Group #13118476 — subgroups

Subgroups — #13118476

Order 1664.

800 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 213 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	C₂³×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#88198	C₂	1	2	yes	maximal, complemented
3	C₂×#88178	C₂	1	1	yes	maximal, characteristic
4	#11718891	C₂	1	2	yes	maximal, complemented
5	#11718890	C₂	1	4	yes	maximal, complemented
6	#11718871	C₂	1	4	yes	maximal
7	C₄×C₂²⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₃	—	2	8	no	—
9	C₂²×Q₈×C₁₃	—	2	4	no	—
10	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
17	#88198	—	2	8	no	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
19	#88178	—	2	4	no	—
20	C₂×#6101	C₂²	1	2	yes	—
21	C₂×#6101	C₂²	1	8	yes	—
22	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
23	#88071	C₂²	1	2	yes	—
24	#88071	C₂²	1	8	yes	—
25	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₁₃	—	2	32	no	—
27	C₂×Q₈×C₁₃	—	2	8	no	—
28	C₂×Q₈×C₁₃	—	2	4	no	—
29	C₂×Q₈×C₁₃	—	2	4	no	—
30	C₂×Q₈×C₁₃	—	2	4	no	—
31	C₂×Q₈×C₁₃	D₄	1	8	yes	—
32	C₂²×C₄×C₁₃	—	2	4	no	—
33	C₂²×C₄×C₁₃	—	2	8	no	—
34	C₂²×C₄×C₁₃	—	2	4	no	—
35	C₂²×C₄×C₁₃	D₄	1	4	yes	—
36	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₃	C₂³	1	2	yes	—
40	C₂²×Dic₁₃	C₂³	1	2	yes	—
41	C₂²×Dic₁₃	C₂³	1	4	yes	—
42	C₂×Dic₂₆	—	2	2	no	—
43	C₂×Dic₂₆	—	2	4	no	—
44	C₂²⋊Dic₁₃	—	2	4	no	—
45	C₂²⋊Dic₁₃	Q₈	1	8	yes	complemented
46	C₄⋊Dic₁₃	—	2	2	no	—
47	#6101	—	2	4	no	—
48	#6101	—	2	16	no	—
49	C₄×Dic₁₃	—	2	4	no	—
50	#2434	—	13	1	no	maximal, Sylow
51	C₂³×C₁₃	—	2	2	no	—
52	C₂³×C₁₃	—	2	2	no	—
53	C₂³×C₁₃	C₂×D₄	1	2	yes	—
54	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
56	Q₈×C₁₃	—	2	16	no	—
57	Q₈×C₁₃	—	2	16	no	—
58	C₂×C₄×C₁₃	—	2	16	no	—
59	C₂×C₄×C₁₃	—	2	4	no	—
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	8	no	—
63	C₂×C₄×C₁₃	C₂×D₄	1	8	yes	—
64	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
66	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×Dic₁₃	—	2	4	no	—
68	C₂×Dic₁₃	—	2	4	no	—
69	C₂×Dic₁₃	—	2	4	no	—
70	C₂×Dic₁₃	—	2	2	no	—
71	C₂×Dic₁₃	—	2	2	no	—
72	C₂×Dic₁₃	—	2	4	no	—
73	C₂×Dic₁₃	C₂×Q₈	1	8	yes	complemented
74	Dic₂₆	—	4	4	no	—
75	C₂³×Q₈	D₁₃	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	—	13	2	no	—
77	C₂×C₂²⋊Q₈	—	13	1	no	—
78	#395	—	13	4	no	—
79	#391	—	13	2	no	—
80	#385	—	13	4	no	—
81	C₄×C₂²⋊C₄	—	13	1	no	—
82	C₂²×C₁₃	—	2	2	no	—
83	C₂²×C₁₃	—	2	2	no	—
84	C₂²×C₁₃	—	2	8	no	—
85	C₂²×C₁₃	C₄⋊Q₈	1	4	yes	complemented
86	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
88	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
89	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
90	C₄×C₁₃	—	2	8	no	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	#120	1	4	yes	—
