Group #13131746 — subgroups

Subgroups — #13131746

Order 1664.

330 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 148 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	#11719393	C₂	1	1	yes	maximal, characteristic
2	#11719078	C₂	1	4	yes	maximal, complemented
3	#11719077	C₂	1	4	yes	maximal
4	#11719074	C₂	1	4	yes	maximal, complemented
5	#11719049	C₂	1	2	yes	maximal
6	C₁₃×#125	C₄	1	1	yes	characteristic
7	#88150	C₄	1	4	yes	—
8	#88146	C₂²	1	2	yes	—
9	#88142	C₄	1	2	yes	—
10	#88141	C₂²	1	4	yes	—
11	C₄²×D₁₃	C₂²	1	1	yes	characteristic
12	#88124	C₄	1	1	yes	characteristic
13	#88112	C₂²	1	2	yes	—
14	#88112	C₂²	1	2	yes	—
15	#88112	C₂²	1	4	yes	—
16	#88111	C₂²	1	4	yes	—
17	#88110	C₂²	1	4	yes	—
18	C₄⋊#1057	C₂²	1	4	yes	—
19	C₄×#1057	C₂²	1	4	yes	—
20	C₄×#1057	C₂²	1	4	yes	—
21	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
24	C₂×#1057	C₂³	1	4	yes	—
25	C₂×#1057	C₂³	1	4	yes	—
26	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
27	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	#6103	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
31	#6101	C₂×C₄	1	4	yes	—
32	#6101	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₃	C₂³	1	2	yes	—
35	C₄×Dic₁₃	C₂³	1	4	yes	—
36	#2986	—	13	1	no	maximal, Sylow
37	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
41	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
42	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
43	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
44	C₄×D₁₃	—	2	4	no	—
45	C₄×D₁₃	—	2	2	no	—
46	Dic₂₆	—	2	8	no	—
47	#1057	—	2	4	no	—
48	#1057	—	2	4	no	—
49	#533	—	13	1	no	—
50	#446	—	13	4	no	—
51	C₈×Q₈	—	13	4	no	—
52	#433	—	13	4	no	—
53	#431	—	13	2	no	—
54	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
55	C₂×D₁₃	—	2	1	no	—
56	C₂×D₁₃	—	2	2	no	—
57	C₄×C₁₃	—	2	1	no	—
58	C₄×C₁₃	—	2	4	no	—
59	C₄×C₁₃	—	2	2	no	—
60	Dic₁₃	—	2	4	no	—
61	Dic₁₃	#131	1	4	yes	—
62	Dic₁₃	2₋¹⁺⁴	1	2	yes	—
63	C₄⋊Q₈	—	13	2	no	—
64	#125	—	13	1	no	—
65	#125	#261	1	1	yes	characteristic
66	C₂²⋊Q₈	—	13	4	no	—
67	C₄×Q₈	—	13	4	no	—
68	#117	—	13	2	no	—
69	C₂×C₄²	—	13	1	no	—
70	C₄⋊C₈	—	13	4	no	—
71	C₄⋊C₈	—	13	2	no	—
72	C₄⋊C₈	—	13	2	no	—
73	C₄⋊C₈	—	13	4	no	—
74	C₂²⋊C₈	—	13	4	no	—
75	#97	—	13	4	no	—
76	C₄×C₈	—	13	4	no	—
77	C₄×C₈	—	13	4	no	—
78	C₂×C₁₃	#519	1	1	yes	characteristic
79	C₂×C₁₃	C₂×#131	1	2	yes	—
80	D₁₃	—	4	1	no	—
81	C₂×Q₈	—	13	4	no	—
82	C₂²×C₄	—	13	1	no	—
83	C₂²×C₄	—	13	2	no	—
84	C₂×C₈	—	13	4	no	—
85	C₂×C₈	—	13	4	no	—
86	C₄⋊C₄	—	13	2	no	—
87	C₄⋊C₄	—	13	4	no	—
88	C₄⋊C₄	—	13	4	no	—
89	C₄⋊C₄	C₂×#261	1	2	yes	—
90	C₄⋊C₄	C₂×#261	1	4	yes	—
91	C₂²⋊C₄	—	13	4	no	—
92	C₄²	—	13	2	no	—
93	C₄²	—	13	4	no	—
94	C₄²	—	13	1	no	—
95	C₄²	C₂×#261	1	1	yes	characteristic
96	C₁₃	#2986	1	1	yes	characteristic, complemented, Sylow
97	C₂³	—	13	1	no	—
98	Q₈	—	26	8	no	—
99	C₂×C₄	—	13	1	no	—
100	C₂×C₄	—	13	2	no	—
101	C₂×C₄	—	13	4	no	—
102	C₂×C₄	—	26	2	no	—
103	C₂×C₄	—	26	4	no	—
104	C₂×C₄	C₂²×#261	1	1	yes	characteristic
105	C₂×C₄	C₂²×#261	1	2	yes	—
106	C₂×C₄	C₂²×#261	1	4	yes	—
107	C₈	—	26	4	no	—
108	C₈	—	26	4	no	—
109	C₂²	—	26	1	no	—
110	C₂²	—	26	2	no	—
111	C₂²	C₂³×#261	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	4	no	—
115	C₄	—	13	2	no	—
116	C₄	—	13	4	no	—
117	C₄	—	26	4	no	—
118	C₂	—	52	1	no	—
119	C₂	#11719627	1	1	yes	characteristic
120	C₂	D₁₃⋊#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11719393

