Group #13123022 — subgroups

Subgroups — #13123022

Order 1664.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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×#399	C₂	1	1	yes	maximal, characteristic
2	#11718645	C₂	1	6	yes	maximal, complemented
3	#11718614	C₂	1	3	yes	maximal
4	#11718607	C₂	1	3	yes	maximal
5	C₁₃⋊#401	C₂	1	2	yes	maximal, complemented
6	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
11	C₂×#6101	C₂²	1	6	yes	—
12	C₁₃×#95	C₂²	1	1	yes	characteristic
13	#88071	C₂²	1	6	yes	—
14	#88071	C₂²	1	6	yes	—
15	#88071	C₂²	1	3	yes	—
16	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
18	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
19	C₂²×Dic₁₃	—	4	3	no	—
20	C₂²×Dic₁₃	C₂³	1	6	yes	—
21	C₂²×Dic₁₃	C₂³	1	2	yes	—
22	C₁₃×C₄⋊C₄	—	4	3	no	—
23	C₁₃×C₂²⋊C₄	—	4	3	no	—
24	C₂²⋊Dic₁₃	—	2	6	no	—
25	C₄⋊Dic₁₃	—	4	3	no	—
26	#6101	—	4	6	no	—
27	#2834	—	13	1	no	maximal, Sylow
28	C₂³×C₁₃	—	2	3	no	—
29	C₂³×C₁₃	—	2	3	no	—
30	C₂³×C₁₃	#41	1	2	yes	—
31	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₁₃	—	2	6	no	—
33	C₂×C₄×C₁₃	—	2	3	no	—
34	C₂×C₄×C₁₃	—	2	3	no	—
35	C₂×C₄×C₁₃	—	2	6	no	—
36	C₂×Dic₁₃	—	2	6	no	—
37	C₂×Dic₁₃	—	2	6	no	—
38	C₂×Dic₁₃	—	2	6	no	—
39	C₂×Dic₁₃	—	2	6	no	—
40	C₂×Dic₁₃	—	4	4	no	—
41	#401	—	13	2	no	—
42	#399	D₁₃	1	1	yes	characteristic
43	#398	—	13	6	no	—
44	#397	—	13	3	no	—
45	#385	—	13	3	no	—
46	C₂²×C₁₃	—	2	6	no	—
47	C₂²×C₁₃	—	4	1	no	—
48	C₂²×C₁₃	—	4	3	no	—
49	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
50	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
51	C₂²×C₁₃	2₋¹⁺⁴	1	3	yes	—
52	C₄×C₁₃	—	4	3	no	—
53	C₄×C₁₃	—	4	3	no	—
54	Dic₁₃	—	4	6	no	—
55	Dic₁₃	—	4	2	no	—
56	C₂³×C₄	—	13	1	no	—
57	C₂×C₄⋊C₄	—	13	6	no	—
58	C₂×C₄⋊C₄	—	13	3	no	—
59	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
60	C₂×C₂²⋊C₄	—	13	3	no	—
61	C₂×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
62	#95	—	13	6	no	—
63	#95	—	13	3	no	—
64	#95	—	13	6	no	—
65	#95	C₂×D₁₃	1	1	yes	characteristic
66	C₂×C₁₃	—	4	2	no	—
67	C₂×C₁₃	#539	1	3	yes	—
68	C₂×C₁₃	#560	1	1	yes	characteristic
69	C₂×C₁₃	#563	1	3	yes	—
70	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
71	C₂²×C₄	—	13	6	no	—
72	C₂²×C₄	—	13	2	no	—
73	C₂²×C₄	—	52	3	no	—
74	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
75	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
76	C₄⋊C₄	—	4	3	no	—
77	C₄⋊C₄	—	52	6	no	—
78	C₄⋊C₄	—	52	3	no	—
79	C₂²⋊C₄	—	4	3	no	—
80	C₂²⋊C₄	—	26	6	no	—
81	C₁₃	#2834	1	1	yes	characteristic, complemented, Sylow
82	C₂³	—	2	3	no	—
83	C₂³	—	2	3	no	—
84	C₂³	#6131	1	2	yes	—
85	C₂³	C₂³×D₁₃	1	1	yes	characteristic
86	C₂×C₄	—	2	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	6	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	26	6	no	—
91	C₂×C₄	—	26	6	no	—
92	C₂×C₄	—	26	6	no	—
93	C₂×C₄	—	26	6	no	—
94	C₂×C₄	—	52	4	no	—
95	C₂²	—	2	6	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	3	no	—
98	C₂²	#88251	1	3	yes	—
99	C₂²	C₂×#6131	1	1	yes	characteristic
100	C₂²	#88257	1	3	yes	—
101	C₄	—	4	3	no	—
102	C₄	—	4	3	no	—
103	C₄	—	52	2	no	—
104	C₄	—	52	6	no	—
105	C₂	—	4	2	no	—
106	C₂	#11719350	1	3	yes	—
107	C₂	#11719356	1	3	yes	—
108	C₂	#11719425	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#399

C₂

yes

maximal, characteristic

#11718645

C₂

yes

maximal, complemented

#11718614

C₂

yes

maximal

#11718607

C₂

yes

maximal

C₁₃⋊#401

C₂

yes

maximal, complemented

C₂³×Dic₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₁₃×#95

C₂²

yes

characteristic

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#2834

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

#41

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#401

—

#399

D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#41

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

—

—

—

C₂×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#539

yes

—

C₂×C₁₃

#560

yes

characteristic

C₂×C₁₃

#563

yes

—

C₂⁴

C₂²×D₁₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₁₃

#2834

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

#6131

yes

—

C₂³

C₂³×D₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#88251

yes

—

C₂²

C₂×#6131

yes

characteristic

100

C₂²

#88257

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₂

—

106

C₂

#11719350

yes

—

107

C₂

#11719356

yes

—

108

C₂

#11719425

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#399	C₂
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	#399	D₁₃
hypocenter	C₁₃	#2834
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#399

C₂

prime radical

C₂⁴×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
2. #11718645	C₂	semidirect
5. C₁₃⋊#401	C₂	semidirect
81. C₁₃	#2834	semidirect

H ≅ G/N

type

2. #11718645

C₂

semidirect

5. C₁₃⋊#401

C₂

semidirect

81. C₁₃

#2834

semidirect

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

subgroup	index
C₁₃×#399	2
#11718645	2
#11718614	2
#11718607	2
C₁₃⋊#401	2
#2834	13

subgroup

index

C₁₃×#399

C₁₃⋊#401

Subgroups — #13123022

Conjugacy classes of subgroups

Complements

Maximal subgroups