Group #15758231 — subgroups

Subgroups — #15758231

Order 3328.

444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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	#13123903	C₂	1	1	yes	maximal, characteristic
2	#13113428	C₂	1	1	yes	maximal, characteristic
3	#13113257	C₂	1	2	yes	maximal, complemented
4	#13112673	C₂	1	2	yes	maximal, complemented
5	C₁₃×#3131	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	1	yes	characteristic
7	C₁₃×#532	C₂²	1	1	yes	characteristic
8	C₁₃×#488	—	2	2	no	—
9	C₁₃×#487	C₄	1	2	yes	—
10	C₁₃×#486	C₄	1	2	yes	—
11	C₁₃×#416	—	2	2	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718895	—	2	2	no	—
14	#11718673	—	2	2	no	—
15	D₄×#1055	—	2	2	no	—
16	#11718586	C₂²	1	2	yes	—
17	C₄×#6099	C₂²	1	2	yes	—
18	#11718132	—	2	2	no	—
19	Q₈⋊#1055	—	2	2	no	—
20	Q₈⋊#1055	—	2	2	no	—
21	C₂×C₁₃×#41	D₄	1	2	yes	—
22	C₂²×C₈×C₁₃	—	2	1	no	—
23	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
24	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
25	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
26	C₁₃×#125	—	2	1	no	—
27	C₁₃×#124	D₄	1	2	yes	—
28	C₁₃×C₂²⋊Q₈	—	2	2	no	—
29	C₁₃×#121	—	2	2	no	—
30	C₄×Q₈×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₁₃×#117	—	2	1	no	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6099	—	2	2	no	—
35	C₂×#6099	—	2	2	no	—
36	C₂²×#1055	—	2	2	no	—
37	C₁₃×#103	—	2	4	no	—
38	C₁₃×#102	—	2	4	no	—
39	C₄×C₈×C₁₃	—	2	1	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₂²⋊#1055	—	2	4	no	—
42	C₂²⋊#1055	—	2	2	no	—
43	C₄⋊#1055	D₄	1	2	yes	—
44	C₄⋊#1055	D₄	1	2	yes	—
45	#88043	—	2	2	no	—
46	C₄×#1055	—	2	2	no	—
47	C₄×#1055	—	2	2	no	—
48	#9067	—	13	1	no	maximal, Sylow
49	C₁₃×#41	—	4	2	no	—
50	C₁₃×#41	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	C₂²⋊C₄	1	2	yes	—
53	C₂×D₄×C₁₃	—	2	2	no	—
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	4	1	no	—
61	C₂×C₈×C₁₃	—	4	1	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	4	no	—
67	C₁₃×C₂²⋊C₄	—	4	1	no	—
68	C₄²×C₁₃	—	4	1	no	—
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×D₄	1	2	yes	—
71	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
72	#6099	—	2	4	no	—
73	#6099	—	4	2	no	—
74	#6099	—	8	2	no	—
75	C₂×#1055	—	2	2	no	—
76	C₂×#1055	—	2	2	no	—
77	C₂×#1055	—	4	2	no	—
78	C₂×#1055	—	8	2	no	—
79	#3131	D₁₃	1	1	yes	characteristic
80	#2979	—	13	1	no	—
81	#1525	—	13	1	no	—
82	#1485	—	13	2	no	—
83	#1278	—	13	2	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	#123	1	1	yes	characteristic
86	Q₈×C₁₃	—	4	2	no	—
87	Q₈×C₁₃	—	4	2	no	—
88	Q₈×C₁₃	—	4	2	no	—
89	D₄×C₁₃	—	4	2	no	—
90	D₄×C₁₃	—	4	2	no	—
91	D₄×C₁₃	—	4	2	no	—
92	D₄×C₁₃	—	8	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#121	1	2	yes	—
105	C₈×C₁₃	—	4	2	no	—
106	#1055	—	4	4	no	—
107	#1055	—	4	2	no	—
108	#1055	—	4	2	no	—
109	#533	C₂×D₁₃	1	1	yes	characteristic
110	#532	C₂×D₁₃	1	1	yes	characteristic
111	#488	—	2	2	no	—
112	#487	Dic₁₃	1	2	yes	—
113	#486	Dic₁₃	1	2	yes	—
114	#436	—	26	2	no	—
115	C₈×D₄	—	26	2	no	—
116	#431	—	13	2	no	—
117	#416	—	2	2	no	—
118	#408	—	26	2	no	—
119	C₄×M₁₆	—	13	2	no	—
120	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
121	#329	—	26	2	no	—
122	Q₈⋊C₈	—	26	2	no	—
123	Q₈⋊C₈	—	26	2	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	4	4	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	#386	1	1	yes	characteristic
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	4	2	no	—
132	C₄×C₁₃	—	8	2	no	—
133	C₄×C₁₃	#443	1	2	yes	—
134	C₄×C₁₃	#444	1	2	yes	—
135	C₄×C₁₃	#455	1	1	yes	characteristic
136	C₄×C₁₃	#455	1	1	yes	characteristic
137	C₂×#41	C₂²⋊D₁₃	1	2	yes	—
138	C₂×M₁₆	—	26	2	no	—
139	C₂×M₁₆	—	26	2	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	26	2	no	—
142	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
143	C₄⋊Q₈	C₂×Dic₁₃	1	1	yes	characteristic
144	#127	C₂×Dic₁₃	1	1	yes	characteristic
145	#125	—	2	1	no	—
146	#124	C₂²⋊D₁₃	1	2	yes	—
147	C₂²⋊Q₈	—	2	2	no	—
148	#121	—	2	2	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×D₄	—	2	2	no	—
151	#117	—	2	1	no	—
152	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
153	C₄⋊C₈	—	13	2	no	—
154	C₄⋊C₈	—	13	2	no	—
155	#103	—	2	4	no	—
156	#102	—	2	4	no	—
157	C₂²⋊C₈	—	26	4	no	—
158	C₂²⋊C₈	—	26	2	no	—
159	#97	—	26	2	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	26	2	no	—
162	C₄×C₈	—	26	2	no	—
163	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
164	C₂×C₁₃	—	4	1	no	—
165	C₂×C₁₃	—	8	2	no	—
166	C₂×C₁₃	#1876	1	2	yes	—
167	C₂×C₁₃	#1885	1	1	yes	characteristic
168	#41	—	4	2	no	—
169	#41	—	4	2	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	C₂²⋊Dic₁₃	1	2	yes	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	C₂²⋊Dic₁₃	1	2	yes	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	#6129	1	2	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
178	M₁₆	—	26	4	no	—
179	M₁₆	—	52	2	no	—
180	M₁₆	—	104	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	26	2	no	—
185	C₂×C₈	—	26	2	no	—
186	C₂×C₈	—	52	2	no	—
187	C₂×C₈	—	104	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	4	1	no	—
194	C₄²	—	4	1	no	—
195	C₄²	D₄×D₁₃	1	2	yes	—
196	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
198	C₁₃	#9067	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	2	2	no	—
200	C₂³	#88189	1	1	yes	characteristic
201	Q₈	—	4	2	no	—
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	D₄	—	4	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	4	2	no	—
207	D₄	—	8	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
217	C₂×C₄	#88191	1	1	yes	characteristic
218	C₂×C₄	#88193	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
220	C₈	—	4	2	no	—
221	C₈	—	52	4	no	—
222	C₈	—	52	2	no	—
223	C₈	—	52	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	2	no	—
228	C₂²	#11718881	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	8	2	no	—
233	C₄	#11718722	1	1	yes	characteristic
234	C₄	#11718722	1	1	yes	characteristic
235	C₄	#11718857	1	2	yes	—
236	C₄	#11718858	1	2	yes	—
237	C₂	—	4	1	no	—
238	C₂	—	8	2	no	—
239	C₂	#13115165	1	2	yes	—
240	C₂	#13115225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13123903

