Group #13120190 — subgroups

Subgroups — #13120190

Order 1664.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718882	C₂	1	2	yes	maximal
4	#11718871	C₂	1	4	yes	maximal, complemented
5	#11718634	C₂	1	2	yes	maximal, complemented
6	#11718610	C₂	1	2	yes	maximal, complemented
7	#11718608	C₂	1	2	yes	maximal, complemented
8	#11718597	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
11	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₁₃	—	2	4	no	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
17	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×#6101	C₂²	1	2	yes	—
21	C₂×#6101	C₂²	1	4	yes	—
22	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
23	C₄×Dic₂₆	—	2	4	no	—
24	#88071	C₂²	1	2	yes	—
25	#88071	C₂²	1	2	yes	—
26	#88071	C₂²	1	4	yes	—
27	C₂⁴×C₁₃	C₂³	1	2	yes	—
28	C₂×D₄×C₁₃	—	2	2	no	—
29	C₂×D₄×C₁₃	—	2	2	no	—
30	C₂×D₄×C₁₃	Q₈	1	4	yes	complemented
31	C₂²×C₄×C₁₃	—	2	4	no	—
32	C₂²×C₄×C₁₃	—	4	4	no	—
33	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₃	C₂³	1	4	yes	—
38	C₂²×Dic₁₃	C₂³	1	2	yes	—
39	C₂²×Dic₁₃	C₂³	1	2	yes	—
40	C₂×Dic₂₆	—	2	2	no	—
41	C₂×Dic₂₆	—	2	4	no	—
42	C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₁₃×C₂²⋊C₄	—	2	4	no	—
44	C₄²×C₁₃	D₄	1	4	yes	—
45	C₂²⋊Dic₁₃	—	2	8	no	—
46	C₂²⋊Dic₁₃	—	4	4	no	—
47	C₄⋊Dic₁₃	—	2	2	no	—
48	C₄⋊Dic₁₃	—	2	4	no	—
49	#6101	—	2	4	no	—
50	#6101	—	4	4	no	—
51	C₄×Dic₁₃	—	2	4	no	—
52	#2580	—	13	1	no	maximal, Sylow
53	C₂³×C₁₃	—	2	2	no	—
54	C₂³×C₁₃	—	2	4	no	—
55	C₂³×C₁₃	—	2	2	no	—
56	C₂³×C₁₃	—	2	4	no	—
57	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
58	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
59	D₄×C₁₃	—	2	8	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	—	4	8	no	—
65	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₁₃	#41	1	2	yes	—
70	C₂×Dic₁₃	—	2	2	no	—
71	C₂×Dic₁₃	—	2	2	no	—
72	C₂×Dic₁₃	—	2	2	no	—
73	C₂×Dic₁₃	—	2	4	no	—
74	C₂×Dic₁₃	—	2	4	no	—
75	C₂×Dic₁₃	—	2	2	no	—
76	C₂×Dic₁₃	—	2	4	no	—
77	C₂×Dic₁₃	—	2	2	no	—
78	C₂×Dic₁₃	#41	1	4	yes	—
79	Dic₂₆	—	2	8	no	—
80	C₂×C₄×Q₈	—	13	1	no	—
81	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic
82	#399	—	13	2	no	—
83	#393	—	13	2	no	—
84	#390	—	13	2	no	—
85	#389	—	13	2	no	—
86	#385	—	13	4	no	—
87	#382	—	13	1	no	—
88	C₂²×C₁₃	—	2	4	no	—
89	C₂²×C₁₃	—	2	8	no	—
90	C₂²×C₁₃	—	4	2	no	—
91	C₂²×C₁₃	—	4	2	no	—
92	C₂²×C₁₃	—	4	4	no	—
93	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
94	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
96	C₂²×C₁₃	C₂×#41	1	2	yes	—
97	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
98	C₄×C₁₃	—	2	2	no	—
99	C₄×C₁₃	—	2	2	no	—
100	C₄×C₁₃	—	4	2	no	—
101	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
102	Dic₁₃	—	2	4	no	—
