Group #13123105 — subgroups

Subgroups — #13123105

Order 1664.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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×#384	C₂	1	1	yes	maximal, characteristic
2	#11718647	C₂	1	3	yes	maximal
3	#11718645	C₂	1	6	yes	maximal
4	#11718640	C₂	1	3	yes	maximal
5	#11718598	C₂	1	2	yes	maximal
6	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	3	yes	—
11	C₂×#6101	C₂²	1	6	yes	—
12	#88071	C₂²	1	6	yes	—
13	#88071	C₂²	1	6	yes	—
14	#88071	C₂²	1	3	yes	—
15	#88071	C₂²	1	3	yes	—
16	C₂²×C₄×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	C₂³	1	6	yes	—
20	C₂²×Dic₁₃	C₂³	1	2	yes	—
21	C₂×Dic₂₆	—	4	3	no	—
22	C₁₃×C₄⋊C₄	—	4	3	no	—
23	C₁₃×C₄⋊C₄	—	4	3	no	—
24	C₄²×C₁₃	Q₈	1	4	yes	complemented
25	C₄⋊Dic₁₃	—	4	3	no	—
26	#6101	—	4	6	no	—
27	#2843	—	13	1	no	maximal, Sylow
28	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₁₃	—	2	3	no	—
30	C₂×C₄×C₁₃	—	2	3	no	—
31	C₂×C₄×C₁₃	—	2	6	no	—
32	C₂×C₄×C₁₃	—	2	3	no	—
33	C₂×C₄×C₁₃	—	2	3	no	—
34	C₂×C₄×C₁₃	C₂×Q₈	1	6	yes	—
35	C₂×Dic₁₃	—	2	6	no	—
36	C₂×Dic₁₃	—	2	6	no	—
37	C₂×Dic₁₃	—	2	12	no	—
38	Dic₂₆	—	4	4	no	—
39	#400	—	13	3	no	—
40	#398	—	13	6	no	—
41	#395	—	13	3	no	—
42	#384	D₁₃	1	1	yes	characteristic
43	#383	—	13	2	no	—
44	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
45	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₁₃	2₋¹⁺⁴	1	3	yes	—
47	C₄×C₁₃	—	2	6	no	—
48	C₄×C₁₃	—	4	1	no	—
49	C₄×C₁₃	—	4	3	no	—
50	Dic₁₃	—	4	2	no	—
51	Dic₁₃	—	4	6	no	—
52	C₂²×Q₈	—	13	1	no	—
53	C₂×C₄⋊C₄	—	13	6	no	—
54	C₂×C₄⋊C₄	—	13	3	no	—
55	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
56	C₂×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
57	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
58	#95	—	13	6	no	—
59	#95	—	13	6	no	—
60	#95	—	13	3	no	—
61	#95	—	13	3	no	—
62	C₂×C₁₃	#544	1	3	yes	—
63	C₂×C₁₃	#560	1	1	yes	characteristic
64	C₂×C₁₃	#563	1	3	yes	—
65	C₂×Q₈	—	52	3	no	—
66	C₂²×C₄	—	13	6	no	—
67	C₂²×C₄	—	13	2	no	—
68	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
69	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
70	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
71	C₄⋊C₄	—	4	3	no	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	52	6	no	—
74	C₄⋊C₄	—	52	3	no	—
75	C₄²	Dic₂₆	1	4	yes	complemented
76	C₁₃	#2843	1	1	yes	characteristic, complemented, Sylow
77	C₂³	C₂³×D₁₃	1	1	yes	characteristic
78	Q₈	—	52	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	26	6	no	—
85	C₂×C₄	—	26	6	no	—
86	C₂×C₄	—	26	12	no	—
87	C₂×C₄	C₂×Dic₂₆	1	6	yes	—
88	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
89	C₂²	#88251	1	3	yes	—
90	C₂²	#88254	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	3	no	—
93	C₄	—	4	1	no	—
94	C₄	—	52	2	no	—
95	C₄	—	52	6	no	—
96	C₂	#11719205	1	3	yes	—
97	C₂	#11719425	1	1	yes	characteristic
98	C₂	#11719440	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#384

C₂

yes

maximal, characteristic

#11718647

C₂

yes

maximal

#11718645

C₂

yes

maximal

#11718640

C₂

yes

maximal

#11718598

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×#6101

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

Q₈

yes

complemented

C₄⋊Dic₁₃

—

#6101

—

#2843

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

—

—

—

D₁₃

yes

characteristic

#383

—

C₂²×C₁₃

C₂²×Q₈

yes

characteristic

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

—

C₂×C₁₃

#544

yes

—

C₂×C₁₃

#560

yes

characteristic

C₂×C₁₃

#563

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₂₆

yes

complemented

C₁₃

#2843

yes

characteristic, complemented, Sylow

C₂³

C₂³×D₁₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₂₆

yes

—

C₂²

C₂²×Dic₂₆

yes

characteristic

C₂²

#88251

yes

—

C₂²

#88254

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

#11719205

yes

—

C₂

#11719425

yes

characteristic

C₂

#11719440

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#384	C₂
hypercenter	#384	D₁₃
hypocenter	C₁₃	#2843
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₁₃×#384

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
24. C₄²×C₁₃	Q₈	semidirect
75. C₄²	Dic₂₆	semidirect
76. C₁₃	#2843	semidirect

H ≅ G/N

type

24. C₄²×C₁₃

Q₈

semidirect

75. C₄²

Dic₂₆

semidirect

76. C₁₃

#2843

semidirect

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

subgroup	index
C₁₃×#384	2
#11718647	2
#11718645	2
#11718640	2
#11718598	2
#2843	13

subgroup

index

C₁₃×#384

Subgroups — #13123105

Conjugacy classes of subgroups

Complements

Maximal subgroups