Group #13119789 — subgroups

Subgroups — #13119789

Order 1664.

676 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 197 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×#88178	C₂	1	4	yes	maximal, complemented
3	C₂×#88123	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718882	C₂	1	2	yes	maximal
5	#11718637	C₂	1	2	yes	maximal, complemented
6	#11718610	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
10	C₂²×Dic₂₆	C₂²	1	2	yes	—
11	C₄×D₄×C₁₃	—	2	4	no	—
12	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Dic₁₃	C₂²	1	8	yes	—
16	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	#88178	—	2	16	no	—
20	C₂×#6101	C₂²	1	8	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
22	#88123	—	2	4	no	—
23	#88071	C₂²	1	2	yes	—
24	C₂⁴×C₁₃	C₂³	1	2	yes	—
25	C₂×D₄×C₁₃	—	2	2	no	—
26	C₂×D₄×C₁₃	—	2	2	no	—
27	C₂×D₄×C₁₃	Q₈	1	4	yes	complemented
28	C₂²×C₄×C₁₃	—	2	8	no	—
29	C₂²×C₄×C₁₃	—	2	4	no	—
30	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₃	C₂³	1	4	yes	—
35	C₂²×Dic₁₃	C₂³	1	4	yes	—
36	C₂×Dic₂₆	—	2	4	no	—
37	C₂×Dic₂₆	—	2	8	no	—
38	C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₁₃×C₂²⋊C₄	—	2	4	no	—
40	C₄²×C₁₃	D₄	1	4	yes	—
41	C₂²⋊Dic₁₃	—	2	16	no	—
42	C₄⋊Dic₁₃	—	2	4	no	—
43	C₄⋊Dic₁₃	—	2	4	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	—	2	4	no	—
49	C₂³×C₁₃	—	2	2	no	—
50	C₂³×C₁₃	—	2	2	no	—
51	C₂³×C₁₃	—	2	4	no	—
52	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
53	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
54	D₄×C₁₃	—	2	8	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	4	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	4	8	no	—
59	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₃	#41	1	2	yes	—
65	C₂×Dic₁₃	—	2	4	no	—
66	C₂×Dic₁₃	—	2	4	no	—
67	C₂×Dic₁₃	—	2	8	no	—
68	C₂×Dic₁₃	—	2	4	no	—
69	C₂×Dic₁₃	C₂×D₄	1	8	yes	—
70	Dic₂₆	—	4	8	no	—
71	C₂×C₄⋊Q₈	—	13	1	no	—
72	C₂×C₂²⋊Q₈	—	13	4	no	—
73	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic
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	—	2	4	no	—
79	C₂²×C₁₃	—	2	8	no	—
80	C₂²×C₁₃	—	4	2	no	—
81	C₂²×C₁₃	—	4	2	no	—
82	C₂²×C₁₃	—	4	4	no	—
83	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
84	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	4	2	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	4	no	—
94	C₂²×Q₈	—	13	2	no	—
95	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
96	C₂³×C₄	C₂×D₁₃	1	2	yes	—
97	C₄⋊Q₈	—	26	4	no	—
98	C₂²⋊Q₈	—	26	16	no	—
99	C₄×D₄	—	2	4	no	—
100	C₂×C₄⋊C₄	—	13	2	no	—
101	C₂×C₄⋊C₄	—	13	8	no	—
102	C₂×C₄⋊C₄	—	13	2	no	—
103	C₂×C₄⋊C₄	—	13	2	no	—
104	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	—	13	8	no	—
106	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
107	C₂×C₄²	—	13	2	no	—
108	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
109	#95	—	13	2	no	—
110	C₂×C₁₃	—	4	4	no	—
111	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
113	C₂×C₁₃	#536	1	1	yes	characteristic
114	C₂×C₁₃	#547	1	2	yes	—
115	C₂×C₁₃	D₄×Q₈	1	2	yes	—
116	C₂⁴	C₂²×D₁₃	1	2	yes	—
117	C₂×Q₈	—	26	8	no	—
118	C₂×Q₈	—	26	4	no	—
119	C₂×D₄	—	2	2	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	Dic₂₆	1	4	yes	complemented
122	C₂²×C₄	—	2	8	no	—
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	—	13	4	no	—
125	C₂²×C₄	—	13	4	no	—
126	C₂²×C₄	C₂²×D₁₃	1	2	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	2	no	—
131	C₄⋊C₄	—	13	8	no	—
132	C₄⋊C₄	—	26	16	no	—
133	C₄⋊C₄	—	26	4	no	—
134	C₄⋊C₄	—	26	4	no	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₂²⋊C₄	—	26	16	no	—
137	C₄²	—	26	4	no	—
138	C₄²	C₂²⋊D₁₃	1	4	yes	—
139	C₁₃	#2542	1	1	yes	characteristic, complemented, Sylow
140	C₂³	—	2	2	no	—
141	C₂³	—	2	4	no	—
142	C₂³	—	2	4	no	—
143	C₂³	—	2	2	no	—
144	C₂³	C₂×Dic₂₆	1	4	yes	—
145	C₂³	C₂³×D₁₃	1	1	yes	characteristic
146	Q₈	—	52	8	no	—
147	D₄	—	2	8	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	13	8	no	—
153	C₂×C₄	—	26	4	no	—
154	C₂×C₄	—	26	4	no	—
155	C₂×C₄	—	26	8	no	—
156	C₂×C₄	—	26	4	no	—
157	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
158	C₂×C₄	#6127	1	2	yes	—
159	C₂×C₄	D₄×D₁₃	1	4	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	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	4	4	no	—
167	C₂²	—	4	2	no	—
168	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
169	C₂²	C₂×#6127	1	1	yes	characteristic
170	C₂²	C₂×D₄×D₁₃	1	2	yes	—
171	C₂²	#88257	1	2	yes	—
172	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	4	2	no	—
175	C₄	—	26	8	no	—
176	C₄	—	52	4	no	—
177	C₄	#88178	1	4	yes	complemented
178	C₄	#88194	1	4	yes	—
179	C₂	—	4	4	no	—
180	C₂	D₄×Dic₂₆	1	2	yes	—
181	C₂	#11719256	1	2	yes	—
182	C₂	C₂×#88178	1	1	yes	characteristic
183	C₂	C₂×#88194	1	1	yes	characteristic
184	C₂	#11719530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₄×C₁₃

