Group #13120669 — subgroups

Subgroups — #13120669

Order 1664.

472 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 157 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×#88126	C₂	1	1	yes	maximal, characteristic
2	C₂×#88124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718655	C₂	1	4	yes	maximal
5	#11718648	C₂	1	2	yes	maximal
6	#11718646	C₂	1	2	yes	maximal, complemented
7	#11718644	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#6103	C₂²	1	2	yes	—
13	C₂×#6103	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
16	C₂×#6101	C₂²	1	4	yes	—
17	C₂×#6101	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
19	#88126	—	2	4	no	—
20	#88124	—	2	4	no	—
21	#88071	C₂²	1	4	yes	—
22	#88071	C₂²	1	4	yes	—
23	C₂³×D₁₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₃	C₂³	1	2	yes	—
28	C₂²×Dic₁₃	C₂³	1	4	yes	—
29	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×D₁₃	—	2	2	no	—
31	C₂×C₄×D₁₃	—	2	2	no	—
32	C₂×C₄×D₁₃	Q₈	1	4	yes	—
33	C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₁₃×C₄⋊C₄	—	4	4	no	—
35	C₄²×C₁₃	D₄	1	4	yes	—
36	#6103	—	2	4	no	—
37	#6103	—	4	4	no	—
38	C₄⋊Dic₁₃	—	2	4	no	—
39	C₄⋊Dic₁₃	—	4	4	no	—
40	#6101	—	2	8	no	—
41	#6101	—	2	4	no	—
42	C₄×Dic₁₃	—	2	2	no	—
43	#2629	—	13	1	no	maximal, Sylow
44	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
45	C₂²×D₁₃	—	2	1	no	—
46	C₂²×D₁₃	—	2	1	no	—
47	C₂²×D₁₃	—	2	2	no	—
48	C₂²×D₁₃	—	2	2	no	—
49	C₂²×D₁₃	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	—	2	4	no	—
55	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
56	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₃	#41	1	2	yes	—
59	C₂×Dic₁₃	—	2	1	no	—
60	C₂×Dic₁₃	—	2	1	no	—
61	C₂×Dic₁₃	—	2	2	no	—
62	C₂×Dic₁₃	—	2	4	no	—
63	C₂×Dic₁₃	—	2	2	no	—
64	C₂×Dic₁₃	—	2	4	no	—
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	—	2	4	no	—
67	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
68	C₄×D₁₃	—	2	8	no	—
69	C₂×#125	—	13	1	no	—
70	C₂×#117	—	13	1	no	—
71	#398	—	13	4	no	—
72	#396	—	13	4	no	—
73	#389	—	13	2	no	—
74	#384	D₁₃	1	1	yes	characteristic, complemented
75	#380	—	13	2	no	—
76	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
79	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
80	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
81	C₂×D₁₃	—	2	2	no	—
82	C₂×D₁₃	—	2	4	no	—
83	C₂×D₁₃	—	4	2	no	—
84	C₂×D₁₃	—	4	1	no	—
85	C₂×D₁₃	—	4	1	no	—
86	C₄×C₁₃	—	2	4	no	—
87	C₄×C₁₃	—	4	4	no	—
88	C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
89	Dic₁₃	—	2	2	no	—
90	Dic₁₃	—	4	2	no	—
91	Dic₁₃	—	4	4	no	—
92	C₂³×C₄	—	13	1	no	—
93	#125	—	26	4	no	—
94	#117	—	26	4	no	—
95	C₂×C₄⋊C₄	—	13	2	no	—
96	C₂×C₄⋊C₄	—	13	4	no	—
97	C₂×C₄⋊C₄	—	13	2	no	—
98	C₂×C₄⋊C₄	—	13	4	no	—
99	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
100	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
101	C₂×C₂²⋊C₄	—	13	2	no	—
102	C₂×C₂²⋊C₄	—	13	4	no	—
103	C₂×C₄²	—	13	1	no	—
104	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
105	#95	—	13	4	no	—
106	#95	—	13	4	no	—
107	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₁₃	#537	1	2	yes	—
109	C₂×C₁₃	#540	1	1	yes	characteristic
110	C₂×C₁₃	#541	1	1	yes	characteristic
111	C₂×C₁₃	#544	1	2	yes	—
112	D₁₃	—	4	2	no	—
113	C₂⁴	—	13	1	no	—
114	C₂²×C₄	—	13	1	no	—
115	C₂²×C₄	—	13	4	no	—
116	C₂²×C₄	—	13	2	no	—
117	C₂²×C₄	—	13	4	no	—
118	C₂²×C₄	—	26	2	no	—
119	C₂²×C₄	—	26	2	no	—
120	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
121	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
122	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	4	4	no	—
125	C₄⋊C₄	—	26	4	no	—
126	C₄⋊C₄	—	26	8	no	—
127	C₄⋊C₄	—	26	4	no	—
128	C₄⋊C₄	—	52	4	no	—
129	C₂²⋊C₄	—	26	4	no	—
130	C₂²⋊C₄	—	52	4	no	—
131	C₄²	—	26	2	no	—
132	C₄²	D₅₂	1	4	yes	—
133	C₁₃	#2629	1	1	yes	characteristic, complemented, Sylow
134	C₂³	—	13	2	no	—
135	C₂³	—	26	1	no	—
136	C₂³	—	26	1	no	—
137	C₂³	—	26	2	no	—
138	C₂³	—	26	2	no	—
139	C₂³	C₂³×D₁₃	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	13	2	no	—
146	C₂×C₄	—	26	1	no	—
147	C₂×C₄	—	26	1	no	—
148	C₂×C₄	—	26	8	no	—
149	C₂×C₄	—	26	2	no	—
150	C₂×C₄	—	26	4	no	—
151	C₂×C₄	—	26	4	no	—
152	C₂×C₄	—	26	2	no	—
153	C₂×C₄	—	26	2	no	—
154	C₂×C₄	—	26	4	no	—
155	C₂×C₄	C₂×D₅₂	1	4	yes	—
156	C₂×C₄	C₂×D₅₂	1	2	yes	—
157	C₂×C₄	#6129	1	2	yes	—
158	C₂×C₄	Q₈×D₁₃	1	2	yes	—
159	C₂²	—	26	4	no	—
160	C₂²	—	26	2	no	—
161	C₂²	—	52	1	no	—
162	C₂²	—	52	1	no	—
163	C₂²	—	52	2	no	—
164	C₂²	C₂²×D₅₂	1	1	yes	characteristic
165	C₂²	C₂×#6129	1	1	yes	characteristic
166	C₂²	#88251	1	2	yes	—
167	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
168	C₂²	#88254	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	4	4	no	—
171	C₄	—	26	2	no	—
172	C₄	—	52	2	no	—
173	C₄	—	52	4	no	—
174	C₄	#88151	1	4	yes	—
175	C₂	—	52	2	no	—
176	C₂	C₂×#88151	1	1	yes	characteristic
177	C₂	#11719215	1	2	yes	—
178	C₂	#11719424	1	1	yes	characteristic
179	C₂	#11719434	1	2	yes	—
180	C₂	#11719436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88126

