Group #13123152 — subgroups

Subgroups — #13123152

Order 1664.

582 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 149 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×#88130	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄²×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718331	C₂	1	3	yes	maximal, complemented
5	#11718329	C₂	1	3	yes	maximal, complemented
6	#11718326	C₂	1	3	yes	maximal
7	#11718314	C₂	1	3	yes	maximal, complemented
8	C₂²×D₅₂	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₁₃	C₂²	1	3	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6103	C₂²	1	3	yes	—
12	C₂×#6103	C₂²	1	3	yes	—
13	C₂×#6103	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
15	C₂×#6101	C₂²	1	3	yes	—
16	C₂×C₄×Dic₁₃	C₂²	1	3	yes	—
17	#88130	—	2	4	no	—
18	C₄²×D₁₃	—	2	4	no	—
19	C₁₃×#95	C₂²	1	3	yes	—
20	C₁₃×#95	C₂²	1	3	yes	—
21	#88071	C₂²	1	3	yes	—
22	#88071	C₂²	1	3	yes	—
23	C₂³×D₁₃	C₂³	1	1	yes	characteristic
24	C₂³×D₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
28	C₂²×Dic₁₃	C₂³	1	3	yes	—
29	C₂²×Dic₁₃	C₂³	1	3	yes	—
30	C₂×D₅₂	—	2	6	no	—
31	C₂×C₄×D₁₃	—	2	6	no	—
32	C₂×C₄×D₁₃	—	2	6	no	—
33	C₂×C₄×D₁₃	—	2	6	no	—
34	C₄²×C₁₃	D₄	1	4	yes	complemented
35	#6103	—	2	6	no	—
36	#6103	—	2	6	no	—
37	#6103	—	2	6	no	—
38	C₄⋊Dic₁₃	—	2	6	no	—
39	#6101	—	2	6	no	—
40	C₄×Dic₁₃	—	2	6	no	—
41	#2852	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
43	C₂²×D₁₃	—	2	1	no	—
44	C₂²×D₁₃	—	2	3	no	—
45	C₂²×D₁₃	—	2	1	no	—
46	C₂²×D₁₃	—	2	3	no	—
47	C₂²×D₁₃	—	2	3	no	—
48	C₂²×D₁₃	#41	1	6	yes	—
49	C₂×C₄×C₁₃	—	2	3	no	—
50	C₂×C₄×C₁₃	—	2	3	no	—
51	C₂×C₄×C₁₃	—	2	3	no	—
52	C₂×C₄×C₁₃	—	2	3	no	—
53	C₂×C₄×C₁₃	—	2	3	no	—
54	C₂×C₄×C₁₃	—	2	3	no	—
55	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
56	C₂×Dic₁₃	—	2	3	no	—
57	C₂×Dic₁₃	—	2	3	no	—
58	C₂×Dic₁₃	—	2	3	no	—
59	C₂×Dic₁₃	—	2	3	no	—
60	C₂×Dic₁₃	—	2	3	no	—
61	C₂×Dic₁₃	#41	1	6	yes	—
62	D₅₂	—	4	4	no	—
63	C₄×D₁₃	—	2	24	no	—
64	C₂×#126	—	13	1	no	—
65	C₂²×C₄²	—	13	1	no	—
66	#388	—	13	3	no	—
67	#387	—	13	3	no	—
68	#386	—	13	3	no	—
69	#385	—	13	3	no	—
70	#383	D₁₃	1	1	yes	characteristic, complemented
71	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₁₃	C₂×#41	1	3	yes	—
73	C₂²×C₁₃	C₂×#41	1	3	yes	—
74	C₂×D₁₃	—	2	6	no	—
75	C₂×D₁₃	—	2	6	no	—
76	C₂×D₁₃	—	4	3	no	—
77	C₂×D₁₃	—	4	1	no	—
78	C₂×D₁₃	—	4	3	no	—
79	C₂×D₁₃	#126	1	4	yes	complemented
80	C₄×C₁₃	—	2	6	no	—
81	C₄×C₁₃	—	4	1	no	—
82	C₄×C₁₃	—	4	3	no	—
83	Dic₁₃	—	2	6	no	—
84	Dic₁₃	—	4	3	no	—
85	C₂²×D₄	—	13	1	no	—
86	C₂³×C₄	—	13	3	no	—
87	#126	—	26	4	no	—
88	C₂×C₄⋊C₄	—	13	3	no	—
89	C₂×C₄⋊C₄	—	13	3	no	—
90	C₂×C₂²⋊C₄	—	13	3	no	—
91	C₂×C₂²⋊C₄	—	13	3	no	—
92	C₂×C₂²⋊C₄	—	13	3	no	—
93	C₂×C₄²	—	13	3	no	—
94	C₂×C₄²	—	26	4	no	—
95	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
96	#95	—	13	3	no	—
97	#95	—	13	3	no	—
98	#95	C₂×D₁₃	1	3	yes	—
99	#95	C₂×D₁₃	1	3	yes	—
100	C₂×C₁₃	#525	1	3	yes	—
101	C₂×C₁₃	C₂×#126	1	1	yes	characteristic
102	C₂×C₁₃	#529	1	3	yes	—
103	D₁₃	—	2	4	no	—
104	D₁₃	—	8	1	no	—
105	C₂⁴	—	13	1	no	—
106	C₂⁴	—	13	1	no	—
107	C₂×D₄	—	26	6	no	—
108	C₂²×C₄	—	13	3	no	—
109	C₂²×C₄	—	13	3	no	—
110	C₂²×C₄	—	26	6	no	—
111	C₂²×C₄	—	26	6	no	—
112	C₂²×C₄	—	26	6	no	—
113	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
115	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
116	C₄⋊C₄	—	26	6	no	—
117	C₄⋊C₄	—	26	6	no	—
118	C₂²⋊C₄	—	26	6	no	—
119	C₂²⋊C₄	—	26	6	no	—
120	C₂²⋊C₄	—	26	6	no	—
121	C₄²	—	26	6	no	—
122	C₄²	D₅₂	1	4	yes	complemented
123	C₁₃	#2852	1	1	yes	characteristic, complemented, Sylow
124	C₂³	—	13	6	no	—
125	C₂³	—	26	1	no	—
126	C₂³	—	26	3	no	—
127	C₂³	—	26	1	no	—
128	C₂³	—	26	3	no	—
129	C₂³	—	26	3	no	—
130	C₂³	C₂³×D₁₃	1	1	yes	characteristic
131	D₄	—	52	4	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	2	3	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	13	6	no	—
139	C₂×C₄	—	26	24	no	—
140	C₂×C₄	—	26	3	no	—
141	C₂×C₄	—	26	3	no	—
142	C₂×C₄	—	26	3	no	—
143	C₂×C₄	—	26	3	no	—
144	C₂×C₄	—	26	3	no	—
145	C₂×C₄	C₂×D₅₂	1	6	yes	—
146	C₂²	—	13	4	no	—
147	C₂²	—	26	6	no	—
148	C₂²	—	26	6	no	—
149	C₂²	—	52	3	no	—
150	C₂²	—	52	1	no	—
151	C₂²	—	52	3	no	—
152	C₂²	C₂²×D₅₂	1	1	yes	characteristic
153	C₂²	#41×D₁₃	1	3	yes	—
154	C₂²	#41×D₁₃	1	3	yes	—
155	C₄	—	2	6	no	—
156	C₄	—	4	1	no	—
157	C₄	—	4	3	no	—
158	C₄	—	26	6	no	—
159	C₄	—	52	3	no	—
160	C₂	—	26	4	no	—
161	C₂	—	104	1	no	—
162	C₂	#11719229	1	3	yes	—
163	C₂	D₁₃×#126	1	1	yes	characteristic
164	C₂	#11719411	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88130