C₂

yes

maximal, characteristic

C₂×#88178

C₂

yes

maximal, complemented

C₂×#88123

C₂

yes

maximal, characteristic, complemented

#11718882

C₂

yes

maximal

#11718637

C₂

yes

maximal, complemented

#11718610

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₂³×C₄×C₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

—

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×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

#88178

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88123

—

#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

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

complemented

#6101

—

C₄×Dic₁₃

—

#2542

—

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×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

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂×C₄×D₄

D₁₃

yes

—

—

—

C₄×C₄⋊C₄

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

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

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊Q₈

yes

complemented

C₄×C₁₃

#127

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×D₄

C₂×D₁₃

yes

characteristic

C₂³×C₄

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₄×D₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂×D₁₃

yes

characteristic

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

107

C₂×C₄²

—

108

C₂×C₄²

C₂×D₁₃

yes

characteristic

109

#95

—

110

C₂×C₁₃

—

111

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

characteristic

112

C₂×C₁₃

C₂×#127

yes

characteristic

113

C₂×C₁₃

#536

yes

characteristic

114

C₂×C₁₃

#547

yes

—

115

C₂×C₁₃

D₄×Q₈

yes

—

116

C₂⁴

C₂²×D₁₃

yes

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

Dic₂₆

yes

complemented

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

C₂×Dic₂₆

yes

—

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

C₂×Dic₂₆

yes

—

158

C₂×C₄

#6127

yes

—

159

C₂×C₄

D₄×D₁₃

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

C₂²×Dic₂₆

yes

characteristic

169

C₂²

C₂×#6127

yes

characteristic

170

C₂²

C₂×D₄×D₁₃

yes

—

171

C₂²

#88257

yes

—

172

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#88178

yes

complemented

178

C₄

#88194

yes

—

179

C₂

—

180

C₂

D₄×Dic₂₆

yes

—

181

C₂

#11719256

yes

—

182

C₂

C₂×#88178

yes

characteristic

183

C₂

C₂×#88194

yes

characteristic

184

C₂

#11719530

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	#2542
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×#88178	C₂	semidirect
3. C₂×#88123	C₂	semidirect
5. #11718637	C₂	semidirect
6. #11718610	C₂	semidirect
7. C₄×C₄⋊Dic₁₃	C₂	semidirect
27. C₂×D₄×C₁₃	Q₈	semidirect
44. C₄⋊Dic₁₃	D₄	semidirect
90. C₄×C₁₃	C₂²⋊Q₈	semidirect
121. C₂×D₄	Dic₂₆	semidirect
139. C₁₃	#2542	semidirect
177. C₄	#88178	semidirect

H ≅ G/N

type

2. C₂×#88178

C₂

semidirect

3. C₂×#88123

C₂

semidirect

5. #11718637

C₂

semidirect

6. #11718610

C₂

semidirect

7. C₄×C₄⋊Dic₁₃

C₂

semidirect

27. C₂×D₄×C₁₃

Q₈

semidirect

44. C₄⋊Dic₁₃

D₄

semidirect

90. C₄×C₁₃

C₂²⋊Q₈

semidirect

121. C₂×D₄

Dic₂₆

semidirect

139. C₁₃

#2542

semidirect

177. C₄

#88178

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×#88178	2
C₂×#88123	2
#11718882	2
#11718637	2
#11718610	2
C₄×C₄⋊Dic₁₃	2
#2542	13

subgroup

index

C₂×C₄×D₄×C₁₃

C₄×C₄⋊Dic₁₃

Subgroups — #13119789

Conjugacy classes of subgroups

Complements

Maximal subgroups