C₂

yes

maximal, characteristic

#11719078

C₂

yes

maximal, complemented

#11719077

C₂

yes

maximal

#11719074

C₂

yes

maximal, complemented

#11719049

C₂

yes

maximal

C₁₃×#125

C₄

yes

characteristic

#88150

C₄

yes

—

#88146

C₂²

yes

—

#88142

C₄

yes

—

#88141

C₂²

yes

—

C₄²×D₁₃

C₂²

yes

characteristic

#88124

C₄

yes

characteristic

#88112

C₂²

yes

—

#88112

C₂²

yes

—

#88112

C₂²

yes

—

#88111

C₂²

yes

—

#88110

C₂²

yes

—

C₄⋊#1057

C₂²

yes

—

C₄×#1057

C₂²

yes

—

C₄×#1057

C₂²

yes

—

C₂×C₄×D₁₃

C₂×C₄

yes

—

C₂×C₄×D₁₃

C₂³

yes

characteristic

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×#1057

C₂³

yes

—

C₂×#1057

C₂³

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

#6103

C₂×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₄

yes

—

#6101

C₂×C₄

yes

—

#6101

C₂×C₄

yes

—

C₄×Dic₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₃

C₂³

yes

—

C₄×Dic₁₃

C₂³

yes

—

#2986

—

maximal, Sylow

C₂²×D₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂²×C₄

yes

—

C₂×Dic₁₃

C₂⁴

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

—

—

—

—

C₈×Q₈

—

#433

—

#431

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

#131

yes

—

Dic₁₃

2₋¹⁺⁴

yes

—

C₄⋊Q₈

—

—

yes

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

#519

yes

characteristic

C₂×C₁₃

C₂×#131

yes

—

D₁₃

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#261

yes

—

C₄⋊C₄

C₂×#261

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#261

yes

characteristic

C₁₃

#2986

yes

characteristic, complemented, Sylow

C₂³

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂²×#261

yes

characteristic

105

C₂×C₄

C₂²×#261

yes

—

106

C₂×C₄

C₂²×#261

yes

—

107

C₈

—

108

C₈

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×#261

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₂

—

119

C₂

#11719627

yes

characteristic

120

C₂

D₁₃⋊#131

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#125	C₄
prime radical	C₂²×D₁₃	C₂²×C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	#125	#261
hypocenter	C₁₃	#2986
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#261

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×#125

C₄

prime radical

C₂²×D₁₃

C₂²×C₄

cosocle

C₂×Dic₁₃

C₂⁴

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
2. #11719078	C₂	semidirect
4. #11719074	C₂	semidirect
96. C₁₃	#2986	semidirect

H ≅ G/N

type

2. #11719078

C₂

semidirect

4. #11719074

C₂

semidirect

96. C₁₃

#2986

semidirect

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

subgroup	index
#11719393	2
#11719078	2
#11719077	2
#11719074	2
#11719049	2
#2986	13

subgroup

index

Subgroups — #13131746

Conjugacy classes of subgroups

Complements

Maximal subgroups