93	Dic₁₃	—	2	8	no	—
94	Dic₁₃	—	4	2	no	—
95	Dic₁₃	—	4	2	no	—
96	C₂²×Q₈	—	2	8	no	—
97	C₂²×Q₈	—	2	4	no	—
98	C₂²×Q₈	—	13	1	no	—
99	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
100	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
101	C₂³×C₄	C₂×D₁₃	1	2	yes	—
102	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
103	C₄⋊Q₈	—	26	8	no	—
104	C₂²⋊Q₈	—	26	4	no	—
105	C₂×C₄⋊C₄	—	13	2	no	—
106	C₂×C₄⋊C₄	—	13	1	no	—
107	C₂×C₄⋊C₄	—	13	8	no	—
108	C₂×C₂²⋊C₄	—	13	2	no	—
109	C₂×C₂²⋊C₄	—	13	2	no	—
110	C₂×C₄²	—	13	2	no	—
111	#95	—	13	2	no	—
112	#95	—	13	8	no	—
113	C₂×C₁₃	—	2	4	no	—
114	C₂×C₁₃	C₂×#120	1	1	yes	characteristic
115	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₁₃	#536	1	1	yes	characteristic
117	C₂×C₁₃	D₄×Q₈	1	4	yes	—
118	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
119	C₂×Q₈	—	2	32	no	—
120	C₂×Q₈	—	2	8	no	—
121	C₂×Q₈	—	2	4	no	—
122	C₂×Q₈	—	2	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	26	2	no	—
125	C₂×Q₈	—	26	4	no	—
126	C₂×Q₈	C₂²⋊D₁₃	1	8	yes	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	13	4	no	—
131	C₂²×C₄	—	13	2	no	—
132	C₂²×C₄	—	13	2	no	—
133	C₂²×C₄	C₂²⋊D₁₃	1	4	yes	—
134	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
135	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
137	C₄⋊C₄	—	26	4	no	—
138	C₄⋊C₄	—	26	2	no	—
139	C₄⋊C₄	—	26	16	no	—
140	C₂²⋊C₄	—	13	8	no	—
141	C₂²⋊C₄	—	26	4	no	—
142	C₄²	—	26	4	no	—
143	C₁₃	#2434	1	1	yes	characteristic, complemented, Sylow
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	Q₈×D₁₃	1	4	yes	—
147	C₂³	C₂×C₂²⋊D₁₃	1	2	yes	—
148	C₂³	C₂³×D₁₃	1	1	yes	characteristic
149	Q₈	—	2	16	no	—
150	Q₈	—	2	16	no	—
151	Q₈	—	52	4	no	—
152	C₂×C₄	—	2	16	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	13	8	no	—
158	C₂×C₄	—	26	4	no	—
159	C₂×C₄	—	26	4	no	—
160	C₂×C₄	—	26	4	no	—
161	C₂×C₄	—	26	2	no	—
162	C₂×C₄	—	26	2	no	—
163	C₂×C₄	—	26	4	no	—
164	C₂×C₄	C₂×C₂²⋊D₁₃	1	8	yes	—
165	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
167	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
168	C₂²	—	2	8	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	2	no	—
171	C₂²	#88198	1	4	yes	complemented
172	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
173	C₂²	#88254	1	2	yes	—
174	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂²	C₂²×C₂²⋊D₁₃	1	2	yes	—
176	C₄	—	2	8	no	—
177	C₄	—	2	2	no	—
178	C₄	—	26	8	no	—
179	C₄	—	52	2	no	—
180	C₄	—	52	2	no	—
181	C₄	#88207	1	4	yes	—
182	C₂	—	2	4	no	—
183	C₂	C₂×#88198	1	1	yes	characteristic
184	C₂	Q₈×C₂²⋊D₁₃	1	4	yes	—
185	C₂	#11719521	1	1	yes	characteristic
186	C₂	C₂×#88207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Q₈×C₁₃

