Group #13120007 — subgroups

Subgroups — #13120007

Order 1664.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂×D₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#88142	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#390	C₂	1	1	yes	maximal, characteristic
4	#11718882	C₂	1	1	yes	maximal, characteristic
5	#11718638	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718608	C₂	1	4	yes	maximal, complemented
7	#11718603	C₂	1	4	yes	maximal, complemented
8	#11718597	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#6101	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₃	C₂²	1	2	yes	—
12	C₂²×Dic₂₆	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
18	D₄×Dic₁₃	—	2	4	no	—
19	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×#6101	C₂²	1	2	yes	—
22	C₂×#6101	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	#88142	—	2	4	no	—
26	#88071	C₂²	1	2	yes	—
27	#88071	C₂²	1	4	yes	—
28	#88071	C₂²	1	4	yes	—
29	C₂⁴×C₁₃	C₂³	1	2	yes	—
30	C₂×D₄×C₁₃	—	2	2	no	—
31	C₂×D₄×C₁₃	—	2	2	no	—
32	C₂×D₄×C₁₃	Q₈	1	4	yes	complemented
33	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₃	—	2	4	no	—
37	C₂²×Dic₁₃	—	2	8	no	—
38	C₂²×Dic₁₃	C₂³	1	2	yes	—
39	C₂²×Dic₁₃	C₂³	1	2	yes	—
40	C₂²×Dic₁₃	C₂³	1	2	yes	—
41	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₂₆	—	2	4	no	—
44	C₂×Dic₂₆	—	4	4	no	—
45	C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₁₃×C₂²⋊C₄	—	4	4	no	—
47	C₄²×C₁₃	—	2	2	no	—
48	C₂²⋊Dic₁₃	—	2	4	no	—
49	C₂²⋊Dic₁₃	—	4	4	no	—
50	C₄⋊Dic₁₃	—	2	2	no	—
51	C₄⋊Dic₁₃	—	2	2	no	—
52	#6101	—	2	4	no	—
53	#6101	—	2	4	no	—
54	C₄×Dic₁₃	—	2	2	no	—
55	C₄×Dic₁₃	D₄	1	4	yes	—
56	#2563	—	13	1	no	maximal, Sylow
57	C₂³×C₁₃	—	2	2	no	—
58	C₂³×C₁₃	—	2	4	no	—
59	C₂³×C₁₃	—	2	2	no	—
60	C₂³×C₁₃	—	2	4	no	—
61	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
62	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
63	D₄×C₁₃	—	2	8	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	4	no	—
69	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₁₃	#41	1	2	yes	—
72	C₂×Dic₁₃	—	2	2	no	—
73	C₂×Dic₁₃	—	2	2	no	—
74	C₂×Dic₁₃	—	2	2	no	—
75	C₂×Dic₁₃	—	2	2	no	—
76	C₂×Dic₁₃	—	2	2	no	—
77	C₂×Dic₁₃	—	2	4	no	—
78	C₂×Dic₁₃	—	2	2	no	—
79	C₂×Dic₁₃	—	2	4	no	—
80	C₂×Dic₁₃	—	4	8	no	—
81	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
82	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
83	C₂×Dic₁₃	#41	1	4	yes	—
84	Dic₂₆	—	4	8	no	—
85	C₂×C₄⋊Q₈	—	13	1	no	—
86	C₂×C₄×D₄	—	13	1	no	—
87	#393	—	13	4	no	—
88	#390	—	13	1	no	—
89	#390	D₁₃	1	1	yes	characteristic
90	#382	—	13	1	no	—
91	#382	—	13	1	no	—
92	#380	—	13	4	no	—
93	C₄×C₄⋊C₄	—	13	1	no	—
94	C₂²×C₁₃	—	2	4	no	—
95	C₂²×C₁₃	—	2	8	no	—
96	C₂²×C₁₃	—	4	2	no	—
97	C₂²×C₁₃	—	4	2	no	—
98	C₂²×C₁₃	—	4	4	no	—
99	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
102	C₂²×C₁₃	C₂×#41	1	2	yes	—
103	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
104	C₄×C₁₃	—	4	2	no	—
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
