Group #13110613 — subgroups

Subgroups — C₁₃×#1555 #13110613

Order 1664.

332 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 96 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×#127	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#432	C₂	1	2	yes	maximal, complemented
3	C₁₃×#331	C₂	1	4	yes	maximal, complemented
4	C₂²×D₄×C₁₃	—	2	2	no	—
5	C₂²×D₄×C₁₃	C₄	1	2	yes	—
6	C₁₃×#127	—	2	2	no	—
7	C₁₃×#127	C₄	1	2	yes	—
8	C₁₃×#127	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×C₄⋊C₈	C₂²	1	4	yes	complemented
11	C₁₃×C₂²⋊C₈	—	2	2	no	—
12	C₁₃×#97	—	2	2	no	—
13	C₂⁴×C₁₃	—	2	2	no	—
14	C₂×D₄×C₁₃	—	2	2	no	—
15	C₂×D₄×C₁₃	—	2	4	no	—
16	C₂×D₄×C₁₃	—	2	4	no	—
17	C₂×D₄×C₁₃	—	4	4	no	—
18	C₂×D₄×C₁₃	—	4	8	no	—
19	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃	D₄	1	2	yes	—
22	C₂×C₈×C₁₃	—	2	4	no	—
23	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₃	D₄	1	2	yes	—
25	C₄²×C₁₃	C₂³	1	1	yes	characteristic
26	#1555	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂³×C₁₃	—	2	4	no	—
28	C₂³×C₁₃	—	4	8	no	—
29	C₂³×C₁₃	—	4	4	no	—
30	C₂³×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
31	D₄×C₁₃	—	4	4	no	—
32	D₄×C₁₃	—	4	8	no	—
33	D₄×C₁₃	—	4	4	no	—
34	D₄×C₁₃	—	4	8	no	—
35	C₂×C₄×C₁₃	—	2	2	no	—
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
41	C₈×C₁₃	—	4	4	no	—
42	C₂×#127	C₂×C₁₃	1	1	yes	characteristic
43	#432	C₂×C₁₃	1	2	yes	complemented
44	#331	C₂×C₁₃	1	4	yes	complemented
45	C₂²×C₁₃	—	2	2	no	—
46	C₂²×C₁₃	—	4	4	no	—
47	C₂²×C₁₃	—	4	8	no	—
48	C₂²×C₁₃	—	8	8	no	—
49	C₂²×C₁₃	#100	1	2	yes	—
50	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₄×C₁₃	—	2	1	no	—
52	C₄×C₁₃	—	2	1	no	—
53	C₄×C₁₃	—	2	2	no	—
54	C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
55	C₂²×D₄	—	2	2	no	—
56	C₂²×D₄	C₄×C₁₃	1	2	yes	—
57	#127	—	2	2	no	—
58	#127	C₄×C₁₃	1	2	yes	—
59	#127	C₂²×C₁₃	1	2	yes	—
60	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
61	C₄⋊C₈	C₂²×C₁₃	1	4	yes	complemented
62	C₂²⋊C₈	—	2	2	no	—
63	#97	—	2	2	no	—
64	C₂×C₁₃	—	2	2	no	—
65	C₂×C₁₃	—	8	4	no	—
66	C₂×C₁₃	C₂×#100	1	1	yes	characteristic
67	C₂×C₁₃	#418	1	2	yes	—
68	C₂⁴	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	4	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	4	4	no	—
73	C₂×D₄	—	4	8	no	—
74	C₂×D₄	C₂×C₄×C₁₃	1	4	yes	—
75	C₂²×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
76	C₂²×C₄	D₄×C₁₃	1	2	yes	—
77	C₂×C₈	—	2	4	no	—
78	C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
79	C₄²	D₄×C₁₃	1	2	yes	—
80	C₄²	C₂³×C₁₃	1	1	yes	characteristic
81	C₁₃	#1555	1	1	yes	characteristic, direct product factor, Sylow
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	8	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
95	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
96	C₈	—	4	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	4	no	—
99	C₂²	—	4	8	no	—
100	C₂²	—	8	8	no	—
101	C₂²	C₁₃×#100	1	2	yes	—
102	C₂²	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×C₁₃×#100	1	1	yes	characteristic
110	C₂	C₁₃×#418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#127

