Group #13127842 — subgroups

Subgroups — #13127842

Order 1664.

348 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 141 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	#11719275	C₂	1	2	yes	maximal
2	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	#11718510	C₂	1	6	yes	maximal
4	#11718360	C₂	1	3	yes	maximal
5	C₄×Dic₅₂	C₂	1	3	yes	maximal
6	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	Q₈×Dic₁₃	—	2	2	no	—
8	C₂×Dic₅₂	C₂²	1	3	yes	—
9	C₂×Dic₅₂	C₂²	1	6	yes	—
10	#88144	—	2	6	no	—
11	#88123	C₂²	1	6	yes	—
12	C₄×Dic₂₆	C₂²	1	6	yes	—
13	C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
14	C₄×C₈×C₁₃	C₂²	1	3	yes	—
15	#88058	C₂²	1	1	yes	characteristic
16	#88056	C₂²	1	6	yes	—
17	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂×Dic₂₆	C₂³	1	6	yes	—
19	C₂×C₈×C₁₃	C₂³	1	3	yes	—
20	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×C₄⋊C₄	D₄	1	3	yes	—
22	C₄²×C₁₃	C₂³	1	3	yes	—
23	C₄⋊Dic₁₃	—	2	6	no	—
24	C₄⋊Dic₁₃	—	2	6	no	—
25	C₄⋊Dic₁₃	C₂³	1	2	yes	—
26	#6101	—	2	6	no	—
27	C₄×Dic₁₃	—	2	6	no	—
28	Dic₅₂	—	2	6	no	—
29	Dic₅₂	—	4	6	no	—
30	#3395	—	13	1	no	maximal, Sylow
31	Q₈×C₁₃	Q₁₆	1	4	yes	complemented
32	C₂×C₄×C₁₃	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₁₃	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₁₃	—	2	2	no	—
36	C₂×Dic₁₃	—	2	6	no	—
37	Dic₂₆	—	2	6	no	—
38	Dic₂₆	—	2	6	no	—
39	C₈×C₁₃	—	2	3	no	—
40	C₈×C₁₃	#41	1	2	yes	—
41	#557	—	13	2	no	—
42	#494	—	13	3	no	—
43	#462	—	13	6	no	—
44	C₈×Q₈	D₁₃	1	1	yes	characteristic
45	C₄×Q₁₆	—	13	3	no	—
46	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₁₃	—	2	3	no	—
48	C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
49	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
50	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
51	Dic₁₃	—	4	6	no	—
52	Dic₁₃	—	4	2	no	—
53	C₂×Q₁₆	—	13	3	no	—
54	C₂×Q₁₆	—	13	6	no	—
55	C₄⋊Q₈	—	13	6	no	—
56	#125	—	26	6	no	—
57	C₄×Q₈	—	13	6	no	—
58	C₄×Q₈	—	26	2	no	—
59	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
60	#107	—	13	1	no	—
61	C₄⋊C₈	C₂×D₁₃	1	3	yes	—
62	#103	—	13	6	no	—
63	C₄×C₈	C₂×D₁₃	1	3	yes	—
64	C₂×C₁₃	#550	1	1	yes	characteristic
65	C₂×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
66	C₂×C₁₃	#578	1	1	yes	characteristic
67	C₂×Q₈	—	13	6	no	—
68	C₂×Q₈	D₅₂	1	1	yes	characteristic
69	Q₁₆	—	26	6	no	—
70	Q₁₆	—	52	6	no	—
71	C₂×C₈	C₂²×D₁₃	1	3	yes	—
72	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
73	C₄⋊C₄	—	13	2	no	—
74	C₄⋊C₄	—	26	6	no	—
75	C₄⋊C₄	—	26	6	no	—
76	C₄⋊C₄	—	26	6	no	—
77	C₄⋊C₄	D₅₂	1	3	yes	—
78	C₄²	—	26	6	no	—
79	C₄²	C₂²×D₁₃	1	3	yes	—
80	C₁₃	#3395	1	1	yes	characteristic, complemented, Sylow
81	Q₈	—	26	6	no	—
82	Q₈	—	26	6	no	—
83	Q₈	Dic₅₂	1	4	yes	complemented
84	C₂×C₄	—	26	2	no	—
85	C₂×C₄	—	26	6	no	—
86	C₂×C₄	C₂×D₅₂	1	3	yes	—
87	C₂×C₄	C₂×D₅₂	1	3	yes	—
88	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	#6131	1	2	yes	—
91	C₂²	C₂²×D₅₂	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	52	2	no	—
94	C₄	—	52	6	no	—
95	C₄	C₂×Dic₅₂	1	6	yes	—
96	C₄	C₂×#6131	1	1	yes	characteristic
97	C₄	#88256	1	1	yes	characteristic
98	C₂	#11719282	1	1	yes	characteristic
99	C₂	C₂²×Dic₅₂	1	1	yes	characteristic
100	C₂	#11719458	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11719275

