Group #13125572 — subgroups

Subgroups — #13125572

Order 1664.

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

normal subgroups only

This group has 446 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	#11719421	C₂	1	2	yes	maximal, complemented
2	C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718820	C₂	1	4	yes	maximal, complemented
4	#11718813	C₂	1	4	yes	maximal, complemented
5	#11718731	C₂	1	2	yes	maximal, complemented
6	C₄×#6094	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#6129	—	2	2	no	—
9	C₂×C₁₃×D₈	C₂²	1	4	yes	—
10	C₁₃×#127	C₂²	1	2	yes	—
11	#88192	—	2	2	no	—
12	#88192	C₂²	1	4	yes	—
13	#88190	—	2	2	no	—
14	D₄×Dic₁₃	—	2	2	no	—
15	D₄×Dic₁₃	C₂²	1	4	yes	—
16	C₂×#6094	C₂²	1	2	yes	—
17	C₄²×D₁₃	C₂²	1	1	yes	characteristic
18	#88123	C₂²	1	2	yes	—
19	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#88072	C₂²	1	8	yes	—
21	#88057	C₂²	1	4	yes	—
22	#88055	C₂²	1	2	yes	—
23	C₄×#1055	C₂²	1	1	yes	characteristic
24	C₂×D₄×C₁₃	—	2	2	no	—
25	C₂×D₄×C₁₃	—	2	2	no	—
26	C₂×D₄×C₁₃	C₂³	1	4	yes	—
27	C₂×C₂²⋊D₁₃	—	2	4	no	—
28	C₂²×Dic₁₃	—	2	4	no	—
29	#6129	—	4	8	no	—
30	C₂×C₄×D₁₃	D₄	1	2	yes	—
31	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₂₆	—	2	2	no	—
33	C₁₃×D₈	—	2	8	no	—
34	C₂×C₈×C₁₃	C₂³	1	2	yes	—
35	C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²⋊Dic₁₃	—	2	4	no	—
37	C₂²⋊Dic₁₃	—	2	4	no	—
38	C₄⋊Dic₁₃	—	2	2	no	—
39	C₄⋊Dic₁₃	C₂³	1	4	yes	—
40	C₄×Dic₁₃	D₄	1	2	yes	—
41	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂×#1055	C₂³	1	2	yes	—
43	#6094	—	2	4	no	—
44	#3184	—	13	1	no	maximal, Sylow
45	C₂³×C₁₃	—	2	4	no	—
46	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
47	D₄×C₁₃	—	2	8	no	—
48	D₄×C₁₃	—	4	4	no	—
49	D₄×C₁₃	—	4	4	no	—
50	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂²⋊D₁₃	—	4	8	no	—
53	C₂×Dic₁₃	—	2	4	no	—
54	C₂×Dic₁₃	—	4	8	no	—
55	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
57	C₄×D₁₃	—	2	4	no	—
58	C₄×D₁₃	—	2	2	no	—
59	Dic₂₆	—	4	4	no	—
60	C₈×C₁₃	#41	1	4	yes	complemented
61	#1055	—	2	2	no	—
62	#532	—	13	2	no	—
63	#498	—	13	1	no	—
64	#493	D₁₃	1	1	yes	characteristic, complemented
65	#486	—	13	2	no	—
66	#469	—	13	4	no	—
67	#442	—	13	4	no	—
68	C₄×M₁₆	—	13	1	no	—
69	C₂²×C₁₃	—	4	4	no	—
70	C₂²×C₁₃	—	4	4	no	—
71	C₂²×C₁₃	—	4	4	no	—
72	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×D₁₃	—	2	1	no	—
74	C₂×D₁₃	—	2	2	no	—
75	C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
76	C₄×C₁₃	C₂×#41	1	2	yes	—
77	Dic₁₃	—	2	1	no	—
78	Dic₁₃	—	2	2	no	—
79	Dic₁₃	—	4	4	no	—
80	C₂×#41	—	26	2	no	—
81	C₂×D₈	C₂×D₁₃	1	4	yes	—
82	C₂×M₁₆	—	13	2	no	—
83	C₄⋊Q₈	—	13	2	no	—
84	#127	C₂×D₁₃	1	2	yes	—
85	#124	—	26	2	no	—
86	#121	—	13	4	no	—
87	#121	—	26	2	no	—
88	C₄×D₄	—	13	4	no	—
89	C₄×D₄	—	26	2	no	—
90	C₂×C₄²	—	13	1	no	—
91	#106	—	13	4	no	—
92	#102	—	13	8	no	—
93	#97	—	13	2	no	—
94	C₄×C₈	—	13	1	no	—
95	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
96	C₂×C₁₃	—	8	4	no	—
97	C₂×C₁₃	#532	1	1	yes	characteristic
98	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
99	D₁₃	—	4	1	no	—
100	#41	—	52	8	no	—
101	C₂×Q₈	—	26	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	26	4	no	—
105	C₂×D₄	C₂²×D₁₃	1	4	yes	—
106	C₂²×C₄	—	13	2	no	—
107	C₂²×C₄	—	13	1	no	—
108	C₂²×C₄	—	26	4	no	—
109	D₈	—	2	8	no	—
110	M₁₆	—	26	4	no	—
111	C₂×C₈	—	13	2	no	—
112	C₂×C₈	C₂²×D₁₃	1	2	yes	—
113	C₄⋊C₄	—	13	4	no	—
114	C₄⋊C₄	—	26	2	no	—
115	C₂²⋊C₄	—	26	4	no	—
116	C₂²⋊C₄	—	26	4	no	—
117	C₄²	—	13	2	no	—
118	C₄²	—	13	1	no	—
119	C₄²	C₂²×D₁₃	1	1	yes	characteristic
120	C₁₃	#3184	1	1	yes	characteristic, complemented, Sylow
121	C₂³	—	2	4	no	—
122	C₂³	—	13	1	no	—
123	Q₈	—	52	4	no	—
124	D₄	—	2	8	no	—
125	D₄	—	4	4	no	—
126	D₄	—	4	4	no	—
127	D₄	—	52	8	no	—
128	C₂×C₄	—	13	1	no	—
129	C₂×C₄	—	13	2	no	—
130	C₂×C₄	—	26	4	no	—
131	C₂×C₄	—	26	2	no	—
132	C₂×C₄	—	26	4	no	—
133	C₂×C₄	—	52	8	no	—
134	C₂×C₄	D₄×D₁₃	1	2	yes	—
135	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
136	C₈	—	26	2	no	—
137	C₈	#6129	1	4	yes	complemented
138	C₂²	—	4	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	26	2	no	—
142	C₂²	—	26	1	no	—
143	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
144	C₄	—	26	2	no	—
145	C₄	—	26	1	no	—
146	C₄	—	52	4	no	—
147	C₄	#88165	1	4	yes	—
148	C₄	C₂×#6129	1	2	yes	—
149	C₂	—	8	4	no	—
150	C₂	—	52	1	no	—
151	C₂	#11719421	1	1	yes	characteristic
152	C₂	C₂×#88165	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719421

