Group #15767359 — subgroups

Subgroups — #15767359

Order 3328.

494 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 135 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	#13119865	C₂	1	1	yes	maximal, characteristic
2	#13115597	C₂	1	2	yes	maximal
3	#13112719	C₂	1	1	yes	maximal, characteristic
4	#13112676	C₂	1	2	yes	maximal
5	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#88142	—	2	2	no	—
7	C₂×C₁₃×#106	—	2	2	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#384	C₂²	1	1	yes	characteristic
10	C₁₃×#384	C₂²	1	1	yes	characteristic
11	#11718640	—	2	2	no	—
12	#11718634	—	2	2	no	—
13	C₂×#88048	—	2	2	no	—
14	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	#11718121	—	2	2	no	—
17	C₂²×Dic₂₆	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	1	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
21	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
26	C₂×#6101	—	2	4	no	—
27	C₂×C₄×Dic₁₃	—	2	2	no	—
28	C₂²×#1055	—	2	2	no	—
29	#88142	—	4	4	no	—
30	#88142	—	4	4	no	—
31	C₁₃×#106	—	4	4	no	—
32	C₄×C₈×C₁₃	—	2	1	no	—
33	C₄×C₈×C₁₃	Q₈	1	2	yes	complemented
34	#88071	—	2	2	no	—
35	#88071	—	2	2	no	—
36	#88048	—	4	4	no	—
37	C₄⋊#1055	—	2	2	no	—
38	C₄⋊#1055	Q₈	1	4	yes	complemented
39	#10574	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	4	1	no	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	2	no	—
45	C₂²×Dic₁₃	—	2	2	no	—
46	C₂×Dic₂₆	—	4	8	no	—
47	C₂×Dic₂₆	—	8	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	—	4	2	no	—
53	C₁₃×C₄⋊C₄	—	4	2	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₁₃×C₄⋊C₄	—	4	2	no	—
57	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
59	C₄²×C₁₃	#41	1	1	yes	characteristic
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄⋊Dic₁₃	—	2	2	no	—
62	C₄⋊Dic₁₃	SD₁₆	1	4	yes	—
63	C₄⋊Dic₁₃	SD₁₆	1	4	yes	—
64	#6101	—	4	8	no	—
65	C₄×Dic₁₃	—	4	2	no	—
66	C₄×Dic₁₃	—	4	2	no	—
67	C₂×#1055	—	2	4	no	—
68	C₂×#1055	—	4	2	no	—
69	C₂×#1055	—	4	2	no	—
70	#2551	—	13	1	no	—
71	#1948	—	13	2	no	—
72	#1844	D₁₃	1	1	yes	characteristic
73	#1296	—	13	1	no	—
74	#1279	—	13	2	no	—
75	C₂³×C₁₃	#120	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	8	2	no	—
81	C₂×C₄×C₁₃	—	8	1	no	—
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×C₄×C₁₃	#137	1	2	yes	—
90	C₂×Dic₁₃	—	2	4	no	—
91	C₂×Dic₁₃	—	2	4	no	—
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	—	4	2	no	—
94	C₂×Dic₁₃	—	4	2	no	—
95	C₂×Dic₁₃	—	4	2	no	—
96	Dic₂₆	—	8	8	no	—
97	C₈×C₁₃	—	2	4	no	—
98	#1055	—	4	4	no	—
99	C₂×C₄⋊Q₈	—	26	2	no	—
100	C₂×#106	—	2	2	no	—
101	C₂×#106	—	26	2	no	—
102	C₂×C₄⋊C₈	—	13	2	no	—
103	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
104	#395	—	26	2	no	—
105	#389	—	26	2	no	—
106	#384	C₂×D₁₃	1	1	yes	characteristic
107	#384	C₂×D₁₃	1	1	yes	characteristic
108	C₄×C₄⋊C₄	—	13	2	no	—
109	#340	—	26	2	no	—
110	C₂²×C₁₃	#395	1	1	yes	characteristic
111	C₂²×C₁₃	#448	1	2	yes	—
112	C₂²×C₁₃	#450	1	2	yes	—
113	C₂²×C₁₃	#453	1	1	yes	characteristic