103	Dic₁₃	—	4	2	no	—
104	Dic₁₃	—	4	4	no	—
105	C₂²×Q₈	—	13	1	no	—
106	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
107	C₂³×C₄	C₂×D₁₃	1	2	yes	—
108	C₄×Q₈	—	26	4	no	—
109	C₄×D₄	—	2	4	no	—
110	C₂×C₄⋊C₄	—	13	2	no	—
111	C₂×C₄⋊C₄	—	13	1	no	—
112	C₂×C₄⋊C₄	—	13	2	no	—
113	C₂×C₄⋊C₄	—	13	4	no	—
114	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
115	C₂×C₂²⋊C₄	—	13	4	no	—
116	C₂×C₂²⋊C₄	—	13	4	no	—
117	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
118	C₂×C₄²	—	13	2	no	—
119	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
120	#95	—	13	2	no	—
121	#95	—	13	2	no	—
122	#95	—	13	4	no	—
123	C₂×C₁₃	—	4	4	no	—
124	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₁₃	#525	1	1	yes	characteristic
126	C₂×C₁₃	#535	1	1	yes	characteristic
127	C₂×C₁₃	#554	1	2	yes	—
128	C₂×C₁₃	#559	1	2	yes	—
129	C₂⁴	C₂²×D₁₃	1	2	yes	—
130	C₂×Q₈	—	26	4	no	—
131	C₂×Q₈	—	26	2	no	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	Dic₂₆	1	4	yes	complemented
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	—	13	2	no	—
138	C₂²×C₄	—	13	4	no	—
139	C₂²×C₄	—	13	2	no	—
140	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
141	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	26	4	no	—
146	C₄⋊C₄	—	26	4	no	—
147	C₄⋊C₄	—	26	2	no	—
148	C₄⋊C₄	—	52	4	no	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	26	8	no	—
151	C₂²⋊C₄	—	52	4	no	—
152	C₄²	—	26	4	no	—
153	C₄²	C₂²⋊D₁₃	1	4	yes	—
154	C₁₃	#2580	1	1	yes	characteristic, complemented, Sylow
155	C₂³	—	2	2	no	—
156	C₂³	—	2	4	no	—
157	C₂³	—	2	4	no	—
158	C₂³	—	2	2	no	—
159	C₂³	C₂×Dic₂₆	1	4	yes	—
160	C₂³	C₂³×D₁₃	1	1	yes	characteristic
161	Q₈	—	26	8	no	—
162	D₄	—	2	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	—	13	4	no	—
169	C₂×C₄	—	26	2	no	—
170	C₂×C₄	—	26	2	no	—
171	C₂×C₄	—	26	4	no	—
172	C₂×C₄	—	26	2	no	—
173	C₂×C₄	—	26	4	no	—
174	C₂×C₄	—	26	2	no	—
175	C₂×C₄	—	26	4	no	—
176	C₂×C₄	—	26	2	no	—
177	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
178	C₂×C₄	#6127	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
180	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	8	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	4	no	—
186	C₂²	—	4	2	no	—
187	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
188	C₂²	C₂×#6127	1	1	yes	characteristic
189	C₂²	#88251	1	2	yes	—
190	C₂²	#41×D₁₃	1	2	yes	—
191	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	26	4	no	—
196	C₄	—	52	2	no	—
197	C₄	—	52	4	no	—
198	C₄	#88178	1	4	yes	—
199	C₂	—	4	4	no	—
200	C₂	#11719244	1	2	yes	—
201	C₂	#11719264	1	2	yes	—
202	C₂	C₂×#88178	1	1	yes	characteristic
203	C₂	#11719510	1	1	yes	characteristic
204	C₂	#11719533	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₄×C₁₃

