Group #13114736 — subgroups

Subgroups — #13114736

Order 1664.

272 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×#88058	C₂	1	1	yes	maximal, characteristic
3	C₂×#88055	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₃	C₂	1	2	yes	maximal
5	#11718121	C₂	1	2	yes	maximal
6	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₃	—	2	2	no	—
10	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#1055	C₂²	1	1	yes	characteristic
12	C₁₃×#106	C₄	1	4	yes	complemented
13	#88071	—	2	2	no	—
14	#88058	C₄	1	4	yes	complemented
15	#88055	C₄	1	4	yes	complemented
16	C₂²×C₄×C₁₃	—	2	2	no	—
17	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₃	—	2	2	no	—
19	C₂²×Dic₁₃	D₄	1	2	yes	—
20	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₃	—	2	2	no	—
26	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
27	C₄×Dic₁₃	—	4	4	no	—
28	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₃	Q₈	1	2	yes	—
30	C₂×#1055	—	2	1	no	—
31	C₂×#1055	—	2	1	no	—
32	C₂×#1055	C₂×C₄	1	2	yes	—
33	#1776	—	13	1	no	maximal, Sylow
34	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃	—	2	4	no	—
36	C₂×C₄×C₁₃	—	4	2	no	—
37	C₂×C₄×C₁₃	—	4	2	no	—
38	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
44	C₂×Dic₁₃	—	2	2	no	—
45	C₂×Dic₁₃	—	2	2	no	—
46	C₂×Dic₁₃	—	2	4	no	—
47	C₂×Dic₁₃	—	4	2	no	—
48	C₂×Dic₁₃	—	4	2	no	—
49	C₂×Dic₁₃	C₄⋊C₄	1	4	yes	—
50	C₈×C₁₃	C₄²	1	4	yes	complemented
51	#1055	—	2	2	no	—
52	C₂×#107	—	13	1	no	—
53	C₂×#106	D₁₃	1	1	yes	characteristic
54	C₂×#97	—	13	1	no	—
55	C₄×C₄⋊C₄	—	13	2	no	—
56	#340	—	13	2	no	—
57	C₂²×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
59	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
60	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₂²×C₁₃	#137	1	2	yes	—
62	C₄×C₁₃	—	4	4	no	—
63	C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
64	C₄×C₁₃	#117	1	1	yes	characteristic
65	C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
66	C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
67	Dic₁₃	—	2	4	no	—
68	Dic₁₃	—	4	4	no	—
69	C₂²×C₈	—	13	1	no	—
70	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
71	C₂×C₄⋊C₄	—	13	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
73	C₂×C₄²	—	13	1	no	—
74	C₂×C₄²	—	26	2	no	—
75	#107	—	13	4	no	—
76	#106	Dic₁₃	1	4	yes	complemented
77	#97	—	13	4	no	—
78	#95	—	26	2	no	—
79	C₂×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₁₃	#428	1	2	yes	—
81	C₂×C₁₃	#440	1	2	yes	—
82	C₂×C₁₃	#441	1	1	yes	characteristic
83	C₂×C₁₃	#442	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	13	2	no	—
86	C₂²×C₄	—	26	2	no	—
87	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
88	C₂×C₈	—	13	2	no	—
89	C₂×C₈	—	26	1	no	—
90	C₂×C₈	—	26	1	no	—
91	C₂×C₈	C₄×D₁₃	1	2	yes	—
92	C₂×C₈	C₄×D₁₃	1	2	yes	—
93	C₂×C₈	C₂×Dic₁₃	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	13	4	no	—
96	C₄⋊C₄	—	26	2	no	—
97	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
98	C₄²	—	13	2	no	—
99	C₄²	—	13	2	no	—
100	C₄²	—	52	4	no	—
101	C₁₃	#1776	1	1	yes	characteristic, complemented, Sylow
102	C₂³	D₄×D₁₃	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	13	4	no	—
107	C₂×C₄	—	26	2	no	—
108	C₂×C₄	—	26	2	no	—
109	C₂×C₄	—	26	4	no	—
110	C₂×C₄	—	52	2	no	—
111	C₂×C₄	—	52	2	no	—
112	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
114	C₂×C₄	#6129	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₃	1	1	yes	characteristic
116	C₂×C₄	#6131	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
118	C₈	—	26	2	no	—
119	C₈	C₄×Dic₁₃	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₁₃	1	1	yes	characteristic
121	C₂²	#88147	1	1	yes	characteristic
122	C₂²	#88168	1	2	yes	—
123	C₂²	#88169	1	2	yes	—
124	C₂²	D₄×Dic₁₃	1	1	yes	characteristic
125	C₄	—	4	4	no	—
126	C₄	—	26	4	no	—
127	C₄	—	52	4	no	—
128	C₄	#88141	1	1	yes	characteristic
129	C₄	#88146	1	1	yes	characteristic
130	C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
131	C₄	Q₈×Dic₁₃	1	1	yes	characteristic
132	C₂	#11718514	1	1	yes	characteristic
133	C₂	#11718522	1	2	yes	—
134	C₂	#11718530	1	1	yes	characteristic
135	C₂	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
136	C₂	#11718825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#106

