Group #13113428 — subgroups

Subgroups — #13113428

Order 1664.

270 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 85 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×#532	C₂	1	1	yes	maximal, characteristic
2	C₄×#6099	C₂	1	2	yes	maximal, complemented
3	#11718132	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#41	C₄	1	2	yes	—
5	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₁₃×#127	C₄	1	1	yes	characteristic
7	C₁₃×#124	C₂²	1	2	yes	—
8	C₁₃×#121	—	2	2	no	—
9	C₄×D₄×C₁₃	—	2	2	no	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6099	—	2	2	no	—
12	#88043	C₂²	1	4	yes	—
13	C₄×#1055	—	2	2	no	—
14	C₁₃×#41	—	4	4	no	—
15	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
16	C₂×D₄×C₁₃	—	2	2	no	—
17	C₂×D₄×C₁₃	—	2	2	no	—
18	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₃	—	2	2	no	—
20	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃	D₄	1	2	yes	—
22	C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₁₃×C₂²⋊C₄	—	2	4	no	—
24	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₃	D₄	1	2	yes	—
26	C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	#6099	—	2	8	no	—
28	C₂×#1055	—	2	4	no	—
29	#1525	—	13	1	no	maximal, Sylow
30	C₂³×C₁₃	—	2	2	no	—
31	C₂³×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
32	Q₈×C₁₃	—	4	2	no	—
33	D₄×C₁₃	—	4	2	no	—
34	D₄×C₁₃	—	4	4	no	—
35	D₄×C₁₃	—	4	4	no	—
36	C₂×C₄×C₁₃	—	2	2	no	—
37	C₂×C₄×C₁₃	—	2	4	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	—	4	4	no	—
40	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	#1055	—	2	8	no	—
45	#532	D₁₃	1	1	yes	characteristic
46	C₄×M₁₆	—	13	2	no	—
47	#329	—	13	4	no	—
48	C₂²×C₁₃	—	2	1	no	—
49	C₂²×C₁₃	—	2	2	no	—
50	C₂²×C₁₃	—	4	2	no	—
51	C₂²×C₁₃	—	4	4	no	—
52	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₃	—	2	1	no	—
54	C₄×C₁₃	—	2	2	no	—
55	C₄×C₁₃	—	4	2	no	—
56	C₄×C₁₃	#100	1	2	yes	—
57	C₄×C₁₃	#104	1	4	yes	—
58	C₂×#41	Dic₁₃	1	2	yes	—
59	C₂×M₁₆	—	26	2	no	—
60	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
61	#127	Dic₁₃	1	1	yes	characteristic
62	#124	C₂×D₁₃	1	2	yes	—
63	#121	—	2	2	no	—
64	C₄×D₄	—	2	2	no	—
65	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
66	#97	—	13	4	no	—
67	C₄×C₈	—	26	2	no	—
68	C₂×C₁₃	—	4	1	no	—
69	C₂×C₁₃	—	8	2	no	—
70	C₂×C₁₃	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₃	C₂×#104	1	2	yes	—
72	#41	—	4	4	no	—
73	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
80	M₁₆	—	26	8	no	—
81	C₂×C₈	—	26	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	2	4	no	—
84	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₃	1	2	yes	—
86	C₄²	C₂²×D₁₃	1	1	yes	characteristic
87	C₁₃	#1525	1	1	yes	characteristic, complemented, Sylow
88	C₂³	—	2	2	no	—
89	C₂³	C₂²⋊Dic₁₃	1	1	yes	characteristic
90	Q₈	—	4	2	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	4	no	—
93	D₄	—	4	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
99	C₂×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
100	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
102	C₈	—	26	8	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	#88073	1	2	yes	—
112	C₄	#88077	1	4	yes	—
113	C₂	—	4	1	no	—
114	C₂	—	8	2	no	—
115	C₂	C₂×#88073	1	1	yes	characteristic
116	C₂	C₂×#88077	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#532

C₂

yes

maximal, characteristic

C₄×#6099

C₂

yes

maximal, complemented

#11718132

C₂

yes

maximal, complemented

C₂×C₁₃×#41

C₄

yes

—

C₁₃×C₄⋊Q₈

C₄

yes

characteristic

C₁₃×#127

C₄

yes

characteristic

C₁₃×#124

C₂²

yes

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6099

—

#88043

C₂²

yes

—

C₄×#1055

—

C₁₃×#41

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×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

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

#1055

—

#532

D₁₃

yes

characteristic

C₄×M₁₆

—

#329

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#100

yes

—

C₄×C₁₃

#104

yes

—

C₂×#41

Dic₁₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₁₃

yes

characteristic

#127

Dic₁₃

yes

characteristic

#124

C₂×D₁₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#100

yes

characteristic

C₂×C₁₃

C₂×#104

yes

—

#41

—

C₂×Q₈

C₂×Dic₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₁₃

yes

characteristic

C₄²

C₂²⋊D₁₃

yes

—

C₄²

C₂²×D₁₃

yes

characteristic

C₁₃

#1525

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

C₂²⋊Dic₁₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₃

yes

—

C₂×C₄

C₂²⋊Dic₁₃

yes

characteristic

100

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#88073

yes

—

112

C₄

#88077

yes

—

113

C₂

—

114

C₂

—

115

C₂

C₂×#88073

yes

characteristic

116

C₂

C₂×#88077

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃
Fitting, prime radical	C₁₃×#532	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
cosocle	C₄²×C₁₃	C₂³
hypercenter	#532	D₁₃
hypocenter	C₁₃	#1525
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃

Fitting, prime radical

C₁₃×#532

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

cosocle

C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
2. C₄×#6099	C₂	semidirect
3. #11718132	C₂	semidirect
87. C₁₃	#1525	semidirect

H ≅ G/N

type

2. C₄×#6099

C₂

semidirect

3. #11718132

C₂

semidirect

87. C₁₃

#1525

semidirect

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

subgroup	index
C₁₃×#532	2
C₄×#6099	2
#11718132	2
#1525	13

subgroup

index

C₁₃×#532

C₄×#6099

#11718132

#1525

Subgroups — #13113428

Conjugacy classes of subgroups

Complements

Maximal subgroups