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₂

yes

maximal, characteristic, complemented

#11718882

C₂

yes

maximal

#11718871

C₂

yes

maximal, complemented

#11718634

C₂

yes

maximal, complemented

#11718610

C₂

yes

maximal, complemented

#11718608

C₂

yes

maximal, complemented

#11718597

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₂³×C₄×C₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₄×D₄×C₁₃

—

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

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₂₆

—

#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

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

#2580

—

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×D₄

yes

—

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

#41

yes

—

Dic₂₆

—

C₂×C₄×Q₈

—

C₂×C₄×D₄

D₁₃

yes

—

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

100

C₄×C₁₃

—

101

C₄×C₁₃

C₂²⋊Q₈

yes

—

102

Dic₁₃

—

103

Dic₁₃

—

104

Dic₁₃

—

105

C₂²×Q₈

—

106

C₂²×D₄

C₂×D₁₃

yes

characteristic

107

C₂³×C₄

C₂×D₁₃

yes

—

108

C₄×Q₈

—

109

C₄×D₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

115

C₂×C₂²⋊C₄

—

116

C₂×C₂²⋊C₄

—

117

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

118

C₂×C₄²

—

119

C₂×C₄²

C₂×D₁₃

yes

characteristic

120

#95

—

121

#95

—

122

#95

—

123

C₂×C₁₃

—

124

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

characteristic

125

C₂×C₁₃

#525

yes

characteristic

126

C₂×C₁₃

#535

yes

characteristic

127

C₂×C₁₃

#554

yes

—

128

C₂×C₁₃

#559

yes

—

129

C₂⁴

C₂²×D₁₃

yes

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

Dic₂₆

yes

complemented

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²×D₁₃

yes

—

141

C₂²×C₄

C₂²×D₁₃

yes

characteristic

142

C₂²×C₄

C₂²×D₁₃

yes

characteristic

143

C₂²×C₄

C₂²×D₁₃

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₂²⋊C₄

—

150

C₂²⋊C₄

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₄²

C₂²⋊D₁₃

yes

—

154

C₁₃

#2580

yes

characteristic, complemented, Sylow

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

C₂×Dic₂₆

yes

—

160

C₂³

C₂³×D₁₃

yes

characteristic

161

Q₈

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

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

C₂×Dic₂₆

yes

—

178

C₂×C₄

#6127

yes

—

179

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

180

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

181

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₂²×Dic₂₆

yes

characteristic

188

C₂²

C₂×#6127

yes

characteristic

189

C₂²

#88251

yes

—

190

C₂²

#41×D₁₃

yes

—

191

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

#88178

yes

—

199

C₂

—

200

C₂

#11719244

yes

—

201

C₂

#11719264

yes

—

202

C₂

C₂×#88178

yes

characteristic

203

C₂

#11719510

yes

characteristic

204

C₂

#11719533

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₄×D₄×C₁₃	C₂
prime radical	C₂²×D₄×C₁₃	C₂²
hypercenter	C₂×C₄×D₄	D₁₃
hypocenter	C₁₃	#2580
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₄×D₄×C₁₃

C₂

prime radical

C₂²×D₄×C₁₃

C₂²

hypercenter

C₂×C₄×D₄

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
2. C₂×C₄×Dic₂₆	C₂	semidirect
4. #11718871	C₂	semidirect
5. #11718634	C₂	semidirect
6. #11718610	C₂	semidirect
7. #11718608	C₂	semidirect
8. #11718597	C₂	semidirect
30. C₂×D₄×C₁₃	Q₈	semidirect
134. C₂×D₄	Dic₂₆	semidirect
154. C₁₃	#2580	semidirect

H ≅ G/N

type

2. C₂×C₄×Dic₂₆

C₂

semidirect

4. #11718871

C₂

semidirect

5. #11718634

C₂

semidirect

6. #11718610

C₂

semidirect

7. #11718608

C₂

semidirect

8. #11718597

C₂

semidirect

30. C₂×D₄×C₁₃

Q₈

semidirect

134. C₂×D₄

Dic₂₆

semidirect

154. C₁₃

#2580

semidirect

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

subgroup	index
C₂×C₄×D₄×C₁₃	2
C₂×C₄×Dic₂₆	2
#11718882	2
#11718871	2
#11718634	2
#11718610	2
#11718608	2
#11718597	2
#2580	13

subgroup

index

C₂×C₄×D₄×C₁₃

C₂×C₄×Dic₂₆

Subgroups — #13120190

Conjugacy classes of subgroups

Complements

Maximal subgroups