C₂

yes

maximal, characteristic

C₂×#88124

C₂

yes

maximal, characteristic, complemented

C₁₃×#384

C₂

yes

maximal, characteristic, complemented

#11718655

C₂

yes

maximal

#11718648

C₂

yes

maximal

#11718646

C₂

yes

maximal, complemented

#11718644

C₂

yes

maximal, complemented

C₂²×C₄×D₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

—

C₂²

yes

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

Q₈

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

#6103

—

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

C₄×Dic₁₃

—

#2629

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

C₂×Q₈

yes

—

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×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×Q₈

yes

—

C₄×D₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic, complemented

#380

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

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

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂³×C₄

—

#125

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

100

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

101

C₂×C₂²⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

—

104

C₂×C₄²

C₂×D₁₃

yes

characteristic

105

#95

—

106

#95

—

107

C₂×C₁₃

C₂×C₂²⋊Q₈

yes

characteristic

108

C₂×C₁₃

#537

yes

—

109

C₂×C₁₃

#540

yes

characteristic

110

C₂×C₁₃

#541

yes

characteristic

111

C₂×C₁₃

#544

yes

—

112

D₁₃

—

113

C₂⁴

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²×D₁₃

yes

—

121

C₂²×C₄

C₂²×D₁₃

yes

—

122

C₂²×C₄

C₂²×D₁₃

yes

characteristic

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₂²⋊C₄

—

130

C₂²⋊C₄

—

131

C₄²

—

132

C₄²

D₅₂

yes

—

133

C₁₃

#2629

yes

characteristic, complemented, Sylow

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

C₂³×D₁₃

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂×D₅₂

yes

—

156

C₂×C₄

C₂×D₅₂

yes

—

157

C₂×C₄

#6129

yes

—

158

C₂×C₄

Q₈×D₁₃

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂²×D₅₂

yes

characteristic

165

C₂²

C₂×#6129

yes

characteristic

166

C₂²

#88251

yes

—

167

C₂²

C₂×Q₈×D₁₃

yes

characteristic

168

C₂²

#88254

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#88151

yes

—

175

C₂

—

176

C₂

C₂×#88151

yes

characteristic

177

C₂

#11719215

yes

—

178

C₂

#11719424

yes

characteristic

179

C₂

#11719434

yes

—

180

C₂

#11719436

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#384	C₂
prime radical	C₂³×D₁₃	C₂³
hypercenter	#384	D₁₃
hypocenter	C₁₃	#2629
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#384

C₂

prime radical

C₂³×D₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
2. C₂×#88124	C₂	semidirect
3. C₁₃×#384	C₂	semidirect
6. #11718646	C₂	semidirect
7. #11718644	C₂	semidirect
74. #384	D₁₃	semidirect
133. C₁₃	#2629	semidirect

H ≅ G/N

type

2. C₂×#88124

C₂

semidirect

3. C₁₃×#384

C₂

semidirect

6. #11718646

C₂

semidirect

7. #11718644

C₂

semidirect

74. #384

D₁₃

semidirect

133. C₁₃

#2629

semidirect

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

subgroup	index
C₂×#88126	2
C₂×#88124	2
C₁₃×#384	2
#11718655	2
#11718648	2
#11718646	2
#11718644	2
#2629	13

subgroup

index

C₁₃×#384

Subgroups — #13120669

Conjugacy classes of subgroups

Complements

Maximal subgroups