C₂

yes

maximal, complemented

C₁₃×#493

C₂

yes

maximal, characteristic, complemented

#11718820

C₂

yes

maximal, complemented

#11718813

C₂

yes

maximal, complemented

#11718731

C₂

yes

maximal, complemented

C₄×#6094

C₂

yes

maximal, characteristic, complemented

#11718341

C₂

yes

maximal, characteristic, complemented

C₂×#6129

—

C₂×C₁₃×D₈

C₂²

yes

—

C₁₃×#127

C₂²

yes

—

#88192

—

#88192

C₂²

yes

—

#88190

—

D₄×Dic₁₃

—

D₄×Dic₁₃

C₂²

yes

—

C₂×#6094

C₂²

yes

—

C₄²×D₁₃

C₂²

yes

characteristic

#88123

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

characteristic

#88072

C₂²

yes

—

#88057

C₂²

yes

—

#88055

C₂²

yes

—

C₄×#1055

C₂²

yes

characteristic

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂³

yes

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

#6129

—

C₂×C₄×D₁₃

D₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

—

C₁₃×D₈

—

C₂×C₈×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

—

C₄×Dic₁₃

D₄

yes

—

C₄×Dic₁₃

C₂³

yes

characteristic

C₂×#1055

C₂³

yes

—

#6094

—

#3184

—

maximal, Sylow

C₂³×C₁₃

—

C₂²×D₁₃

C₂×D₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

C₂×D₄

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

yes

—

—

—

D₁₃

yes

characteristic, complemented

#486

—

#469

—

#442

—

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₃

C₂×#41

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×#41

—

C₂×D₈

C₂×D₁₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

C₂×D₁₃

yes

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

—

#106

—

#102

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#532

yes

characteristic

C₂×C₁₃

C₂×AGL(1,ℤ/8)

yes

—

D₁₃

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂²×D₁₃

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

D₈

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₁₃

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₂²×D₁₃

yes

characteristic

120

C₁₃

#3184

yes

characteristic, complemented, Sylow

121

C₂³

—

122

C₂³

—

123

Q₈

—

124

D₄

—

125

D₄

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₁₃

yes

—

135

C₂×C₄

C₂³×D₁₃

yes

characteristic

136

C₈

—

137

C₈

#6129

yes

complemented

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×D₄×D₁₃

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#88165

yes

—

148

C₄

C₂×#6129

yes

—

149

C₂

—

150

C₂

—

151

C₂

#11719421

yes

characteristic

152

C₂

C₂×#88165

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#493	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	#493	D₁₃
hypocenter	C₁₃	#3184
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₁₃×#493

C₂

socle

C₂²×C₁₃

C₂²×D₄

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #11719421	C₂	semidirect
2. C₁₃×#493	C₂	semidirect
3. #11718820	C₂	semidirect
4. #11718813	C₂	semidirect
5. #11718731	C₂	semidirect
6. C₄×#6094	C₂	semidirect
7. #11718341	C₂	semidirect
60. C₈×C₁₃	#41	semidirect
64. #493	D₁₃	semidirect
120. C₁₃	#3184	semidirect
137. C₈	#6129	semidirect

H ≅ G/N

type

1. #11719421

C₂

semidirect

2. C₁₃×#493

C₂

semidirect

3. #11718820

C₂

semidirect

4. #11718813

C₂

semidirect

5. #11718731

C₂

semidirect

6. C₄×#6094

C₂

semidirect

7. #11718341

C₂

semidirect

60. C₈×C₁₃

#41

semidirect

64. #493

D₁₃

semidirect

120. C₁₃

#3184

semidirect

137. C₈

#6129

semidirect

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

subgroup	index
#11719421	2
C₁₃×#493	2
#11718820	2
#11718813	2
#11718731	2
C₄×#6094	2
#11718341	2
#3184	13

subgroup

index

C₁₃×#493

Subgroups — #13125572

Conjugacy classes of subgroups

Complements

Maximal subgroups