C₂

yes

maximal, characteristic

C₂×#88198

C₂

yes

maximal, complemented

C₂×#88178

C₂

yes

maximal, characteristic

#11718891

C₂

yes

maximal, complemented

#11718890

C₂

yes

maximal, complemented

#11718871

C₂

yes

maximal

C₄×C₂²⋊Dic₁₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

—

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂³×C₄×C₁₃

C₂²

yes

—

C₂³×C₄×C₁₃

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

#88198

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

#88178

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

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

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

Q₈

yes

complemented

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

#2434

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

—

C₂³×C₁₃

C₂×Q₈

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

Q₈×C₁₃

—

Q₈×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×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

complemented

Dic₂₆

—

C₂³×Q₈

D₁₃

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

#395

—

#391

—

#385

—

C₄×C₂²⋊C₄

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄⋊Q₈

yes

complemented

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#120

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

—

100

C₂²×Q₈

C₂×D₁₃

yes

—

101

C₂³×C₄

C₂×D₁₃

yes

—

102

C₂³×C₄

C₂×D₁₃

yes

characteristic

103

C₄⋊Q₈

—

104

C₂²⋊Q₈

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₂²⋊C₄

—

110

C₂×C₄²

—

111

#95

—

112

#95

—

113

C₂×C₁₃

—

114

C₂×C₁₃

C₂×#120

yes

characteristic

115

C₂×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

116

C₂×C₁₃

#536

yes

characteristic

117

C₂×C₁₃

D₄×Q₈

yes

—

118

C₂⁴

C₂²×D₁₃

yes

characteristic

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

C₂²⋊D₁₃

yes

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

C₂²⋊D₁₃

yes

—

134

C₂²×C₄

C₂²×D₁₃

yes

—

135

C₂²×C₄

C₂²×D₁₃

yes

characteristic

136

C₂²×C₄

C₂²×D₁₃

yes

characteristic

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₂²⋊C₄

—

141

C₂²⋊C₄

—

142

C₄²

—

143

C₁₃

#2434

yes

characteristic, complemented, Sylow

144

C₂³

—

145

C₂³

—

146

C₂³

Q₈×D₁₃

yes

—

147

C₂³

C₂×C₂²⋊D₁₃

yes

—

148

C₂³

C₂³×D₁₃

yes

characteristic

149

Q₈

—

150

Q₈

—

151

Q₈

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

165

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

166

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

167

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#88198

yes

complemented

172

C₂²

C₂×Q₈×D₁₃

yes

—

173

C₂²

#88254

yes

—

174

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

175

C₂²

C₂²×C₂²⋊D₁₃

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#88207

yes

—

182

C₂

—

183

C₂

C₂×#88198

yes

characteristic

184

C₂

Q₈×C₂²⋊D₁₃

yes

—

185

C₂

#11719521

yes

characteristic

186

C₂

C₂×#88207

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂³×Q₈×C₁₃	C₂
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	C₂³×Q₈	D₁₃
hypocenter	C₁₃	#2434
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂³×Q₈×C₁₃

C₂

prime radical

C₂⁴×C₁₃

C₂³

hypercenter

C₂³×Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. C₂×#88198	C₂	semidirect
4. #11718891	C₂	semidirect
5. #11718890	C₂	semidirect
45. C₂²⋊Dic₁₃	Q₈	semidirect
73. C₂×Dic₁₃	C₂×Q₈	semidirect
85. C₂²×C₁₃	C₄⋊Q₈	semidirect
143. C₁₃	#2434	semidirect
171. C₂²	#88198	semidirect

H ≅ G/N

type

2. C₂×#88198

C₂

semidirect

4. #11718891

C₂

semidirect

5. #11718890

C₂

semidirect

45. C₂²⋊Dic₁₃

Q₈

semidirect

73. C₂×Dic₁₃

C₂×Q₈

semidirect

85. C₂²×C₁₃

C₄⋊Q₈

semidirect

143. C₁₃

#2434

semidirect

171. C₂²

#88198

semidirect

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

subgroup	index
C₂³×Q₈×C₁₃	2
C₂×#88198	2
C₂×#88178	2
#11718891	2
#11718890	2
#11718871	2
C₄×C₂²⋊Dic₁₃	2
#2434	13

subgroup

index

C₂³×Q₈×C₁₃

C₄×C₂²⋊Dic₁₃

Subgroups — #13118476

Conjugacy classes of subgroups

Complements

Maximal subgroups