Group #13081622 — subgroups

Subgroups — #13081622

Order 1408.

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	#10623101	C₂	1	2	yes	maximal
2	C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
3	#10622462	C₂	1	6	yes	maximal
4	#10622312	C₂	1	3	yes	maximal
5	C₄×Dic₄₄	C₂	1	3	yes	maximal
6	C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
7	Q₈×Dic₁₁	—	2	2	no	—
8	C₂×Dic₄₄	C₂²	1	3	yes	—
9	C₂×Dic₄₄	C₂²	1	6	yes	—
10	#66943	—	2	6	no	—
11	#66922	C₂²	1	6	yes	—
12	C₄×Dic₂₂	C₂²	1	6	yes	—
13	C₁₁×C₄⋊C₈	C₂²	1	3	yes	—
14	C₄×C₈×C₁₁	C₂²	1	3	yes	—
15	#66882	C₂²	1	1	yes	characteristic
16	#66880	C₂²	1	6	yes	—
17	C₂×Q₈×C₁₁	D₄	1	1	yes	characteristic
18	C₂×Dic₂₂	C₂³	1	6	yes	—
19	C₂×C₈×C₁₁	C₂³	1	3	yes	—
20	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
21	C₁₁×C₄⋊C₄	D₄	1	3	yes	—
22	C₄²×C₁₁	C₂³	1	3	yes	—
23	C₄⋊Dic₁₁	—	2	6	no	—
24	C₄⋊Dic₁₁	—	2	6	no	—
25	C₄⋊Dic₁₁	C₂³	1	2	yes	—
26	Dic₁₁⋊C₄	—	2	6	no	—
27	C₄×Dic₁₁	—	2	6	no	—
28	Dic₄₄	—	2	6	no	—
29	Dic₄₄	—	4	6	no	—
30	#3395	—	11	1	no	maximal, Sylow
31	Q₈×C₁₁	Q₁₆	1	4	yes	complemented
32	C₂×C₄×C₁₁	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₁₁	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₁₁	—	2	2	no	—
36	C₂×Dic₁₁	—	2	6	no	—
37	Dic₂₂	—	2	6	no	—
38	Dic₂₂	—	2	6	no	—
39	C₈×C₁₁	—	2	3	no	—
40	C₈×C₁₁	#41	1	2	yes	—
41	#557	—	11	2	no	—
42	#494	—	11	3	no	—
43	#462	—	11	6	no	—
44	C₈×Q₈	D₁₁	1	1	yes	characteristic
45	C₄×Q₁₆	—	11	3	no	—
46	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₁₁	—	2	3	no	—
48	C₄×C₁₁	C₂×Q₁₆	1	6	yes	—
49	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
50	C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
51	Dic₁₁	—	4	6	no	—
52	Dic₁₁	—	4	2	no	—
53	C₂×Q₁₆	—	11	3	no	—
54	C₂×Q₁₆	—	11	6	no	—
55	C₄⋊Q₈	—	11	6	no	—
56	#125	—	22	6	no	—
57	C₄×Q₈	—	11	6	no	—
58	C₄×Q₈	—	22	2	no	—
59	C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
60	#107	—	11	1	no	—
61	C₄⋊C₈	C₂×D₁₁	1	3	yes	—
62	#103	—	11	6	no	—
63	C₄×C₈	C₂×D₁₁	1	3	yes	—
64	C₂×C₁₁	#550	1	1	yes	characteristic
65	C₂×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
66	C₂×C₁₁	#578	1	1	yes	characteristic
67	C₂×Q₈	—	11	6	no	—
68	C₂×Q₈	D₄₄	1	1	yes	characteristic
69	Q₁₆	—	22	6	no	—
70	Q₁₆	—	44	6	no	—
71	C₂×C₈	C₂²×D₁₁	1	3	yes	—
72	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
73	C₄⋊C₄	—	11	2	no	—
74	C₄⋊C₄	—	22	6	no	—
75	C₄⋊C₄	—	22	6	no	—
76	C₄⋊C₄	—	22	6	no	—
77	C₄⋊C₄	D₄₄	1	3	yes	—
78	C₄²	—	22	6	no	—
79	C₄²	C₂²×D₁₁	1	3	yes	—
80	C₁₁	#3395	1	1	yes	characteristic, complemented, Sylow
81	Q₈	—	22	6	no	—
82	Q₈	—	22	6	no	—
83	Q₈	Dic₄₄	1	4	yes	complemented
84	C₂×C₄	—	22	2	no	—
85	C₂×C₄	—	22	6	no	—
86	C₂×C₄	C₂×D₄₄	1	3	yes	—
87	C₂×C₄	C₂×D₄₄	1	3	yes	—
88	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	#4343	1	2	yes	—
91	C₂²	C₂²×D₄₄	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	44	2	no	—
94	C₄	—	44	6	no	—
95	C₄	C₂×Dic₄₄	1	6	yes	—
96	C₄	C₂×#4343	1	1	yes	characteristic
97	C₄	#67042	1	1	yes	characteristic
98	C₂	#10623108	1	1	yes	characteristic
99	C₂	C₂²×Dic₄₄	1	1	yes	characteristic
100	C₂	#10623284	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10623101