C₂

yes

maximal, characteristic

#13113428

C₂

yes

maximal, characteristic

#13113257

C₂

yes

maximal, complemented

#13112673

C₂

yes

maximal, complemented

C₁₃×#3131

C₂

yes

maximal, characteristic

C₁₃×#533

C₂²

yes

characteristic

C₁₃×#532

C₂²

yes

characteristic

C₁₃×#488

—

C₁₃×#487

C₄

yes

—

C₁₃×#486

C₄

yes

—

C₁₃×#416

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

—

C₂²

yes

—

C₄×#6099

C₂²

yes

—

#11718132

—

Q₈⋊#1055

—

Q₈⋊#1055

—

C₂×C₁₃×#41

D₄

yes

—

C₂²×C₈×C₁₃

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

—

C₁₃×C₄⋊Q₈

C₂×C₄

yes

characteristic

C₁₃×#127

C₂×C₄

yes

characteristic

C₁₃×#125

—

C₁₃×#124

D₄

yes

—

C₁₃×C₂²⋊Q₈

—

C₁₃×#121

—

C₄×Q₈×C₁₃

—

C₄×D₄×C₁₃

—

C₁₃×#117

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₂×#6099

—

C₂²×#1055

—

C₁₃×#103

—

C₁₃×#102

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₂²⋊#1055

—

C₂²⋊#1055

—

C₄⋊#1055

D₄

yes

—

C₄⋊#1055

D₄

yes

—

—

—

—

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂²⋊C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