114	C₂²×C₁₃	#456	1	1	yes	characteristic
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	8	2	no	—
118	C₄×C₁₃	—	16	1	no	—
119	C₄×C₁₃	#475	1	2	yes	—
120	C₄×C₁₃	#476	1	2	yes	—
121	C₄×C₁₃	#498	1	2	yes	—
122	C₄×C₁₃	#501	1	2	yes	—
123	Dic₁₃	—	4	4	no	—
124	Dic₁₃	—	4	4	no	—
125	C₂²×Q₈	—	26	2	no	—
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	26	2	no	—
128	C₄⋊Q₈	—	52	4	no	—
129	C₄⋊Q₈	—	52	4	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	13	2	no	—
134	C₂×C₄⋊C₄	—	13	2	no	—
135	C₂×C₄⋊C₄	—	26	4	no	—
136	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	2	yes	—
137	C₂×C₄²	—	26	2	no	—
138	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
139	#106	—	4	4	no	—
140	#106	—	52	4	no	—
141	C₄⋊C₈	—	13	4	no	—
142	C₄⋊C₈	—	26	2	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	Dic₂₆	1	2	yes	complemented
145	#95	—	26	2	no	—
146	#95	—	26	2	no	—
147	C₂×C₁₃	#1623	1	1	yes	characteristic
148	C₂×C₁₃	#1629	1	1	yes	characteristic
149	C₂×C₁₃	#1651	1	2	yes	—
150	C₂×C₁₃	#2058	1	2	yes	—
151	C₂×C₁₃	#2060	1	1	yes	characteristic
152	C₂×Q₈	—	52	8	no	—
153	C₂×Q₈	—	104	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	4	1	no	—
156	C₂²×C₄	—	26	2	no	—
157	C₂²×C₄	—	26	2	no	—
158	C₂²×C₄	D₄×D₁₃	1	2	yes	—
159	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	26	4	no	—
164	C₂×C₈	—	52	2	no	—
165	C₂×C₈	—	52	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	13	4	no	—
173	C₄⋊C₄	—	13	4	no	—
174	C₄⋊C₄	—	26	2	no	—
175	C₄⋊C₄	—	26	2	no	—
176	C₄⋊C₄	—	52	8	no	—
177	C₄²	—	52	2	no	—
178	C₄²	—	52	2	no	—
179	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
180	C₄²	#6127	1	1	yes	characteristic
181	C₄²	Q₈×D₁₃	1	2	yes	—
182	C₁₃	#10574	1	1	yes	characteristic, complemented, Sylow
183	C₂³	#88191	1	1	yes	characteristic
184	Q₈	—	104	8	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	26	4	no	—
192	C₂×C₄	—	26	4	no	—
193	C₂×C₄	—	52	2	no	—
194	C₂×C₄	—	52	2	no	—
195	C₂×C₄	—	52	2	no	—
196	C₂×C₄	—	52	2	no	—
197	C₂×C₄	#88142	1	2	yes	—
198	C₂×C₄	#88150	1	2	yes	—
199	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
200	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
201	C₂×C₄	#88168	1	2	yes	—
202	C₂×C₄	#88169	1	2	yes	—
203	C₂×C₄	#88178	1	1	yes	characteristic
204	C₂×C₄	#88198	1	1	yes	characteristic
205	C₈	—	2	4	no	—
206	C₈	—	52	4	no	—
207	C₂²	#11718640	1	1	yes	characteristic
208	C₂²	#11718738	1	1	yes	characteristic
209	C₂²	#11718756	1	1	yes	characteristic
210	C₂²	#11718830	1	2	yes	—
211	C₂²	#11718831	1	2	yes	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	—	16	1	no	—
216	C₄	—	52	4	no	—
217	C₄	—	52	4	no	—
218	C₄	#11718515	1	2	yes	—
219	C₄	#11718516	1	2	yes	—
220	C₄	#11718517	1	2	yes	—
221	C₄	#11718531	1	2	yes	—
222	C₂	#13113912	1	1	yes	characteristic
223	C₂	#13113950	1	1	yes	characteristic
224	C₂	#13114096	1	2	yes	—
225	C₂	#13116167	1	2	yes	—
226	C₂	#13116181	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13119865

