₉

#121

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

#120

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

#10523

—

maximal, Sylow

C₂²×C₅₉

—

100

C₂²×C₅₉

—

101

C₂²×C₅₉

#394

yes

characteristic

102

C₂²×C₅₉

#467

yes

characteristic

103

C₂²×C₅₉

#470

yes

characteristic

104

C₄×C₅₉

—

105

C₄×C₅₉

—

106

C₄×C₅₉

—

107

C₄×C₅₉

—

108

C₄×C₅₉

#454

yes

characteristic

109

C₄×C₅₉

#455

yes

characteristic

110

C₄×C₅₉

#462

yes

—

111

C₄×C₅₉

#464

yes

—

112

Dic₅₉

—

113

Dic₅₉

—

114

#3249

—

115

#1994

—

116

#1831

D₅₉

yes

characteristic

117

#1550

—

118

#1292

—

119

C₂×C₅₉

—

120

C₂×C₅₉

—

121

C₂×C₅₉

#2040

yes

characteristic

122

C₂×C₅₉

#2050

yes

characteristic

123

C₂×C₅₉

#2056

yes

characteristic

124

#533

—

125

#477

—

118

—

126

#467

—

118

—

127

#462

—

118

—

128

#423

—

129

C₂×C₄⋊C₈

C₂×D₅₉

yes

characteristic

130

C₂×C₄⋊C₈

C₂×D₅₉

yes

characteristic

131

C₄³

C₂×D₅₉

yes

characteristic

132

#339

—

118

—

133

#339

—

118

—

134

#336

—

135

#333

—

118

—

136

#332

—

118

—

137

C₅₉

#10523

yes

characteristic, complemented, Sylow

138

C₂×Q₁₆

—

118

—

139

C₂×M₁₆

—

118

—

140

C₂²×C₈

—

141

C₂²×C₈

C₂²⋊D₅₉

yes

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

#125

—

118

—

145

C₂²⋊Q₈

—

118

—

146

C₂²⋊Q₈

—

118

—

147

C₄×Q₈

—

118

—

148

C₄×Q₈

—

118

—

149

#117

—

118

—

150

C₂×C₄²

—

151

C₂×C₄²

—

152

C₂×C₄²

D₂₃₆

yes

characteristic

153

C₂×C₄²

D₂₃₆

yes

characteristic

154

C₂×C₄²

C₂²×D₅₉

yes

characteristic

155

#107

—

118

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

118

—

159

C₄⋊C₈

—

118

—

160

#103

—

118

—

161

C₂×Q₈

—

118

—

162

C₂×Q₈

—

118

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

166

C₂²×C₄

#11745685

yes

characteristic

167

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

168

Q₁₆

—

472

—

169

M₁₆

—

236

—

170

M₁₆

—

236

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

236

—

176

C₄⋊C₄

—

118

—

177

C₄⋊C₄

—

118

—

178

C₄⋊C₄

—

118

—

179

C₄⋊C₄

—

118

—

180

C₄⋊C₄

—

236

—

181

C₄⋊C₄

—

236

—

182

C₂²⋊C₄

—

236

—

183

C₄²

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

188

C₄²

—

236

—

189

C₄²

Dic₂₃₆

yes

complemented

190

C₄²

D₄×D₅₉

yes

—

191

C₄²

#11745689

yes

—

192

C₂³

#14273701

yes

characteristic

193

Q₈

—

472

—

194

Q₈

—

236

—

195

Q₈

—

236

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

236

—

206

C₂×C₄

—

236

—

207

C₂×C₄

#14273695

yes

characteristic

208

C₂×C₄

#14273702

yes

—

209

C₂×C₄

#14273712

yes

—

210

C₂×C₄

#14273716

yes

characteristic

211

C₂×C₄

C₂×Dic₂₃₆

yes

characteristic

212

C₂×C₄

#14273749

yes

characteristic

213

C₂×C₄

#14273760

yes

—

214

C₈

—

215

C₈

—

216

C₈

—

472

—

217

C₂²

—

218

C₂²

—

219

C₂²

#16837319

yes

characteristic

220

C₂²

#16837643

yes

characteristic

221

C₂²

#16837660

yes

characteristic

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

472

—

227

C₄

—

472

—

228

C₄

#16837574

yes

characteristic

229

C₄

#16837579

yes

characteristic

230

C₄

#16837613

yes

—

231

C₄

#16837625

yes

—

232

C₂

—

233

C₂

—

234

C₂

#21425325

yes

characteristic

235

C₂

#21425383

yes

characteristic

236

C₂

#21425419

yes

characteristic

	subgroup	G/H
center	C₂²	#16837319
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1831	C₂
socle	C₂²×C₅₉	#394
prime radical	C₂³×C₅₉	#121
hypercenter	#1831	D₅₉
hypocenter	C₅₉	#10523
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1831

subgroup

G/H

center

C₂²

#16837319

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1831

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1831

N	H ≅ G/N	type
58. C₄²×C₅₉	Q₁₆	semidirect
137. C₅₉	#10523	semidirect
189. C₄²	Dic₂₃₆	semidirect

H ≅ G/N

type

58. C₄²×C₅₉

Q₁₆

semidirect

137. C₅₉

#10523

semidirect

189. C₄²

Dic₂₃₆

semidirect

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

subgroup	index
#21435441	2
#21425073	2
#21422784	2
#21421971	2
C₅₉×#1831	2
#10523	59

subgroup

index

C₅₉×#1831

Subgroups — #28492822

Conjugacy classes of subgroups

Complements

Maximal subgroups