Group #13125636 — subgroups

Subgroups — #13125636

Order 1664.

514 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88184	C₂	1	2	yes	maximal, complemented
2	C₁₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718796	C₂	1	4	yes	maximal, complemented
4	#11718396	C₂	1	4	yes	maximal, complemented
5	#11718393	C₂	1	4	yes	maximal, complemented
6	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₂²⋊D₁₃	—	2	2	no	—
8	C₂²×D₅₂	C₂²	1	2	yes	complemented
9	C₂×C₁₃×M₁₆	C₂²	1	2	yes	complemented
10	#88184	—	2	2	no	—
11	#88184	—	2	2	no	—
12	#88184	C₂²	1	4	yes	—
13	#88184	C₂²	1	4	yes	complemented
14	C₂×#6103	—	2	2	no	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	complemented
16	C₂×D₁₀₄	C₂²	1	4	yes	complemented
17	C₁₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
18	#88061	C₂²	1	8	yes	complemented
19	#88057	C₂²	1	4	yes	complemented
20	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×D₁₃	—	2	2	no	—
22	C₂²×C₄×C₁₃	—	4	2	no	—
23	C₂²×C₄×C₁₃	D₄	1	2	yes	—
24	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₃	—	2	4	no	—
28	C₂×C₂²⋊D₁₃	—	2	4	no	—
29	C₂×C₂²⋊D₁₃	—	4	8	no	—
30	C₂²×Dic₁₃	—	2	2	no	—
31	C₂×D₅₂	—	2	2	no	—
32	C₂×D₅₂	—	4	4	no	—
33	C₂×D₅₂	C₂³	1	4	yes	—
34	C₁₃×M₁₆	—	4	2	no	—
35	C₂×C₈×C₁₃	C₂³	1	4	yes	—
36	#6103	—	2	4	no	—
37	#6103	—	2	4	no	—
38	C₄⋊Dic₁₃	—	2	2	no	—
39	C₄⋊Dic₁₃	C₂³	1	4	yes	—
40	D₁₀₄	—	4	4	no	—
41	#3192	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	—	2	4	no	—
43	C₂³×C₁₃	—	4	1	no	—
44	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂³×C₁₃	C₂×D₄	1	2	yes	—
46	C₂²×D₁₃	—	2	4	no	—
47	C₂²×D₁₃	—	4	4	no	—
48	C₂²×D₁₃	—	4	4	no	—
49	C₂²×D₁₃	—	4	4	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	4	no	—
52	C₂×C₄×C₁₃	—	4	1	no	—
53	C₂×C₄×C₁₃	—	4	2	no	—
54	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₂²⋊D₁₃	—	4	8	no	—
58	C₂²⋊D₁₃	—	4	8	no	—
59	C₂×Dic₁₃	—	2	4	no	—
60	C₂×Dic₁₃	—	4	4	no	—
61	D₅₂	—	2	8	no	—
62	D₅₂	—	4	4	no	—
63	C₈×C₁₃	—	2	4	no	—
64	C₂×#121	—	13	2	no	—
65	#481	—	13	4	no	—
66	#469	—	13	4	no	—
67	#447	—	13	4	no	—
68	#407	D₁₃	1	1	yes	characteristic, complemented
69	C₂²×C₁₃	—	2	2	no	—
70	C₂²×C₁₃	—	2	2	no	—
71	C₂²×C₁₃	—	2	1	no	—
72	C₂²×C₁₃	—	2	2	no	—
73	C₂²×C₁₃	—	4	4	no	—
74	C₂²×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
75	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₁₃	—	4	4	no	—