C₂

yes

maximal, characteristic

C₁₃×#432

C₂

yes

maximal, complemented

C₁₃×#331

C₂

yes

maximal, complemented

C₂²×D₄×C₁₃

—

C₂²×D₄×C₁₃

C₄

yes

—

C₁₃×#127

—

C₁₃×#127

C₄

yes

—

C₁₃×#127

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₁₃×C₄⋊C₈

C₂²

yes

complemented

C₁₃×C₂²⋊C₈

—

C₁₃×#97

—

C₂⁴×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

#1555

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₈×C₁₃

—

C₂×#127

C₂×C₁₃

yes

characteristic

#432

C₂×C₁₃

yes

complemented

#331

C₂×C₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#100

yes

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂²×D₄

—

C₂²×D₄

C₄×C₁₃

yes

—

#127

—

#127

C₄×C₁₃

yes

—

#127

C₂²×C₁₃

yes

—

C₂×C₄²

C₂²×C₁₃

yes

characteristic

C₄⋊C₈

C₂²×C₁₃

yes

complemented

C₂²⋊C₈

—

#97

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#100

yes

characteristic

C₂×C₁₃

#418

yes

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₁₃

yes

—

C₂²×C₄

C₂×C₄×C₁₃

yes

characteristic

C₂²×C₄

D₄×C₁₃

yes

—

C₂×C₈

—

C₄²

C₂×C₄×C₁₃

yes

characteristic

C₄²

D₄×C₁₃

yes

—

C₄²

C₂³×C₁₃

yes

characteristic

C₁₃

#1555

yes

characteristic, direct product factor, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

C₁₃×C₂²⋊C₄

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₃×C₂²⋊C₄

yes

—

C₂×C₄

C₁₃×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂²×C₄×C₁₃

yes

characteristic

C₂×C₄

C₂×D₄×C₁₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

C₁₃×#100

yes

—

102

C₂²

C₂×C₁₃×C₂²⋊C₄

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₁₃×AGL(1,ℤ/8)

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×C₁₃×#100

yes

characteristic

110

C₂

C₁₃×#418

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₁₃

Frattini, cosocle

C₄²

C₂³×C₁₃

prime radical

C₂×C₁₃×#127

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

N	H ≅ G/N	type
2. C₁₃×#432	C₂	semidirect
3. C₁₃×#331	C₂	semidirect
10. C₁₃×C₄⋊C₈	C₂²	semidirect
26. #1555	C₁₃	direct (N × H)
43. #432	C₂×C₁₃	semidirect
44. #331	C₂×C₁₃	semidirect
61. C₄⋊C₈	C₂²×C₁₃	semidirect
81. C₁₃	#1555	direct (N × H)

H ≅ G/N

type

2. C₁₃×#432

C₂

semidirect

3. C₁₃×#331

C₂

semidirect

10. C₁₃×C₄⋊C₈

C₂²

semidirect

26. #1555

C₁₃

direct (N × H)

43. #432

C₂×C₁₃

semidirect

44. #331

C₂×C₁₃

semidirect

61. C₄⋊C₈

C₂²×C₁₃

semidirect

81. C₁₃

#1555

direct (N × H)

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

subgroup	index
C₂×C₁₃×#127	2
C₁₃×#432	2
C₁₃×#331	2
#1555	13

subgroup

index

C₂×C₁₃×#127

C₁₃×#432

C₁₃×#331

#1555

Subgroups — C₁₃×#1555 #13110613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C13×#1555 #13110613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#1555 #13110613