Group #13123319 — subgroups

Subgroups — #13123319

Order 1664.

704 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 289 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×C₁₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	#11719524	C₂	1	2	yes	maximal, complemented
3	#11718898	C₂	1	6	yes	maximal, complemented
4	C₂×#88076	C₂	1	6	yes	maximal, complemented
5	C₁₃×2₋¹⁺⁴	—	2	8	no	—
6	C₂×C₁₃×#41	C₄	1	6	yes	—
7	C₂×C₁₃×#41	C₂²	1	3	yes	—
8	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
9	C₂²×Q₈×C₁₃	C₄	1	2	yes	—
10	C₂²×Q₈×C₁₃	C₂²	1	3	yes	—
11	#88202	—	2	8	no	—
12	C₂×#6099	C₂²	1	12	yes	—
13	#88076	C₂²	1	12	yes	—
14	#88073	C₂²	1	4	yes	—
15	C₁₃×#41	—	2	24	no	—
16	C₁₃×#41	—	2	12	no	—
17	C₁₃×#41	D₄	1	8	yes	—
18	C₂×Q₈×C₁₃	—	2	8	no	—
19	C₂×Q₈×C₁₃	—	2	12	no	—
20	C₂×Q₈×C₁₃	C₂×C₄	1	6	yes	—
21	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Q₈×C₁₃	C₂³	1	3	yes	—
23	C₂×D₄×C₁₃	C₂×C₄	1	6	yes	—
24	C₂×D₄×C₁₃	C₂×C₄	1	3	yes	—
25	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂×C₄	1	6	yes	—
27	C₂²×C₄×C₁₃	C₂×C₄	1	6	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
29	#6099	—	2	12	no	—
30	#6099	C₂³	1	8	yes	—
31	C₂×#1055	—	2	12	no	—
32	#2889	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
34	C₂³×C₁₃	C₂²×C₄	1	3	yes	—
35	Q₈×C₁₃	—	2	12	no	—
36	Q₈×C₁₃	—	2	6	no	—
37	Q₈×C₁₃	C₂²⋊C₄	1	4	yes	—
38	D₄×C₁₃	—	2	12	no	—
39	D₄×C₁₃	—	2	2	no	—
40	D₄×C₁₃	C₂²⋊C₄	1	12	yes	—
41	C₂×C₄×C₁₃	—	2	12	no	—
42	C₂×C₄×C₁₃	—	2	12	no	—
43	C₂×C₄×C₁₃	C₂²×C₄	1	6	yes	—
44	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₃	C₂×D₄	1	12	yes	—
46	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
47	#1055	—	2	8	no	—
48	C₂×2₋¹⁺⁴	D₁₃	1	1	yes	characteristic
49	#568	—	13	2	no	—
50	#413	—	13	6	no	—
51	C₂×#101	—	13	6	no	—
52	C₂²×C₁₃	—	2	2	no	—
53	C₂²×C₁₃	—	4	3	no	—
54	C₂²×C₁₃	—	4	2	no	—
55	C₂²×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
56	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
57	C₄×C₁₃	—	2	6	no	—
58	C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
59	C₄×C₁₃	C₂²×D₄	1	2	yes	—
60	2₋¹⁺⁴	—	2	8	no	—
61	C₂×#41	Dic₁₃	1	6	yes	—
62	C₂×#41	C₂×D₁₃	1	3	yes	—
63	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
64	C₂²×Q₈	Dic₁₃	1	2	yes	—
65	C₂²×Q₈	C₂×D₁₃	1	3	yes	—
66	#131	—	26	8	no	—
67	C₂×M₁₆	—	13	12	no	—
68	#101	—	13	12	no	—
69	#100	—	13	4	no	—
70	C₂×C₁₃	—	2	1	no	—
71	C₂×C₁₃	—	2	6	no	—
72	C₂×C₁₃	—	4	2	no	—
73	C₂×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
74	#41	—	2	24	no	—
75	#41	—	2	12	no	—
76	#41	C₂²⋊D₁₃	1	8	yes	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	—	2	12	no	—
79	C₂×Q₈	C₂×Dic₁₃	1	6	yes	—
80	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
81	C₂×Q₈	C₂²×D₁₃	1	3	yes	—
82	C₂×D₄	C₂×Dic₁₃	1	6	yes	—
83	C₂×D₄	C₂×Dic₁₃	1	3	yes	—
84	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₁₃	1	6	yes	—
86	C₂²×C₄	C₂×Dic₁₃	1	6	yes	—
87	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
88	M₁₆	—	13	8	no	—
89	M₁₆	—	26	12	no	—
90	C₂×C₈	—	26	12	no	—
91	C₁₃	#2889	1	1	yes	characteristic, complemented, Sylow
92	C₂³	C₂²×Dic₁₃	1	2	yes	—
93	C₂³	C₂²×Dic₁₃	1	3	yes	—
94	Q₈	—	2	12	no	—
95	Q₈	—	2	6	no	—
96	Q₈	C₂²⋊Dic₁₃	1	4	yes	—
97	D₄	—	2	12	no	—
98	D₄	—	2	2	no	—
99	D₄	C₂²⋊Dic₁₃	1	12	yes	—
100	C₂×C₄	—	2	12	no	—
101	C₂×C₄	—	2	12	no	—
102	C₂×C₄	C₂²×Dic₁₃	1	6	yes	—
103	C₂×C₄	C₂²×Dic₁₃	1	3	yes	—
104	C₂×C₄	C₂×C₂²⋊D₁₃	1	12	yes	—
105	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
106	C₈	—	26	8	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	3	no	—
109	C₂²	—	4	2	no	—
110	C₂²	C₂×C₂²⋊Dic₁₃	1	6	yes	—
111	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
112	C₄	—	2	6	no	—
113	C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
114	C₄	C₂²×C₂²⋊D₁₃	1	2	yes	—
115	C₂	—	2	1	no	—
116	C₂	—	2	6	no	—
117	C₂	—	4	2	no	—
118	C₂	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×2₋¹⁺⁴

