Group #13119790 — subgroups

Subgroups — #13119790

Order 1664.

676 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 189 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×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88200	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
4	#11718653	C₂	1	4	yes	maximal, complemented
5	#11718649	C₂	1	2	yes	maximal
6	#11718637	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×D₅₂	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
11	C₁₃×C₄⋊Q₈	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	#88200	—	2	16	no	—
16	C₂×#6103	C₂²	1	2	yes	—
17	C₂×#6103	C₂²	1	8	yes	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×#6101	C₂²	1	8	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
22	C₄×D₅₂	—	2	4	no	—
23	#88071	C₂²	1	2	yes	—
24	C₂³×D₁₃	C₂³	1	2	yes	—
25	C₂×Q₈×C₁₃	—	2	8	no	—
26	C₂×Q₈×C₁₃	—	2	4	no	—
27	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₂³	1	2	yes	—
32	C₂²×Dic₁₃	C₂³	1	4	yes	—
33	C₂×D₅₂	—	2	2	no	—
34	C₂×D₅₂	—	2	2	no	—
35	C₂×D₅₂	Q₈	1	4	yes	complemented
36	C₂×C₄×D₁₃	—	2	4	no	—
37	C₂×C₄×D₁₃	—	2	8	no	—
38	C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₄²×C₁₃	D₄	1	4	yes	—
41	#6103	—	2	4	no	—
42	#6103	—	2	16	no	—
43	C₄⋊Dic₁₃	—	2	2	no	—
44	C₄⋊Dic₁₃	D₄	1	8	yes	complemented
45	#6101	—	2	16	no	—
46	C₄×Dic₁₃	—	2	4	no	—
47	#2542	—	13	1	no	maximal, Sylow
48	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
49	C₂²×D₁₃	—	2	4	no	—
50	C₂²×D₁₃	—	2	2	no	—
51	C₂²×D₁₃	—	2	2	no	—
52	C₂²×D₁₃	—	2	4	no	—
53	C₂²×D₁₃	C₂×Q₈	1	4	yes	—
54	Q₈×C₁₃	—	4	8	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	8	no	—
57	C₂×C₄×C₁₃	—	2	4	no	—
58	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₁₃	#41	1	2	yes	—
64	C₂×Dic₁₃	—	2	4	no	—
65	C₂×Dic₁₃	—	2	4	no	—
66	C₂×Dic₁₃	—	2	4	no	—
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	C₂×D₄	1	8	yes	—
69	D₅₂	—	2	8	no	—
70	C₄×D₁₃	—	4	8	no	—
71	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
72	C₂×C₂²⋊Q₈	—	13	4	no	—
73	C₂×C₄×D₄	—	13	1	no	—
74	#399	—	13	4	no	—
75	#390	—	13	2	no	—
76	#384	—	13	2	no	—
77	C₄×C₄⋊C₄	—	13	1	no	—
78	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
79	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
82	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
83	C₂×D₁₃	—	2	4	no	—
84	C₂×D₁₃	—	2	8	no	—
85	C₂×D₁₃	—	4	2	no	—
86	C₂×D₁₃	—	4	2	no	—
87	C₂×D₁₃	—	4	4	no	—
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	4	4	no	—
90	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	complemented
91	C₄×C₁₃	#127	1	4	yes	—
92	Dic₁₃	—	2	8	no	—
93	Dic₁₃	—	4	2	no	—
94	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
95	C₂²×D₄	—	13	1	no	—
96	C₂³×C₄	—	13	2	no	—
97	C₄⋊Q₈	—	2	4	no	—
98	C₂²⋊Q₈	—	26	16	no	—
99	C₄×D₄	—	26	4	no	—
100	C₂×C₄⋊C₄	—	13	2	no	—
101	C₂×C₄⋊C₄	—	13	1	no	—
102	C₂×C₄⋊C₄	—	13	8	no	—
103	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
104	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
105	C₂×C₂²⋊C₄	—	13	2	no	—
106	C₂×C₂²⋊C₄	—	13	8	no	—
107	C₂×C₄²	—	13	2	no	—
108	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
109	#95	—	13	2	no	—
110	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
112	C₂×C₁₃	#536	1	1	yes	characteristic
113	C₂×C₁₃	#547	1	2	yes	—
114	C₂×C₁₃	D₄×Q₈	1	2	yes	—
115	D₁₃	—	4	4	no	—
116	C₂⁴	—	13	2	no	—
117	C₂×Q₈	—	2	8	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×D₄	—	13	4	no	—
120	C₂×D₄	—	26	2	no	—
121	C₂×D₄	—	26	2	no	—
122	C₂²×C₄	—	13	4	no	—
123	C₂²×C₄	—	13	2	no	—
124	C₂²×C₄	—	26	8	no	—
125	C₂²×C₄	—	26	4	no	—
126	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
127	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	13	8	no	—
133	C₄⋊C₄	—	26	16	no	—
134	C₄⋊C₄	—	26	2	no	—
135	C₂²⋊C₄	—	26	16	no	—
136	C₂²⋊C₄	—	26	4	no	—
137	C₄²	—	26	4	no	—
138	C₄²	C₂²⋊D₁₃	1	4	yes	—
139	C₁₃	#2542	1	1	yes	characteristic, complemented, Sylow
140	C₂³	—	13	4	no	—
141	C₂³	—	26	4	no	—
142	C₂³	—	26	2	no	—
143	C₂³	—	26	4	no	—
144	C₂³	—	26	2	no	—
145	C₂³	C₂³×D₁₃	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	D₄	—	26	8	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	13	8	no	—
152	C₂×C₄	—	26	4	no	—
153	C₂×C₄	—	26	4	no	—
154	C₂×C₄	—	26	4	no	—
155	C₂×C₄	—	26	2	no	—
156	C₂×C₄	—	52	8	no	—
157	C₂×C₄	D₄×D₁₃	1	4	yes	—
158	C₂×C₄	Q₈×D₁₃	1	2	yes	—
159	C₂×C₄	#6131	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
163	C₂²	—	26	8	no	—
164	C₂²	—	26	4	no	—
165	C₂²	—	52	4	no	—
166	C₂²	—	52	2	no	—
167	C₂²	—	52	2	no	—
168	C₂²	C₂×D₄×D₁₃	1	2	yes	—
169	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
170	C₂²	C₂×#6131	1	1	yes	characteristic
171	C₂²	#88254	1	2	yes	—
172	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	4	4	no	—
175	C₄	—	26	8	no	—
176	C₄	—	52	2	no	—
177	C₄	#88194	1	4	yes	—
178	C₄	#88200	1	4	yes	complemented
179	C₂	—	52	4	no	—
180	C₂	#11719431	1	2	yes	—
181	C₂	#11719432	1	2	yes	—
182	C₂	C₂×#88194	1	1	yes	characteristic
183	C₂	C₂×#88200	1	1	yes	characteristic
184	C₂	#11719521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂×#88200