107	Dic₁₃	—	2	2	no	—
108	Dic₁₃	—	4	2	no	—
109	Dic₁₃	—	4	2	no	—
110	Dic₁₃	—	4	2	no	—
111	Dic₁₃	#124	1	4	yes	—
112	C₂²×Q₈	—	13	2	no	—
113	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
114	C₂³×C₄	—	13	2	no	—
115	C₄⋊Q₈	—	26	4	no	—
116	C₄×D₄	—	26	4	no	—
117	C₂×C₄⋊C₄	—	13	1	no	—
118	C₂×C₄⋊C₄	—	13	2	no	—
119	C₂×C₄⋊C₄	—	13	1	no	—
120	C₂×C₄⋊C₄	—	13	2	no	—
121	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	—	13	2	no	—
123	C₂×C₂²⋊C₄	—	13	4	no	—
124	C₂×C₂²⋊C₄	C₂×D₁₃	1	4	yes	—
125	C₂×C₄²	—	13	1	no	—
126	C₂×C₄²	—	13	1	no	—
127	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
128	#95	—	13	2	no	—
129	#95	—	13	4	no	—
130	#95	—	13	4	no	—
131	C₂×C₁₃	—	4	4	no	—
132	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₁₃	C₂×#124	1	1	yes	characteristic
134	C₂×C₁₃	#535	1	1	yes	characteristic
135	C₂×C₁₃	#554	1	2	yes	—
136	C₂×C₁₃	#555	1	2	yes	—
137	C₂⁴	C₂²×D₁₃	1	2	yes	—
138	C₂×Q₈	—	26	4	no	—
139	C₂×Q₈	—	52	4	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	Dic₂₆	1	4	yes	complemented
143	C₂²×C₄	—	13	1	no	—
144	C₂²×C₄	—	13	1	no	—
145	C₂²×C₄	—	13	2	no	—
146	C₂²×C₄	—	13	2	no	—
147	C₂²×C₄	—	13	2	no	—
148	C₂²×C₄	—	26	4	no	—
149	C₂²×C₄	—	26	8	no	—
150	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
151	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
152	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	26	4	no	—
155	C₄⋊C₄	—	26	4	no	—
156	C₄⋊C₄	—	26	2	no	—
157	C₄⋊C₄	—	26	2	no	—
158	C₂²⋊C₄	—	4	4	no	—
159	C₂²⋊C₄	—	26	4	no	—
160	C₂²⋊C₄	—	52	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	13	4	no	—
163	C₄²	—	26	2	no	—
164	C₁₃	#2563	1	1	yes	characteristic, complemented, Sylow
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	4	no	—
169	C₂³	C₂×Dic₂₆	1	4	yes	—
170	C₂³	C₂³×D₁₃	1	1	yes	characteristic
171	Q₈	—	52	8	no	—
172	D₄	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	13	2	no	—
179	C₂×C₄	—	13	4	no	—
180	C₂×C₄	—	13	2	no	—
181	C₂×C₄	—	26	2	no	—
182	C₂×C₄	—	26	2	no	—
183	C₂×C₄	—	26	2	no	—
184	C₂×C₄	—	26	2	no	—
185	C₂×C₄	—	26	2	no	—
186	C₂×C₄	—	26	4	no	—
187	C₂×C₄	—	26	2	no	—
188	C₂×C₄	—	26	4	no	—
189	C₂×C₄	—	52	8	no	—
190	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
191	C₂×C₄	D₄×D₁₃	1	2	yes	—
192	C₂×C₄	#6129	1	2	yes	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	8	no	—
195	C₂²	—	4	4	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
199	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
200	C₂²	C₂×#6129	1	1	yes	characteristic
201	C₂²	#88251	1	2	yes	—
202	C₂²	#41×D₁₃	1	2	yes	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	13	4	no	—
206	C₄	—	26	2	no	—
207	C₄	—	52	2	no	—
208	C₄	—	52	2	no	—
209	C₄	—	52	2	no	—
210	C₄	#88132	1	4	yes	—
211	C₂	—	4	4	no	—
212	C₂	C₂×#88132	1	1	yes	characteristic
213	C₂	#11719244	1	2	yes	—
214	C₂	#11719321	1	2	yes	—
215	C₂	D₁₃×#124	1	1	yes	characteristic
216	C₂	#11719420	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×Dic₁₃