C₂

yes

maximal, characteristic

C₂×#88058

C₂

yes

maximal, characteristic

C₂×#88055

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₁₃

C₂

yes

maximal

#11718121

C₂

yes

maximal

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

C₂²

yes

characteristic

C₂²×#1055

C₂²

yes

characteristic

C₁₃×#106

C₄

yes

complemented

#88071

—

#88058

C₄

yes

complemented

#88055

C₄

yes

complemented

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

D₄

yes

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×C₄

yes

—

C₄×Dic₁₃

Q₈

yes

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂×C₄

yes

—

#1776

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₄⋊C₄

yes

—

C₈×C₁₃

C₄²

yes

—

—

D₁₃

yes

characteristic

C₂×#97

—

C₄×C₄⋊C₄

—

#340

—

C₂²×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

characteristic

C₂²×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₄²

yes

characteristic

C₄×C₁₃

#117

yes

characteristic

C₄×C₁₃

C₄×Q₈

yes

characteristic

C₄×C₁₃

C₄×Q₈

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

#107

—

#106

Dic₁₃

yes

complemented

#97

—

#95

—

C₂×C₁₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₃

#428

yes

—

C₂×C₁₃

#440

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₃

yes

—

C₂×C₈

C₄×D₁₃

yes

—

C₂×C₈

C₂×Dic₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₁₃

#1776

yes

characteristic, complemented, Sylow

102

C₂³

D₄×D₁₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

113

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

114

C₂×C₄

#6129

yes

characteristic

115

C₂×C₄

Q₈×D₁₃

yes

characteristic

116

C₂×C₄

#6131

yes

characteristic

117

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

118

C₈

—

119

C₈

C₄×Dic₁₃

yes

complemented

120

C₂²

C₄⋊C₄×D₁₃

yes

characteristic

121

C₂²

#88147

yes

characteristic

122

C₂²

#88168

yes

—

123

C₂²

#88169

yes

—

124

C₂²

D₄×Dic₁₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#88141

yes

characteristic

129

C₄

#88146

yes

characteristic

130

C₄

C₂×C₄×Dic₁₃

yes

characteristic

131

C₄

Q₈×Dic₁₃

yes

characteristic

132

C₂

#11718514

yes

characteristic

133

C₂

#11718522

yes

—

134

C₂

#11718530

yes

characteristic

135

C₂

C₄⋊C₄×Dic₁₃

yes

characteristic

136

C₂

#11718825

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₃
commutator	C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#106	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×#106	D₁₃
hypocenter	C₁₃	#1776
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₃

commutator

C₄×C₁₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×#106

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
12. C₁₃×#106	C₄	semidirect
14. #88058	C₄	semidirect
15. #88055	C₄	semidirect
50. C₈×C₁₃	C₄²	semidirect
76. #106	Dic₁₃	semidirect
101. C₁₃	#1776	semidirect
119. C₈	C₄×Dic₁₃	semidirect

H ≅ G/N

type

12. C₁₃×#106

C₄

semidirect

14. #88058

C₄

semidirect

15. #88055

C₄

semidirect

50. C₈×C₁₃

C₄²

semidirect

76. #106

Dic₁₃

semidirect

101. C₁₃

#1776

semidirect

119. C₈

C₄×Dic₁₃

semidirect

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

subgroup	index
C₂×C₁₃×#106	2
C₂×#88058	2
C₂×#88055	2
C₄⋊C₄×Dic₁₃	2
#11718121	2
#1776	13

subgroup

index

C₂×C₁₃×#106

C₄⋊C₄×Dic₁₃

Subgroups — #13114736

Conjugacy classes of subgroups

Complements

Maximal subgroups