Group #13123938 — subgroups

Subgroups — #13123938

Order 1664.

330 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 205 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×#533	C₂	1	1	yes	maximal, characteristic
2	#11718685	C₂	1	4	yes	maximal, complemented
3	Q₈×#1055	C₂	1	4	yes	maximal, complemented
4	#11718661	C₂	1	4	yes	maximal
5	#11718586	C₂	1	2	yes	maximal
6	C₁₃×C₄⋊Q₈	C₄	1	2	yes	—
7	C₁₃×#125	C₄	1	2	yes	—
8	C₁₃×C₂²⋊Q₈	C₄	1	4	yes	—
9	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
10	C₁₃×#117	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²⋊#1055	C₂²	1	4	yes	—
13	C₄⋊#1055	C₂²	1	8	yes	—
14	C₄⋊#1055	C₂²	1	2	yes	—
15	C₄⋊#1055	C₂²	1	2	yes	—
16	#88043	C₂²	1	4	yes	—
17	C₄×#1055	C₂²	1	4	yes	—
18	C₄×#1055	C₂²	1	4	yes	—
19	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
23	C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
24	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
25	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
26	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₃	C₂³	1	2	yes	—
29	C₄²×C₁₃	C₂³	1	4	yes	—
30	C₂×#1055	C₂³	1	4	yes	—
31	C₂×#1055	C₂³	1	4	yes	—
32	#2986	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	Q₈×C₁₃	—	2	8	no	—
35	C₂×C₄×C₁₃	—	2	4	no	—
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₃	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
42	#1055	—	2	4	no	—
43	#1055	—	2	4	no	—
44	#533	D₁₃	1	1	yes	characteristic
45	#446	—	13	4	no	—
46	C₈×Q₈	—	13	4	no	—
47	#433	—	13	4	no	—
48	#431	—	13	2	no	—
49	C₂²×C₁₃	—	2	1	no	—
50	C₂²×C₁₃	—	2	2	no	—
51	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₁₃	—	2	1	no	—
53	C₄×C₁₃	—	2	2	no	—
54	C₄×C₁₃	—	2	8	no	—
55	C₄×C₁₃	#131	1	4	yes	—
56	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
57	C₄⋊Q₈	Dic₁₃	1	2	yes	—
58	#125	Dic₁₃	1	2	yes	—
59	C₂²⋊Q₈	Dic₁₃	1	4	yes	—
60	C₄×Q₈	C₂×D₁₃	1	4	yes	—
61	#117	C₂×D₁₃	1	2	yes	—
62	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
63	C₄⋊C₈	—	13	2	no	—
64	C₄⋊C₈	—	13	2	no	—
65	C₄⋊C₈	—	13	8	no	—
66	C₂²⋊C₈	—	13	4	no	—
67	#97	—	13	4	no	—
68	C₄×C₈	—	13	4	no	—
69	C₄×C₈	—	13	4	no	—
70	C₂×C₁₃	—	4	1	no	—
71	C₂×C₁₃	#519	1	1	yes	characteristic
72	C₂×C₁₃	C₂×#131	1	2	yes	—
73	C₂×Q₈	C₂×Dic₁₃	1	4	yes	—
74	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
76	C₂×C₈	—	13	4	no	—
77	C₂×C₈	—	13	4	no	—
78	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
79	C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
80	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
81	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
82	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
83	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
84	C₄²	C₂²×D₁₃	1	2	yes	—
85	C₄²	C₂²×D₁₃	1	4	yes	—
86	C₁₃	#2986	1	1	yes	characteristic, complemented, Sylow
87	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
88	Q₈	—	2	8	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
93	C₂×C₄	C₂²×Dic₁₃	1	8	yes	—
94	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
95	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
96	C₈	—	26	4	no	—
97	C₈	—	26	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	2	no	—
103	C₄	—	2	8	no	—
104	C₄	#88202	1	4	yes	—
105	C₄	#88254	1	2	yes	—
106	C₂	—	4	1	no	—
107	C₂	#11719517	1	1	yes	characteristic
108	C₂	C₂×#88202	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#533

C₂

yes

maximal, characteristic

#11718685

C₂

yes

maximal, complemented

Q₈×#1055

C₂

yes

maximal, complemented

#11718661

C₂

yes

maximal

#11718586

C₂

yes

maximal

C₁₃×C₄⋊Q₈

C₄

yes

—

C₁₃×#125

C₄

yes

—

C₁₃×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₁₃

C₂²

yes

—

C₁₃×#117

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂²⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

C₄⋊#1055

C₂²

yes

—

#88043

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₄×#1055

C₂²

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₁₃×C₂²⋊C₄

C₂×C₄

yes

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

C₂³

yes

—

C₄²×C₁₃

C₂³

yes

—

C₂×#1055

C₂³

yes

—

C₂×#1055

C₂³

yes

—

#2986

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

—

—

D₁₃

yes

characteristic

#446

—

C₈×Q₈

—

#433

—

#431

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#131

yes

—

C₄×C₁₃

2₋¹⁺⁴

yes

—

C₄⋊Q₈

Dic₁₃

yes

—

#125

Dic₁₃

yes

—

C₂²⋊Q₈

Dic₁₃

yes

—

C₄×Q₈

C₂×D₁₃

yes

—

#117

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#519

yes

characteristic

C₂×C₁₃

C₂×#131

yes

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₄⋊C₄

C₂×Dic₁₃

yes

—

C₂²⋊C₄

C₂×Dic₁₃

yes

—

C₄²

C₂×Dic₁₃

yes

characteristic

C₄²

C₂×Dic₁₃

yes

characteristic

C₄²

C₂²×D₁₃

yes

—

C₄²

C₂²×D₁₃

yes

—

C₁₃

#2986

yes

characteristic, complemented, Sylow

C₂³

C₂²×Dic₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

C₂×C₄

C₂²×Dic₁₃

yes

—

C₂×C₄

C₂²×Dic₁₃

yes

—

C₂×C₄

C₂²×Dic₁₃

yes

—

C₂×C₄

C₂³×D₁₃

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

C₂³×Dic₁₃

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#88202

yes

—

105

C₄

#88254

yes

—

106

C₂

—

107

C₂

#11719517

yes

characteristic

108

C₂

C₂×#88202

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#533	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#533	D₁₃
hypocenter	C₁₃	#2986
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃

commutator, socle

C₂²×C₁₃

C₂³×C₄

Fitting

C₁₃×#533

C₂

prime radical

C₂³×C₁₃

C₂²×C₄

cosocle

C₂×C₄×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
2. #11718685	C₂	semidirect
3. Q₈×#1055	C₂	semidirect
86. C₁₃	#2986	semidirect

H ≅ G/N

type

2. #11718685

C₂

semidirect

3. Q₈×#1055

C₂

semidirect

86. C₁₃

#2986

semidirect

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

subgroup	index
C₁₃×#533	2
#11718685	2
Q₈×#1055	2
#11718661	2
#11718586	2
#2986	13

subgroup

index

C₁₃×#533

Subgroups — #13123938

Conjugacy classes of subgroups

Complements

Maximal subgroups