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₁₃

C₂

yes

maximal, characteristic, complemented

C₁₃×#383

C₂

yes

maximal, characteristic, complemented

#11718331

C₂

yes

maximal, complemented

#11718329

C₂

yes

maximal, complemented

#11718326

C₂

yes

maximal

#11718314

C₂

yes

maximal, complemented

C₂²×D₅₂

C₂²

yes

characteristic

C₂²×C₄×D₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88130

—

C₄²×D₁₃

—

C₁₃×#95

C₂²

yes

—

C₁₃×#95

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

characteristic

C₂³×D₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₄²×C₁₃

D₄

yes

—

—

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

#2852

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

#41

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#41

yes

—

D₅₂

—

C₄×D₁₃

—

C₂×#126

—

C₂²×C₄²

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

C₂×#41

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

#126

yes

complemented

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×D₄

—

C₂³×C₄

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

#95

C₂×D₁₃

yes

—

100

C₂×C₁₃

#525

yes

—

101

C₂×C₁₃

C₂×#126

yes

characteristic

102

C₂×C₁₃

#529

yes

—

103

D₁₃

—

104

D₁₃

—

105

C₂⁴

—

106

C₂⁴

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₁₃

yes

characteristic

114

C₂²×C₄

C₂²×D₁₃

yes

—

115

C₂²×C₄

C₂²×D₁₃

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

D₅₂

yes

complemented

123

C₁₃

#2852

yes

characteristic, complemented, Sylow

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂³×D₁₃

yes

characteristic

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₂×D₅₂

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂²×D₅₂

yes

characteristic

153

C₂²

#41×D₁₃

yes

—

154

C₂²

#41×D₁₃

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₂

—

161

C₂

—

104

—

162

C₂

#11719229

yes

—

163

C₂

D₁₃×#126

yes

characteristic

164

C₂

#11719411

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#383	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	#383	D₁₃
hypocenter	C₁₃	#2852
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#383

C₂

prime radical

C₂²×D₅₂

C₂²

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
1. C₂×#88130	C₂	semidirect
2. C₂×C₄²×D₁₃	C₂	semidirect
3. C₁₃×#383	C₂	semidirect
4. #11718331	C₂	semidirect
5. #11718329	C₂	semidirect
7. #11718314	C₂	semidirect
34. C₄²×C₁₃	D₄	semidirect
70. #383	D₁₃	semidirect
79. C₂×D₁₃	#126	semidirect
122. C₄²	D₅₂	semidirect
123. C₁₃	#2852	semidirect

H ≅ G/N

type

1. C₂×#88130

C₂

semidirect

2. C₂×C₄²×D₁₃

C₂

semidirect

3. C₁₃×#383

C₂

semidirect

4. #11718331

C₂

semidirect

5. #11718329

C₂

semidirect

7. #11718314

C₂

semidirect

34. C₄²×C₁₃

D₄

semidirect

70. #383

D₁₃

semidirect

79. C₂×D₁₃

#126

semidirect

122. C₄²

D₅₂

semidirect

123. C₁₃

#2852

semidirect

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

subgroup	index
C₂×#88130	2
C₂×C₄²×D₁₃	2
C₁₃×#383	2
#11718331	2
#11718329	2
#11718326	2
#11718314	2
#2852	13

subgroup

index

C₂×#88130

C₂×C₄²×D₁₃

C₁₃×#383

Subgroups — #13123152

Conjugacy classes of subgroups

Complements

Maximal subgroups