C₂

yes

maximal, characteristic

C₂×#88142

C₂

yes

maximal, characteristic, complemented

C₁₃×#390

C₂

yes

maximal, characteristic

#11718882

C₂

yes

maximal, characteristic

#11718638

C₂

yes

maximal, characteristic, complemented

#11718608

C₂

yes

maximal, complemented

#11718603

C₂

yes

maximal, complemented

#11718597

C₂

yes

maximal, characteristic, complemented

C₄×#6101

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₂³×Dic₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

D₄×Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

#88142

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

Q₈

yes

complemented

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

—

#2563

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×Q₈

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

#41

yes

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×D₄

—

#393

—

#390

—

#390

D₁₃

yes

—

—

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

100

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

101

C₂²×C₁₃

C₂×#41

yes

characteristic

102

C₂²×C₁₃

C₂×#41

yes

—

103

C₂²×C₁₃

H(5,𝔽₂)

yes

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₄×C₁₃

C₂²⋊Q₈

yes

—

107

Dic₁₃

—

108

Dic₁₃

—

109

Dic₁₃

—

110

Dic₁₃

—

111

Dic₁₃

#124

yes

—

112

C₂²×Q₈

—

113

C₂²×D₄

C₂×D₁₃

yes

characteristic

114

C₂³×C₄

—

115

C₄⋊Q₈

—

116

C₄×D₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

122

C₂×C₂²⋊C₄

—

123

C₂×C₂²⋊C₄

—

124

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

125

C₂×C₄²

—

126

C₂×C₄²

—

127

C₂×C₄²

C₂×D₁₃

yes

characteristic

128

#95

—

129

#95

—

130

#95

—

131

C₂×C₁₃

—

132

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

characteristic

133

C₂×C₁₃

C₂×#124

yes

characteristic

134

C₂×C₁₃

#535

yes

characteristic

135

C₂×C₁₃

#554

yes

—

136

C₂×C₁₃

#555

yes

—

137

C₂⁴

C₂²×D₁₃

yes

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

Dic₂₆

yes

complemented

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

C₂²×D₁₃

yes

—

151

C₂²×C₄

C₂²×D₁₃

yes

—

152

C₂²×C₄

C₂²×D₁₃

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₁₃

#2563

yes

characteristic, complemented, Sylow

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

C₂×Dic₂₆

yes

—

170

C₂³

C₂³×D₁₃

yes

characteristic

171

Q₈

—

172

D₄

—

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

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

C₂×Dic₂₆

yes

—

191

C₂×C₄

D₄×D₁₃

yes

—

192

C₂×C₄

#6129

yes

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

C₂²×Dic₂₆

yes

characteristic

199

C₂²

C₂×D₄×D₁₃

yes

characteristic

200

C₂²

C₂×#6129

yes

characteristic

201

C₂²

#88251

yes

—

202

C₂²

#41×D₁₃

yes

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

#88132

yes

—

211

C₂

—

212

C₂

C₂×#88132

yes

characteristic

213

C₂

#11719244

yes

—

214

C₂

#11719321

yes

—

215

C₂

D₁₃×#124

yes

characteristic

216

C₂

#11719420

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#390	C₂
prime radical	C₂²×D₄×C₁₃	C₂²
hypercenter	#390	D₁₃
hypocenter	C₁₃	#2563
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #390

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#390

C₂

prime radical

C₂²×D₄×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #390

N	H ≅ G/N	type
2. C₂×#88142	C₂	semidirect
5. #11718638	C₂	semidirect
6. #11718608	C₂	semidirect
7. #11718603	C₂	semidirect
8. #11718597	C₂	semidirect
9. C₄×#6101	C₂	semidirect
32. C₂×D₄×C₁₃	Q₈	semidirect
142. C₂×D₄	Dic₂₆	semidirect
164. C₁₃	#2563	semidirect

H ≅ G/N

type

2. C₂×#88142

C₂

semidirect

5. #11718638

C₂

semidirect

6. #11718608

C₂

semidirect

7. #11718603

C₂

semidirect

8. #11718597

C₂

semidirect

9. C₄×#6101

C₂

semidirect

32. C₂×D₄×C₁₃

Q₈

semidirect

142. C₂×D₄

Dic₂₆

semidirect

164. C₁₃

#2563

semidirect

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

subgroup	index
C₂×D₄×Dic₁₃	2
C₂×#88142	2
C₁₃×#390	2
#11718882	2
#11718638	2
#11718608	2
#11718603	2
#11718597	2
C₄×#6101	2
#2563	13

subgroup

index

C₂×D₄×Dic₁₃

C₁₃×#390

Subgroups — #13120007

Conjugacy classes of subgroups

Complements

Maximal subgroups