C₂

yes

maximal

C₈×Q₈×C₁₁

C₂

yes

maximal, characteristic

#10622462

C₂

yes

maximal

#10622312

C₂

yes

maximal

C₄×Dic₄₄

C₂

yes

maximal

C₄×Q₈×C₁₁

C₂²

yes

characteristic

Q₈×Dic₁₁

—

C₂×Dic₄₄

C₂²

yes

—

C₂×Dic₄₄

C₂²

yes

—

#66943

—

#66922

C₂²

yes

—

C₄×Dic₂₂

C₂²

yes

—

C₁₁×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₁₁

C₂²

yes

—

#66882

C₂²

yes

characteristic

#66880

C₂²

yes

—

C₂×Q₈×C₁₁

D₄

yes

characteristic

C₂×Dic₂₂

C₂³

yes

—

C₂×C₈×C₁₁

C₂³

yes

—

C₂×C₈×C₁₁

C₂³

yes

characteristic

C₁₁×C₄⋊C₄

D₄

yes

—

C₄²×C₁₁

C₂³

yes

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

—

C₄⋊Dic₁₁

C₂³

yes

—

Dic₁₁⋊C₄

—

C₄×Dic₁₁

—

Dic₄₄

—

Dic₄₄

—

#3395

—

maximal, Sylow

Q₈×C₁₁

Q₁₆

yes

complemented

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂⁴

yes

characteristic

C₂×Dic₁₁

—

C₂×Dic₁₁

—

Dic₂₂

—

Dic₂₂

—

C₈×C₁₁

—

C₈×C₁₁

#41

yes

—

#557

—

#494

—

#462

—

C₈×Q₈

D₁₁

yes

characteristic

C₄×Q₁₆

—

C₂²×C₁₁

C₂²×D₄

yes

characteristic

C₄×C₁₁

—

C₄×C₁₁

C₂×Q₁₆

yes

—

C₄×C₁₁

C₂×#41

yes

characteristic

C₄×C₁₁

H(5,𝔽₂)

yes

characteristic

Dic₁₁

—

Dic₁₁

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₁

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₁₁

yes

—

#103

—

C₄×C₈

C₂×D₁₁

yes

—

C₂×C₁₁

#550

yes

characteristic

C₂×C₁₁

C₂²×Q₁₆

yes

characteristic

C₂×C₁₁

#578

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄₄

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

C₂²×D₁₁

yes

—

C₂×C₈

C₂²×D₁₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄₄

yes

—

C₄²

—

C₄²

C₂²×D₁₁

yes

—

C₁₁

#3395

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

Q₈

Dic₄₄

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄₄

yes

—

C₂×C₄

C₂×D₄₄

yes

—

C₂×C₄

C₂³×D₁₁

yes

characteristic

C₈

—

C₈

#4343

yes

—

C₂²

C₂²×D₄₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×Dic₄₄

yes

—

C₄

yes

C₄

yes

C₂

yes

C₂

C₂²×Dic₄₄

yes

characteristic

100

C₂

#10623284

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₄
commutator, cosocle	C₂×C₄×C₁₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁
Fitting	C₈×Q₈×C₁₁	C₂
socle, prime radical	C₂²×C₁₁	C₂²×D₄
hypercenter	C₈×Q₈	D₁₁
hypocenter	C₁₁	#3395
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₄₄

commutator, cosocle

C₂×C₄×C₁₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₁

Fitting

C₈×Q₈×C₁₁

C₂

socle, prime radical

C₂²×C₁₁

C₂²×D₄

hypercenter

C₈×Q₈

D₁₁

hypocenter

C₁₁

#3395

LCS

G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
31. Q₈×C₁₁	Q₁₆	semidirect
80. C₁₁	#3395	semidirect
83. Q₈	Dic₄₄	semidirect

H ≅ G/N

type

31. Q₈×C₁₁

Q₁₆

semidirect

80. C₁₁

#3395

semidirect

83. Q₈

Dic₄₄

semidirect

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

subgroup	index
#10623101	2
C₈×Q₈×C₁₁	2
#10622462	2
#10622312	2
C₄×Dic₄₄	2
#3395	11

subgroup

index

C₈×Q₈×C₁₁

C₄×Dic₄₄

Subgroups — #13081622

Conjugacy classes of subgroups

Complements

Maximal subgroups