C₂

yes

maximal, characteristic

#11719524

C₂

yes

maximal, complemented

#11718898

C₂

yes

maximal, complemented

C₂×#88076

C₂

yes

maximal, complemented

C₁₃×2₋¹⁺⁴

—

C₂×C₁₃×#41

C₄

yes

—

C₂×C₁₃×#41

C₂²

yes

—

C₂×C₁₃×#41

C₂²

yes

characteristic

C₂²×Q₈×C₁₃

C₄

yes

—

C₂²×Q₈×C₁₃

C₂²

yes

—

#88202

—

C₂×#6099

C₂²

yes

—

#88076

C₂²

yes

—

#88073

C₂²

yes

—

C₁₃×#41

—

C₁₃×#41

—

C₁₃×#41

D₄

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×Q₈×C₁₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₁₃

C₂³

yes

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

#6099

—

#6099

C₂³

yes

—

C₂×#1055

—

#2889

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

—

C₂³×C₁₃

C₂²×C₄

yes

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

C₂²⋊C₄

yes

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

#1055

—

C₂×2₋¹⁺⁴

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₂×C₂²⋊C₄

yes

—

C₄×C₁₃

C₂²×D₄

yes

—

2₋¹⁺⁴

—

C₂×#41

Dic₁₃

yes

—

C₂×#41

C₂×D₁₃

yes

—

C₂×#41

C₂×D₁₃

yes

characteristic

C₂²×Q₈

Dic₁₃

yes

—

C₂²×Q₈

C₂×D₁₃

yes

—

#131

—

C₂×M₁₆

—

#101

—

#100

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂²×C₂²⋊C₄

yes

—

—

C₂²⋊D₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂×Q₈

C₂×Dic₁₃

yes

characteristic

C₂×Q₈

C₂²×D₁₃

yes

—

C₂×D₄

C₂×Dic₁₃

yes

—

C₂×D₄

C₂×Dic₁₃

yes

—

C₂×D₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₁₃

#2889

yes

characteristic, complemented, Sylow

C₂³

C₂²×Dic₁₃

yes

—

C₂³

C₂²×Dic₁₃

yes

—

Q₈

—

Q₈

—

Q₈

C₂²⋊Dic₁₃

yes

—

D₄

—

D₄

—

D₄

C₂²⋊Dic₁₃

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²×Dic₁₃

yes

—

103

C₂×C₄

C₂²×Dic₁₃

yes

—

104

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

105

C₂×C₄

C₂³×D₁₃

yes

characteristic

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊Dic₁₃

yes

—

111

C₂²

C₂³×Dic₁₃

yes

characteristic

112

C₄

—

113

C₄

C₂×C₂²⋊Dic₁₃

yes

—

114

C₄

C₂²×C₂²⋊D₁₃

yes

—

115

C₂

—

116

C₂

—

117

C₂

—

118

C₂

C₂²×C₂²⋊Dic₁₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×2₋¹⁺⁴	C₂
socle	C₂×C₁₃	C₂²×C₂²⋊C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×2₋¹⁺⁴	D₁₃
hypocenter	C₁₃	#2889
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

subgroup

G/H

center

C₂

C₂²×C₂²⋊Dic₁₃

commutator

C₂²×C₁₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₃

Fitting, prime radical

C₂×C₁₃×2₋¹⁺⁴

C₂

socle

C₂×C₁₃

C₂²×C₂²⋊C₄

cosocle

C₂×C₄×C₁₃

C₂⁴

hypercenter

C₂×2₋¹⁺⁴

D₁₃

hypocenter

C₁₃

#2889

LCS

G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

N	H ≅ G/N	type
2. #11719524	C₂	semidirect
3. #11718898	C₂	semidirect
4. C₂×#88076	C₂	semidirect
91. C₁₃	#2889	semidirect

H ≅ G/N

type

2. #11719524

C₂

semidirect

3. #11718898

C₂

semidirect

4. C₂×#88076

C₂

semidirect

91. C₁₃

#2889

semidirect

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

subgroup	index
C₂×C₁₃×2₋¹⁺⁴	2
#11719524	2
#11718898	2
C₂×#88076	2
#2889	13

subgroup

index

C₂×C₁₃×2₋¹⁺⁴

Subgroups — #13123319

Conjugacy classes of subgroups

Complements

Maximal subgroups