C₂

yes

maximal

C₈×Q₈×C₁₃

C₂

yes

maximal, characteristic

#11718510

C₂

yes

maximal

#11718360

C₂

yes

maximal

C₄×Dic₅₂

C₂

yes

maximal

C₄×Q₈×C₁₃

C₂²

yes

characteristic

Q₈×Dic₁₃

—

C₂×Dic₅₂

C₂²

yes

—

C₂×Dic₅₂

C₂²

yes

—

#88144

—

#88123

C₂²

yes

—

C₄×Dic₂₆

C₂²

yes

—

C₁₃×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

—

#88058

C₂²

yes

characteristic

#88056

C₂²

yes

—

C₂×Q₈×C₁₃

D₄

yes

characteristic

C₂×Dic₂₆

C₂³

yes

—

C₂×C₈×C₁₃

C₂³

yes

—

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

D₄

yes

—

C₄²×C₁₃

C₂³

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

#3395

—

maximal, Sylow

Q₈×C₁₃

Q₁₆

yes

complemented

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

#41

yes

—

#557

—

#494

—

#462

—

C₈×Q₈

D₁₃

yes

characteristic

C₄×Q₁₆

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₂×Q₁₆

yes

—

C₄×C₁₃

C₂×#41

yes

characteristic

C₄×C₁₃

H(5,𝔽₂)

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₃

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₁₃

yes

—

#103

—

C₄×C₈

C₂×D₁₃

yes

—

C₂×C₁₃

#550

yes

characteristic

C₂×C₁₃

C₂²×Q₁₆

yes

characteristic

C₂×C₁₃

#578

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₅₂

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

C₂²×D₁₃

yes

—

C₂×C₈

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₅₂

yes

—

C₄²

—

C₄²

C₂²×D₁₃

yes

—

C₁₃

#3395

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

Q₈

Dic₅₂

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₅₂

yes

—

C₂×C₄

C₂×D₅₂

yes

—

C₂×C₄

C₂³×D₁₃

yes

characteristic

C₈

—

C₈

#6131

yes

—

C₂²

C₂²×D₅₂

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×Dic₅₂

yes

—

C₄

yes

C₄

yes

C₂

yes

C₂

C₂²×Dic₅₂

yes

characteristic

100

C₂

#11719458

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂²×C₁₃	C₂²×D₄
hypercenter	C₈×Q₈	D₁₃
hypocenter	C₁₃	#3395
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₅₂

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₈×Q₈×C₁₃

C₂

socle, prime radical

C₂²×C₁₃

C₂²×D₄

hypercenter

C₈×Q₈

D₁₃

hypocenter

C₁₃

#3395

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
31. Q₈×C₁₃	Q₁₆	semidirect
80. C₁₃	#3395	semidirect
83. Q₈	Dic₅₂	semidirect

H ≅ G/N

type

31. Q₈×C₁₃

Q₁₆

semidirect

80. C₁₃

#3395

semidirect

83. Q₈

Dic₅₂

semidirect

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

subgroup	index
#11719275	2
C₈×Q₈×C₁₃	2
#11718510	2
#11718360	2
C₄×Dic₅₂	2
#3395	13

subgroup

index

C₈×Q₈×C₁₃

C₄×Dic₅₂

Subgroups — #13127842

Conjugacy classes of subgroups

Complements

Maximal subgroups