C₂

yes

maximal, complemented

C₂×C₄×D₅₂

C₂

yes

maximal, characteristic

#11718653

C₂

yes

maximal, complemented

#11718649

C₂

yes

maximal

#11718637

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

characteristic

C₂²×C₄×D₁₃

C₂²

yes

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88200

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄×D₅₂

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×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

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

Q₈

yes

complemented

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

#6103

—

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

complemented

#6101

—

C₄×Dic₁₃

—

#2542

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

C₂×Q₈

yes

—

Q₈×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×D₄

yes

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

D₅₂

—

C₄×D₁₃

—

C₂×C₄⋊Q₈

D₁₃

yes

characteristic, complemented

C₂×C₂²⋊Q₈

—

C₂×C₄×D₄

—

#399

—

#390

—

#384

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

complemented

C₄×C₁₃

#127

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂²×D₄

—

C₂³×C₄

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₄×D₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

104

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

—

107

C₂×C₄²

—

108

C₂×C₄²

C₂×D₁₃

yes

characteristic

109

#95

—

110

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

characteristic

111

C₂×C₁₃

C₂×#127

yes

characteristic

112

C₂×C₁₃

#536

yes

characteristic

113

C₂×C₁₃

#547

yes

—

114

C₂×C₁₃

D₄×Q₈

yes

—

115

D₁₃

—

116

C₂⁴

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₂²×D₁₃

yes

—

127

C₂²×C₄

C₂²×D₁₃

yes

characteristic

128

C₂²×C₄

C₂²×D₁₃

yes

characteristic

129

C₂²×C₄

C₂²×D₁₃

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₂²⋊C₄

—

137

C₄²

—

138

C₄²

C₂²⋊D₁₃

yes

—

139

C₁₃

#2542

yes

characteristic, complemented, Sylow

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂³×D₁₃

yes

characteristic

146

Q₈

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

D₄×D₁₃

yes

—

158

C₂×C₄

Q₈×D₁₃

yes

—

159

C₂×C₄

#6131

yes

—

160

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

161

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

162

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂×D₄×D₁₃

yes

—

169

C₂²

C₂×Q₈×D₁₃

yes

characteristic

170

C₂²

C₂×#6131

yes

characteristic

171

C₂²

#88254

yes

—

172

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#88194

yes

—

178

C₄

#88200

yes

complemented

179

C₂

—

180

C₂

#11719431

yes

—

181

C₂

#11719432

yes

—

182

C₂

C₂×#88194

yes

characteristic

183

C₂

C₂×#88200

yes

characteristic

184

C₂

#11719521

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2542
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

prime radical

C₂²×D₅₂

C₂²

hypercenter

C₂×C₄⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×C₁₃×C₄⋊Q₈	C₂	semidirect
2. C₂×#88200	C₂	semidirect
4. #11718653	C₂	semidirect
6. #11718637	C₂	semidirect
7. C₄×C₄⋊Dic₁₃	C₂	semidirect
35. C₂×D₅₂	Q₈	semidirect
44. C₄⋊Dic₁₃	D₄	semidirect
71. C₂×C₄⋊Q₈	D₁₃	semidirect
90. C₄×C₁₃	C₂²⋊Q₈	semidirect
139. C₁₃	#2542	semidirect
178. C₄	#88200	semidirect

H ≅ G/N

type

1. C₂×C₁₃×C₄⋊Q₈

C₂

semidirect

2. C₂×#88200

C₂

semidirect

4. #11718653

C₂

semidirect

6. #11718637

C₂

semidirect

7. C₄×C₄⋊Dic₁₃

C₂

semidirect

35. C₂×D₅₂

Q₈

semidirect

44. C₄⋊Dic₁₃

D₄

semidirect

71. C₂×C₄⋊Q₈

D₁₃

semidirect

90. C₄×C₁₃

C₂²⋊Q₈

semidirect

139. C₁₃

#2542

semidirect

178. C₄

#88200

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88200	2
C₂×C₄×D₅₂	2
#11718653	2
#11718649	2
#11718637	2
C₄×C₄⋊Dic₁₃	2
#2542	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

C₂×C₄×D₅₂

C₄×C₄⋊Dic₁₃

Subgroups — #13119790

Conjugacy classes of subgroups

Complements

Maximal subgroups