Group #15762951 — subgroups

Subgroups — #15762951

Order 3328.

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

normal subgroups only

This group has 486 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	#13114988	C₂	1	1	yes	maximal, characteristic
2	C₄×#88043	C₂	1	1	yes	maximal, characteristic
3	#13112677	C₂	1	4	yes	maximal
4	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
6	C₁₃×#389	C₄	1	2	yes	—
7	C₁₃×#387	—	2	2	no	—
8	C₁₃×#382	C₄	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
10	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄³×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	—	2	2	no	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×#88043	—	2	2	no	—
15	C₂×C₄×#1055	C₂²	1	2	yes	—
16	C₁₃×#125	—	2	4	no	—
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
19	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₃	—	2	2	no	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	2	yes	—
25	C₂×C₄²×C₁₃	D₄	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×#1055	—	2	2	no	—
28	C₂²×#1055	—	2	2	no	—
29	C₁₃×#95	—	2	2	no	—
30	C₁₃×#95	—	2	2	no	—
31	C₁₃×#95	—	2	2	no	—
32	C₄⋊#1055	—	2	4	no	—
33	C₄⋊#1055	—	4	4	no	—
34	#88043	—	2	8	no	—
35	C₄×#1055	—	2	2	no	—
36	C₄×#1055	Q₈	1	4	yes	—
37	#9876	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	2	2	no	—
40	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₁₃×C₄⋊C₄	—	2	8	no	—
46	C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₄⋊C₄	—	4	4	no	—
49	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
50	C₄²×C₁₃	—	2	4	no	—
51	C₄²×C₁₃	—	2	2	no	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	—	2	2	no	—
54	C₄²×C₁₃	—	2	2	no	—
55	C₄²×C₁₃	—	4	2	no	—
56	C₄²×C₁₃	—	4	2	no	—
57	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
58	C₄²×C₁₃	#41	1	2	yes	—
59	C₂×#1055	—	2	8	no	—
60	C₂×#1055	—	2	4	no	—
61	C₂×#1055	—	4	2	no	—
62	C₂×#1055	—	4	4	no	—
63	C₄×#125	D₁₃	1	1	yes	characteristic
64	#1842	—	13	1	no	—
65	C₄×#97	—	13	1	no	—
66	#1279	—	13	4	no	—
67	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	4	2	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	#104	1	4	yes	—
81	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₁₃	#131	1	2	yes	—
87	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄×C₁₃	#137	1	2	yes	—
89	#1055	—	2	8	no	—
90	#1055	—	8	2	no	—
91	C₂×#125	Dic₁₃	1	1	yes	characteristic
92	C₂×C₄⋊C₈	—	13	2	no	—
93	C₂×C₄⋊C₈	—	26	2	no	—
94	C₂×#97	—	26	2	no	—
95	C₂×C₄×C₈	—	13	2	no	—
96	#389	Dic₁₃	1	2	yes	—
97	#387	—	2	2	no	—
98	#382	Dic₁₃	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	2	2	no	—
100	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
101	C₄³	C₂×D₁₃	1	1	yes	characteristic
102	C₂²×C₁₃	#391	1	1	yes	characteristic
103	C₂²×C₁₃	#407	1	1	yes	characteristic
104	C₂²×C₁₃	#408	1	1	yes	characteristic
105	C₂²×C₁₃	#417	1	2	yes	—
106	C₂²×C₁₃	C₂×#104	1	2	yes	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	2	4	no	—
109	C₄×C₁₃	—	4	2	no	—
110	C₄×C₁₃	—	4	4	no	—
111	C₄×C₁₃	—	8	2	no	—
112	C₄×C₁₃	#446	1	4	yes	—
113	C₄×C₁₃	#490	1	1	yes	characteristic
114	C₄×C₁₃	#491	1	1	yes	characteristic
115	C₄×C₁₃	#501	1	2	yes	—
116	C₂²×C₈	—	26	2	no	—
117	C₂²×C₈	—	26	2	no	—
118	#125	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
123	C₂×C₄²	—	2	2	no	—
124	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
128	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	C₄⋊C₈	—	26	4	no	—
130	C₄⋊C₈	—	52	4	no	—
131	#97	—	26	8	no	—
132	C₄×C₈	—	13	4	no	—
133	C₄×C₈	—	26	2	no	—
134	#95	—	2	2	no	—
135	#95	—	2	2	no	—
136	#95	—	2	2	no	—
137	C₂×C₁₃	#1483	1	2	yes	—
138	C₂×C₁₃	#1489	1	2	yes	—
139	C₂×C₁₃	#1985	1	1	yes	characteristic
140	C₂×C₁₃	#1992	1	1	yes	characteristic
141	C₂×C₁₃	#1994	1	1	yes	characteristic
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
145	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
149	C₂×C₈	—	26	8	no	—
150	C₂×C₈	—	26	4	no	—
151	C₂×C₈	—	52	2	no	—
152	C₂×C₈	—	52	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	#6099	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	4	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	Q₈×D₁₃	1	2	yes	—
166	C₄²	#6131	1	2	yes	—
167	C₁₃	#9876	1	1	yes	characteristic, complemented, Sylow
168	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#88077	1	4	yes	—
182	C₂×C₄	C₂×#6099	1	2	yes	—
183	C₂×C₄	#88198	1	1	yes	characteristic
184	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
185	C₂×C₄	#88200	1	2	yes	—
186	C₂×C₄	#88201	1	1	yes	characteristic
187	C₂×C₄	#88202	1	2	yes	—
188	C₂×C₄	#88203	1	2	yes	—
189	C₂×C₄	#88205	1	2	yes	—
190	C₈	—	26	8	no	—
191	C₈	—	104	2	no	—
192	C₂²	#11718868	1	1	yes	characteristic
193	C₂²	#11718891	1	1	yes	characteristic
194	C₂²	#11718893	1	2	yes	—
195	C₂²	#11718895	1	1	yes	characteristic
196	C₂²	C₂×#88077	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	8	2	no	—
202	C₄	#11718685	1	4	yes	—
203	C₄	#11718726	1	2	yes	—
204	C₄	#11718728	1	1	yes	characteristic
205	C₄	#11718730	1	1	yes	characteristic
206	C₂	#13113249	1	2	yes	—
207	C₂	#13113275	1	2	yes	—
208	C₂	#13115777	1	1	yes	characteristic
209	C₂	#13115806	1	1	yes	characteristic
210	C₂	#13115814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114988