C₂

yes

maximal, characteristic

#13115597

C₂

yes

maximal

#13112719

C₂

yes

maximal, characteristic

#13112676

C₂

yes

maximal

C₁₃×#1844

C₂

yes

maximal, characteristic

C₂×#88142

—

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

—

—

C₄×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

—

#11718121

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

D₄

yes

—

C₂×C₄⋊Dic₁₃

D₄

yes

—

C₂×#6101

—

C₂×C₄×Dic₁₃

—

C₂²×#1055

—

#88142

—

#88142

—

C₁₃×#106

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

Q₈

yes

complemented

#10574

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

SD₁₆

yes

—

C₄⋊Dic₁₃

SD₁₆

yes

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic

#1296

—

#1279

—

C₂³×C₁₃

#120

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊Q₈

—

100

C₂×#106

—

101

C₂×#106

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

104

#395

—

105

#389

—

106

#384

C₂×D₁₃

yes

characteristic

107

#384

C₂×D₁₃

yes

characteristic

108

C₄×C₄⋊C₄

—

109

#340

—

110

C₂²×C₁₃

#395

yes

characteristic

111

C₂²×C₁₃

#448

yes

—

112

C₂²×C₁₃

#450

yes

—

113

C₂²×C₁₃

#453

yes

characteristic

114

C₂²×C₁₃

#456

yes

characteristic

115

C₄×C₁₃

—

116

C₄×C₁₃

—

117

C₄×C₁₃

—

118

C₄×C₁₃

—

119

C₄×C₁₃

#475

yes

—

120

C₄×C₁₃

#476

yes

—

121

C₄×C₁₃

#498

yes

—

122

C₄×C₁₃

#501

yes

—

123

Dic₁₃

—

124

Dic₁₃

—

125

C₂²×Q₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

137

C₂×C₄²

—

138

C₂×C₄²

C₂²×D₁₃

yes

characteristic

139

#106

—

140

#106

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄×C₈

—

144

C₄×C₈

Dic₂₆

yes

complemented

145

#95

—

146

#95

—

147

C₂×C₁₃

#1623

yes

characteristic

148

C₂×C₁₃

#1629

yes

characteristic

149

C₂×C₁₃

#1651

yes

—

150

C₂×C₁₃

#2058

yes

—

151

C₂×C₁₃

#2060

yes

characteristic

152

C₂×Q₈

—

153

C₂×Q₈

—

104

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

D₄×D₁₃

yes

—

159

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄²

—

178

C₄²

—

179

C₄²

C₂×Dic₂₆

yes

characteristic

180

C₄²

#6127

yes

characteristic

181

C₄²

Q₈×D₁₃

yes

—

182

C₁₃

#10574

yes

characteristic, complemented, Sylow

183

C₂³

#88191

yes

characteristic

184

Q₈

—

104

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

#88142

yes

—

198

C₂×C₄

#88150

yes

—

199

C₂×C₄

SD₁₆×D₁₃

yes

—

200

C₂×C₄

SD₁₆×D₁₃

yes

—

201

C₂×C₄

#88168

yes

—

202

C₂×C₄

#88169

yes

—

203

C₂×C₄

#88178

yes

characteristic

204

C₂×C₄

#88198

yes

characteristic

205

C₈

—

206

C₈

—

207

C₂²

#11718640

yes

characteristic

208

C₂²

#11718738

yes

characteristic

209

C₂²

#11718756

yes

characteristic

210

C₂²

#11718830

yes

—

211

C₂²

#11718831

yes

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

#11718515

yes

—

219

C₄

#11718516

yes

—

220

C₄

#11718517

yes

—

221

C₄

#11718531

yes

—

222

C₂

#13113912

yes

characteristic

223

C₂

#13113950

yes

characteristic

224

C₂

#13114096

yes

—

225

C₂

#13116167

yes

—

226

C₂

#13116181

yes

characteristic

	subgroup	G/H
center	C₂³	#88191
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1844	C₂
socle, prime radical	C₂³×C₁₃	#120
hypercenter	#1844	D₁₃
hypocenter	C₁₃	#10574
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#88191

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1844

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
33. C₄×C₈×C₁₃	Q₈	semidirect
38. C₄⋊#1055	Q₈	semidirect
144. C₄×C₈	Dic₂₆	semidirect
182. C₁₃	#10574	semidirect

H ≅ G/N

type

33. C₄×C₈×C₁₃

Q₈

semidirect

38. C₄⋊#1055

Q₈

semidirect

144. C₄×C₈

Dic₂₆

semidirect

182. C₁₃

#10574

semidirect

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

subgroup	index
#13119865	2
#13115597	2
#13112719	2
#13112676	2
C₁₃×#1844	2
#10574	13

subgroup

index

C₁₃×#1844

Subgroups — #15767359

Conjugacy classes of subgroups

Complements

Maximal subgroups