—

—

—

—

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

#123

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

100

C₂×C₄×C₁₃

—

101

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

102

C₂×C₄×C₁₃

C₄×D₄

yes

—

103

C₂×C₄×C₁₃

#120

yes

characteristic

104

C₂×C₄×C₁₃

#121

yes

—

105

C₈×C₁₃

—

106

#1055

—

107

#1055

—

108

#1055

—

109

#533

C₂×D₁₃

yes

characteristic

110

#532

C₂×D₁₃

yes

characteristic

111

#488

—

112

#487

Dic₁₃

yes

—

113

#486

Dic₁₃

yes

—

114

#436

—

115

C₈×D₄

—

116

#431

—

117

#416

—

118

#408

—

119

C₄×M₁₆

—

120

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

121

#329

—

122

Q₈⋊C₈

—

123

Q₈⋊C₈

—

124

C₂²×C₁₃

—

125

C₂²×C₁₃

—

126

C₂²×C₁₃

—

127

C₂²×C₁₃

—

128

C₂²×C₁₃

#386

yes

characteristic

129

C₄×C₁₃

—

130

C₄×C₁₃

—

131

C₄×C₁₃

—

132

C₄×C₁₃

—

133

C₄×C₁₃

#443

yes

—

134

C₄×C₁₃

#444

yes

—

135

C₄×C₁₃

#455

yes

characteristic

136

C₄×C₁₃

#455

yes

characteristic

137

C₂×#41

C₂²⋊D₁₃

yes

—

138

C₂×M₁₆

—

139

C₂×M₁₆

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₄⋊Q₈

C₂×Dic₁₃

yes

—

143

C₄⋊Q₈

C₂×Dic₁₃

yes

characteristic

144

#127

C₂×Dic₁₃

yes

characteristic

145

#125

—

146

#124

C₂²⋊D₁₃

yes

—

147

C₂²⋊Q₈

—

148

#121

—

149

C₄×Q₈

—

150

C₄×D₄

—

151

#117

—

152

C₂×C₄²

C₂²×D₁₃

yes

characteristic

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

#103

—

156

#102

—

157

C₂²⋊C₈

—

158

C₂²⋊C₈

—

159

#97

—

160

C₄×C₈

—

161

C₄×C₈

—

162

C₄×C₈

—

163

C₄×C₈

C₂×Dic₁₃

yes

—

164

C₂×C₁₃

—

165

C₂×C₁₃

—

166

C₂×C₁₃

#1876

yes

—

167

C₂×C₁₃

#1885

yes

characteristic

168

#41

—

169

#41

—

170

C₂×Q₈

—

171

C₂×Q₈

C₂²⋊Dic₁₃

yes

—

172

C₂×D₄

—

173

C₂×D₄

—

174

C₂×D₄

C₂²⋊Dic₁₃

yes

—

175

C₂²×C₄

—

176

C₂²×C₄

#6129

yes

—

177

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

178

M₁₆

—

179

M₁₆

—

180

M₁₆

—

104

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₂×C₈

—

104

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₂²⋊C₄

—

193

C₂²⋊C₄

—

194

C₄²

—

195

C₄²

D₄×D₁₃

yes

—

196

C₄²

C₂²×Dic₁₃

yes

characteristic

197

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

198

C₁₃

#9067

yes

characteristic, complemented, Sylow

199

C₂³

—

200

C₂³

#88189

yes

characteristic

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

D₄

—

205

D₄

—

206

D₄

—

207

D₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

D₄×Dic₁₃

yes

—

217

C₂×C₄

#88191

yes

characteristic

218

C₂×C₄

#88193

yes

—

219

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

220

C₈

—

221

C₈

—

222

C₈

—

223

C₈

—

224

C₂²

—

225

C₂²

—

226

C₂²

—

227

C₂²

—

228

C₂²

#11718881

yes

characteristic

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

#11718722

yes

characteristic

234

C₄

#11718722

yes

characteristic

235

C₄

#11718857

yes

—

236

C₄

#11718858

yes

—

237

C₂

—

238

C₂

—

239

C₂

#13115165

yes

—

240

C₂

#13115225

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88191
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#3131	C₂
socle	C₂²×C₁₃	#386
prime radical	C₁₃×#532	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#3131	D₁₃
hypocenter	C₁₃	#9067
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

subgroup

G/H

center

C₂×C₄

#88191

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#3131

C₂

socle

C₂²×C₁₃

#386

prime radical

C₁₃×#532

C₂²

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

N	H ≅ G/N	type
3. #13113257	C₂	semidirect
4. #13112673	C₂	semidirect
198. C₁₃	#9067	semidirect

H ≅ G/N

type

3. #13113257

C₂

semidirect

4. #13112673

C₂

semidirect

198. C₁₃

#9067

semidirect

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

subgroup	index
#13123903	2
#13113428	2
#13113257	2
#13112673	2
C₁₃×#3131	2
#9067	13

subgroup

index

C₁₃×#3131

Subgroups — #15758231

Conjugacy classes of subgroups

Complements

Maximal subgroups