77	C₂×D₁₃	—	4	4	no	—
78	C₂×D₁₃	—	4	4	no	—
79	C₂×D₁₃	—	8	8	no	—
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
83	Dic₁₃	—	4	4	no	—
84	C₂²×D₄	—	13	2	no	—
85	C₂²×D₄	—	26	2	no	—
86	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
87	C₂×D₈	—	13	4	no	—
88	C₂×M₁₆	C₂×D₁₃	1	2	yes	complemented
89	#121	—	13	4	no	—
90	#121	—	13	4	no	—
91	#121	—	26	2	no	—
92	#121	—	26	2	no	—
93	C₂×C₄⋊C₄	—	13	2	no	—
94	C₂×C₂²⋊C₄	—	26	2	no	—
95	#106	—	13	4	no	—
96	#102	—	13	8	no	—
97	C₂²⋊C₈	C₂×D₁₃	1	4	yes	complemented
98	C₂×C₁₃	—	2	4	no	—
99	C₂×C₁₃	—	4	1	no	—
100	C₂×C₁₃	#534	1	1	yes	characteristic
101	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
102	D₁₃	—	8	4	no	—
103	C₂⁴	—	26	2	no	—
104	C₂⁴	D₅₂	1	1	yes	characteristic
105	C₂×D₄	—	13	4	no	—
106	C₂×D₄	—	26	4	no	—
107	C₂×D₄	—	26	4	no	—
108	C₂×D₄	—	26	2	no	—
109	C₂×D₄	—	52	4	no	—
110	C₂×D₄	—	52	8	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	—	26	2	no	—
113	C₂²×C₄	D₅₂	1	2	yes	—
114	C₂²×C₄	D₅₂	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
116	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
117	D₈	—	52	4	no	—
118	M₁₆	—	4	2	no	—
119	C₂×C₈	C₂²×D₁₃	1	4	yes	—
120	C₄⋊C₄	—	13	4	no	—
121	C₄⋊C₄	—	26	2	no	—
122	C₂²⋊C₄	—	26	4	no	—
123	C₂²⋊C₄	—	26	4	no	—
124	C₁₃	#3192	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	2	4	no	—
126	C₂³	—	4	1	no	—
127	C₂³	—	26	4	no	—
128	C₂³	—	52	4	no	—
129	C₂³	—	52	4	no	—
130	C₂³	—	52	4	no	—
131	C₂³	C₂×D₅₂	1	1	yes	characteristic
132	C₂³	C₂×D₅₂	1	2	yes	—
133	D₄	—	26	8	no	—
134	D₄	—	52	4	no	—
135	D₄	—	52	8	no	—
136	D₄	—	52	8	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	26	4	no	—
142	C₂×C₄	—	52	4	no	—
143	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
144	C₂×C₄	C₂×D₅₂	1	2	yes	—
145	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
146	C₈	—	2	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	4	4	no	—
152	C₂²	—	52	4	no	—
153	C₂²	—	52	4	no	—
154	C₂²	—	52	4	no	—
155	C₂²	—	104	8	no	—
156	C₂²	#88162	1	4	yes	—
157	C₂²	C₂²×D₅₂	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	2	no	—
160	C₄	—	52	4	no	—
161	C₄	#88251	1	2	yes	—
162	C₂	—	2	4	no	—
163	C₂	—	4	1	no	—
164	C₂	—	104	4	no	—
165	C₂	#11719198	1	1	yes	characteristic
166	C₂	C₂×#88162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88184

