Group #13112709 — subgroups

Subgroups — #13112709

Order 1664.

138 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₁₃×#432	C₂	1	1	yes	maximal, characteristic
2	#11718587	C₂	1	1	yes	maximal, characteristic
3	C₄×#6099	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
5	C₂×#6099	—	2	1	no	—
6	C₁₃×C₄⋊C₈	—	2	1	no	—
7	C₁₃×C₂²⋊C₈	C₄	1	2	yes	—
8	C₁₃×#97	—	2	1	no	—
9	C₂²⋊#1055	—	2	1	no	—
10	C₄⋊#1055	C₄	1	2	yes	—
11	#88043	—	2	1	no	—
12	#88043	—	2	1	no	—
13	C₄×#1055	C₄	1	2	yes	—
14	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₁₃	—	2	2	no	—
18	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₄²×C₁₃	D₄	1	1	yes	characteristic
22	#6099	—	2	4	no	—
23	C₂×#1055	—	2	2	no	—
24	C₂×#1055	—	2	2	no	—
25	#1291	—	13	1	no	maximal, Sylow
26	C₂³×C₁₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃	—	2	2	no	—
28	C₂×C₄×C₁₃	—	4	1	no	—
29	C₂×C₄×C₁₃	—	4	1	no	—
30	C₂×C₄×C₁₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
36	C₈×C₁₃	—	4	2	no	—
37	#1055	—	2	4	no	—
38	#1055	—	4	2	no	—
39	#432	—	13	1	no	—
40	#432	D₁₃	1	1	yes	characteristic
41	C₄×M₁₆	—	13	1	no	—
42	C₂²×C₁₃	—	2	1	no	—
43	C₂²×C₁₃	—	2	1	no	—
44	C₂²×C₁₃	—	2	1	no	—
45	C₂²×C₁₃	#95	1	1	yes	characteristic
46	C₄×C₁₃	—	2	1	no	—
47	C₄×C₁₃	—	2	1	no	—
48	C₄×C₁₃	—	2	1	no	—
49	C₄×C₁₃	—	2	1	no	—
50	C₄×C₁₃	—	2	1	no	—
51	C₄×C₁₃	#104	1	2	yes	—
52	C₂×M₁₆	—	26	1	no	—
53	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	—	13	2	no	—
56	C₂²⋊C₈	—	26	1	no	—
57	C₂²⋊C₈	Dic₁₃	1	2	yes	—
58	#97	—	2	1	no	—
59	#97	—	26	1	no	—
60	#97	—	26	1	no	—
61	C₄×C₈	—	13	2	no	—
62	C₂×C₁₃	—	4	1	no	—
63	C₂×C₁₃	#338	1	1	yes	characteristic
64	C₂×C₁₃	#339	1	1	yes	characteristic
65	C₂×C₁₃	#344	1	1	yes	characteristic
66	C₂²×C₄	Dic₂₆	1	1	yes	characteristic
67	C₂²×C₄	D₅₂	1	1	yes	characteristic
68	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
69	M₁₆	—	26	4	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	26	2	no	—
72	C₂×C₈	—	26	2	no	—
73	C₄²	C₄×D₁₃	1	1	yes	characteristic
74	C₄²	C₄×D₁₃	1	1	yes	characteristic
75	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
76	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
77	C₁₃	#1291	1	1	yes	characteristic, complemented, Sylow
78	C₂³	C₄⋊Dic₁₃	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	C₄×Dic₁₃	1	1	yes	characteristic
83	C₂×C₄	#6101	1	1	yes	characteristic
84	C₂×C₄	#6101	1	1	yes	characteristic
85	C₂×C₄	#6103	1	1	yes	characteristic
86	C₂×C₄	#6103	1	1	yes	characteristic
87	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
88	C₈	—	4	2	no	—
89	C₈	—	26	4	no	—
90	C₈	—	52	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	#88071	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	#88065	1	2	yes	—
101	C₂	—	4	1	no	—
102	C₂	#11718125	1	1	yes	characteristic
103	C₂	#11718147	1	1	yes	characteristic
104	C₂	#11718152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#432

C₂

yes

maximal, characteristic

#11718587

C₂

yes

maximal, characteristic

C₄×#6099

C₂

yes

maximal, characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6099

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

C₄

yes

—

C₁₃×#97

—

C₂²⋊#1055

—

C₄⋊#1055

C₄

yes

—

#88043

—

#88043

—

C₄×#1055

C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

Q₈

yes

characteristic

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

—

—

—

C₂³×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄²

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₈×C₁₃

—

—

—

—

D₁₃

yes

characteristic

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#95

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#104

yes

—

C₂×M₁₆

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₁₃

yes

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

Dic₂₆

yes

characteristic

C₂²×C₄

D₅₂

yes

characteristic

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₁₃

yes

characteristic

C₄²

C₄×D₁₃

yes

characteristic

C₄²

C₂²⋊D₁₃

yes

characteristic

C₄²

C₂²⋊D₁₃

yes

characteristic

C₁₃

#1291

yes

characteristic, complemented, Sylow

C₂³

C₄⋊Dic₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#88071

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

#88065

yes

—

101

C₂

—

102

C₂

#11718125

yes

characteristic

103

C₂

#11718147

yes

characteristic

104

C₂

#11718152

yes

characteristic

	subgroup	G/H
center	C₂²	#88071
commutator	C₂×C₄×C₁₃	C₄²
Frattini	C₂×C₄²	C₂×D₁₃
Fitting	C₁₃×#432	C₂
socle	C₂²×C₁₃	#95
prime radical	C₂³×C₁₃	C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂²
hypercenter	#432	D₁₃
hypocenter	C₁₃	#1291
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

#88071

commutator

C₂×C₄×C₁₃

C₄²

Frattini

C₂×C₄²

C₂×D₁₃

Fitting

C₁₃×#432

C₂

socle

C₂²×C₁₃

#95

prime radical

C₂³×C₁₃

C₄⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
77. C₁₃	#1291	semidirect

H ≅ G/N

type

77. C₁₃

#1291

semidirect

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

subgroup	index
C₁₃×#432	2
#11718587	2
C₄×#6099	2
#1291	13

subgroup

index

C₁₃×#432

#11718587

C₄×#6099

#1291

Subgroups — #13112709

Conjugacy classes of subgroups

Complements

Maximal subgroups