C₂

yes

maximal, characteristic

C₄×#88043

C₂

yes

maximal, characteristic

#13112677

C₂

yes

maximal

C₄×C₁₃×#125

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

C₄

yes

characteristic

C₁₃×#389

C₄

yes

—

C₁₃×#387

—

C₁₃×#382

C₄

yes

characteristic

C₄×C₁₃×C₄⋊C₄

—

C₄×C₁₃×C₄⋊C₄

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

C₂²

yes

—

C₂×#88043

—

C₂×C₄×#1055

C₂²

yes

—

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₂²×#1055

—

C₁₃×#95

—

C₁₃×#95

—

C₁₃×#95

—

—

—

—

—

Q₈

yes

—

#9876

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

M₁₆

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

#41

yes

—

—

—

—

—

D₁₃

yes

characteristic

#1842

—

C₄×#97

—

#1279

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#104

yes

—

C₂×C₄×C₁₃

C₄×Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

#1055

—

#1055

—

C₂×#125

Dic₁₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#389

Dic₁₃

yes

—

#387

—

#382

Dic₁₃

yes

characteristic

C₄×C₄⋊C₄

—

100

C₄×C₄⋊C₄

C₂×D₁₃

yes

—

101

C₄³

C₂×D₁₃

yes

characteristic

102

C₂²×C₁₃

#391

yes

characteristic

103

C₂²×C₁₃

#407

yes

characteristic

104

C₂²×C₁₃

#408

yes

characteristic

105

C₂²×C₁₃

#417

yes

—

106

C₂²×C₁₃

C₂×#104

yes

—

107

C₄×C₁₃

—

108

C₄×C₁₃

—

109

C₄×C₁₃

—

110

C₄×C₁₃

—

111

C₄×C₁₃

—

112

C₄×C₁₃

#446

yes

—

113

C₄×C₁₃

#490

yes

characteristic

114

C₄×C₁₃

#491

yes

characteristic

115

C₄×C₁₃

#501

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

#125

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

122

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

123

C₂×C₄²

—

124

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

125

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

126

C₂×C₄²

C₂²⋊D₁₃

yes

—

127

C₂×C₄²

C₂²⋊D₁₃

yes

—

128

C₂×C₄²

C₂²×D₁₃

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

#97

—

132

C₄×C₈

—

133

C₄×C₈

—

134

#95

—

135

#95

—

136

#95

—

137

C₂×C₁₃

#1483

yes

—

138

C₂×C₁₃

#1489

yes

—

139

C₂×C₁₃

#1985

yes

characteristic

140

C₂×C₁₃

#1992

yes

characteristic

141

C₂×C₁₃

#1994

yes

characteristic

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

145

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

146

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

#6099

yes

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

Q₈×D₁₃

yes

—

166

C₄²

#6131

yes

—

167

C₁₃

#9876

yes

characteristic, complemented, Sylow

168

C₂³

C₂×C₂²⋊Dic₁₃

yes

characteristic

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#88077

yes

—

182

C₂×C₄

C₂×#6099

yes

—

183

C₂×C₄

#88198

yes

characteristic

184

C₂×C₄

Q₈×Dic₁₃

yes

—

185

C₂×C₄

#88200

yes

—

186

C₂×C₄

#88201

yes

characteristic

187

C₂×C₄

#88202

yes

—

188

C₂×C₄

#88203

yes

—

189

C₂×C₄

#88205

yes

—

190

C₈

—

191

C₈

—

104

—

192

C₂²

#11718868

yes

characteristic

193

C₂²

#11718891

yes

characteristic

194

C₂²

#11718893

yes

—

195

C₂²

#11718895

yes

characteristic

196

C₂²

C₂×#88077

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

#11718685

yes

—

203

C₄

#11718726

yes

—

204

C₄

#11718728

yes

characteristic

205

C₄

#11718730

yes

characteristic

206

C₂

#13113249

yes

—

207

C₂

#13113275

yes

—

208

C₂

#13115777

yes

characteristic

209

C₂

#13115806

yes

characteristic

210

C₂

#13115814

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#125	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#125	D₁₃
hypocenter	C₁₃	#9876
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₁₃

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₄×C₁₃×#125

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
167. C₁₃	#9876	semidirect

H ≅ G/N

type

167. C₁₃

#9876

semidirect

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

subgroup	index
#13114988	2
C₄×#88043	2
#13112677	2
C₄×C₁₃×#125	2
#9876	13

subgroup

index

C₄×C₁₃×#125

Subgroups — #15762951

Conjugacy classes of subgroups

Complements

Maximal subgroups