C₂

yes

maximal, complemented

C₁₃×#407

C₂

yes

maximal, characteristic, complemented

#11718796

C₂

yes

maximal, complemented

#11718396

C₂

yes

maximal, complemented

#11718393

C₂

yes

maximal, complemented

C₂³×C₄×C₁₃

C₂²

yes

characteristic

C₂²×C₂²⋊D₁₃

—

C₂²×D₅₂

C₂²

yes

complemented

C₂×C₁₃×M₁₆

C₂²

yes

—

—

C₂²

yes

—

#88184

C₂²

yes

complemented

C₂×#6103

—

C₂×C₄⋊Dic₁₃

C₂²

yes

complemented

C₂×D₁₀₄

C₂²

yes

complemented

C₁₃×C₂²⋊C₈

C₂²

yes

complemented

#88061

C₂²

yes

complemented

#88057

C₂²

yes

complemented

C₂⁴×C₁₃

D₄

yes

characteristic

C₂³×D₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₂²⋊D₁₃

—

C₂×C₂²⋊D₁₃

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

C₂³

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₂³

yes

—

#6103

—

#6103

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

—

D₁₀₄

—

#3192

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂³×C₁₃

C₂×D₄

yes

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₃

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

H(5,𝔽₂)

yes

—

Dic₁₃

—

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

C₂×D₁₃

yes

characteristic

C₂×D₈

—

C₂×M₁₆

C₂×D₁₃

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

#106

—

#102

—

C₂²⋊C₈

C₂×D₁₃

yes

complemented

C₂×C₁₃

—

C₂×C₁₃

—

100

C₂×C₁₃

#534

yes

characteristic

101

C₂×C₁₃

C₂×AGL(1,ℤ/8)

yes

—

102

D₁₃

—

103

C₂⁴

—

104

C₂⁴

D₅₂

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

D₅₂

yes

—

114

C₂²×C₄

D₅₂

yes

characteristic

115

C₂²×C₄

C₂²×D₁₃

yes

—

116

C₂²×C₄

C₂²×D₁₃

yes

characteristic

117

D₈

—

118

M₁₆

—

119

C₂×C₈

C₂²×D₁₃

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₁₃

#3192

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

C₂×D₅₂

yes

characteristic

132

C₂³

C₂×D₅₂

yes

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂×D₅₂

yes

characteristic

144

C₂×C₄

C₂×D₅₂

yes

—

145

C₂×C₄

C₂³×D₁₃

yes

characteristic

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

104

—

156

C₂²

#88162

yes

—

157

C₂²

C₂²×D₅₂

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#88251

yes

—

162

C₂

—

163

C₂

—

164

C₂

—

104

—

165

C₂

#11719198

yes

characteristic

166

C₂

C₂×#88162

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#407	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	#407	D₁₃
hypocenter	C₁₃	#3192
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂²×D₅₂

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₁₃×#407

C₂

socle

C₂²×C₁₃

C₂²×D₄

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
1. C₂×#88184	C₂	semidirect
2. C₁₃×#407	C₂	semidirect
3. #11718796	C₂	semidirect
4. #11718396	C₂	semidirect
5. #11718393	C₂	semidirect
8. C₂²×D₅₂	C₂²	semidirect
9. C₂×C₁₃×M₁₆	C₂²	semidirect
13. #88184	C₂²	semidirect
15. C₂×C₄⋊Dic₁₃	C₂²	semidirect
16. C₂×D₁₀₄	C₂²	semidirect
17. C₁₃×C₂²⋊C₈	C₂²	semidirect
18. #88061	C₂²	semidirect
19. #88057	C₂²	semidirect
68. #407	D₁₃	semidirect
88. C₂×M₁₆	C₂×D₁₃	semidirect
97. C₂²⋊C₈	C₂×D₁₃	semidirect
124. C₁₃	#3192	semidirect

H ≅ G/N

type

1. C₂×#88184

C₂

semidirect

2. C₁₃×#407

C₂

semidirect

3. #11718796

C₂

semidirect

4. #11718396

C₂

semidirect

5. #11718393

C₂

semidirect

8. C₂²×D₅₂

C₂²

semidirect

9. C₂×C₁₃×M₁₆

C₂²

semidirect

13. #88184

C₂²

semidirect

15. C₂×C₄⋊Dic₁₃

C₂²

semidirect

16. C₂×D₁₀₄

C₂²

semidirect

17. C₁₃×C₂²⋊C₈

C₂²

semidirect

18. #88061

C₂²

semidirect

19. #88057

C₂²

semidirect

68. #407

D₁₃

semidirect

88. C₂×M₁₆

C₂×D₁₃

semidirect

97. C₂²⋊C₈

C₂×D₁₃

semidirect

124. C₁₃

#3192

semidirect

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

subgroup	index
C₂×#88184	2
C₁₃×#407	2
#11718796	2
#11718396	2
#11718393	2
#3192	13

subgroup

index

C₁₃×#407

Subgroups — #13125636

Conjugacy classes of subgroups

Complements

Maximal subgroups