Group #21447666 — subgroups

Subgroups — D₁₆×Dic₆₁ #21447666

Order 7808.

174 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 67 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	D₈×Dic₆₁	C₂	1	2	yes	maximal, complemented
2	#16839588	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839563	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	D₄×Dic₆₁	—	2	2	no	—
7	C₂×#11758185	C₂²	1	1	yes	characteristic
8	#14519711	C₂²	1	2	yes	—
9	#14519698	—	2	2	no	—
10	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×D₈×C₆₁	C₂²	1	2	yes	—
12	D₁₆×C₆₁	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₆₁	—	2	2	no	—
15	C₂²×Dic₆₁	—	4	2	no	—
16	D₈×C₆₁	—	2	2	no	—
17	D₈×C₆₁	C₂×C₄	1	4	yes	complemented
18	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²⋊Dic₆₁	—	4	2	no	—
20	C₄⋊Dic₆₁	—	2	2	no	—
21	C₄×Dic₆₁	D₄	1	1	yes	characteristic
22	C₂×#92418	D₄	1	1	yes	characteristic
23	C₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
24	#11758185	—	2	1	no	—
25	C₂³×C₆₁	—	4	2	no	—
26	D₄×C₆₁	—	2	4	no	—
27	D₄×C₆₁	—	4	2	no	—
28	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₆₁	—	4	2	no	—
30	C₂×Dic₆₁	—	8	4	no	—
31	C₂×Dic₆₁	D₈	1	1	yes	characteristic
32	C₂×Dic₆₁	D₈	1	1	yes	characteristic
33	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₆₁	#41	1	1	yes	characteristic
35	#92418	—	2	1	no	—
36	C₂²×C₆₁	—	4	4	no	—
37	C₂²×C₆₁	—	8	2	no	—
38	C₂²×C₆₁	—	8	2	no	—
39	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
40	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
41	C₄×C₆₁	#135	1	1	yes	characteristic
42	Dic₆₁	—	2	1	no	—
43	Dic₆₁	—	8	2	no	—
44	Dic₆₁	D₁₆	1	2	yes	direct product factor
45	C₄×D₁₆	—	61	1	no	maximal, Sylow
46	C₂×C₆₁	—	8	4	no	—
47	C₂×C₆₁	C₄×D₈	1	1	yes	characteristic
48	C₂×C₆₁	C₂×D₁₆	1	1	yes	characteristic
49	C₂×C₆₁	#508	1	1	yes	characteristic
50	C₂×D₁₆	D₆₁	1	1	yes	characteristic
51	C₄×D₈	—	61	2	no	—
52	#366	—	61	1	no	—
53	#357	—	61	2	no	—
54	C₄×C₁₆	—	61	1	no	—
55	C₆₁	C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₈	C₂×D₆₁	1	2	yes	—
57	C₄×D₄	—	122	2	no	—
58	D₁₆	Dic₆₁	1	4	yes	direct product factor
59	C₂×C₁₆	—	61	1	no	—
60	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
61	#107	—	61	2	no	—
62	#102	—	122	2	no	—
63	C₄×C₈	—	61	1	no	—
64	C₂×D₄	—	2	2	no	—
65	C₂²×C₄	—	244	2	no	—
66	D₈	—	2	2	no	—
67	D₈	C₂×Dic₆₁	1	4	yes	complemented
68	C₂×C₈	—	61	1	no	—
69	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
70	C₄⋊C₄	—	122	2	no	—
71	C₂²⋊C₄	—	244	2	no	—
72	C₄²	—	61	1	no	—
73	C₁₆	—	122	1	no	—
74	C₁₆	C₂×Dic₆₁	1	2	yes	complemented
75	C₂³	—	4	2	no	—
76	D₄	—	2	4	no	—
77	D₄	—	4	2	no	—
78	C₂×C₄	—	61	1	no	—
79	C₂×C₄	—	61	1	no	—
80	C₂×C₄	—	488	4	no	—
81	C₂×C₄	—	244	2	no	—
82	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
83	C₈	—	122	1	no	—
84	C₈	#11758224	1	1	yes	characteristic
85	C₈	C₂²×Dic₆₁	1	1	yes	characteristic
86	C₂²	—	4	4	no	—
87	C₂²	—	8	2	no	—
88	C₂²	—	8	2	no	—
89	C₂²	D₈×D₆₁	1	1	yes	characteristic
90	C₄	—	61	2	no	—
91	C₄	—	122	1	no	—
92	C₄	—	488	2	no	—
93	C₄	D₄×Dic₆₁	1	1	yes	characteristic
94	C₄	#14519796	1	1	yes	characteristic
95	C₂	—	8	4	no	—
96	C₂	D₈×Dic₆₁	1	1	yes	characteristic
97	C₂	D₁₆×D₆₁	1	1	yes	characteristic
98	C₂	#16840191	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₈×Dic₆₁

C₂

yes

maximal, complemented

#16839588

C₂

yes

maximal, characteristic, complemented

#16839563

C₂

yes

maximal, complemented

C₁₆×Dic₆₁

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₆₁

C₂

yes

maximal, characteristic

D₄×Dic₆₁

—

C₂×#11758185

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519698

—

C₈×Dic₆₁

C₂²

yes

characteristic

C₂×D₈×C₆₁

C₂²

yes

—

D₁₆×C₆₁

C₄

yes

complemented

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

—

C₂²×Dic₆₁

—

D₈×C₆₁

—

D₈×C₆₁

C₂×C₄

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

C₁₆×C₆₁

C₂×C₄

yes

complemented

#11758185

—

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

D₈

yes

characteristic

C₂×Dic₆₁

D₈

yes

characteristic

C₈×C₆₁

C₂²×C₄

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

#135

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

D₁₆

yes

direct product factor

C₄×D₁₆

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₄×D₈

yes

characteristic

C₂×C₆₁

C₂×D₁₆

yes

characteristic

C₂×C₆₁

#508

yes

characteristic

C₂×D₁₆

D₆₁

yes

characteristic

C₄×D₈

—

#366

—

#357

—

C₄×C₁₆

—

C₆₁

C₄×D₁₆

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₆₁

yes

—

C₄×D₄

—

122

—

D₁₆

Dic₆₁

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₁

yes

characteristic

#107

—

#102

—

122

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

244

—

D₈

—

D₈

C₂×Dic₆₁

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

244

—

C₄²

—

C₁₆

—

122

—

C₁₆

C₂×Dic₆₁

yes

complemented

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

488

—

C₂×C₄

—

244

—

C₂×C₄

D₄×D₆₁

yes

characteristic

C₈

—

122

—

C₈

#11758224

yes

characteristic

C₈

C₂²×Dic₆₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

D₈×D₆₁

yes

characteristic

C₄

—

C₄

—

122

—

C₄

—

488

—

C₄

D₄×Dic₆₁

yes

characteristic

C₄

#14519796

yes

characteristic

C₂

—

C₂

D₈×Dic₆₁

yes

characteristic

C₂

D₁₆×D₆₁

yes

characteristic

C₂

#16840191

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₁
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₂×D₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₂×D₁₆	D₆₁
hypocenter	C₆₁	C₄×D₁₆
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

D₈×D₆₁

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₂×D₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₂×D₁₆

D₆₁

hypocenter

C₆₁

C₄×D₁₆

LCS

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. D₈×Dic₆₁	C₂	semidirect
2. #16839588	C₂	semidirect
3. #16839563	C₂	semidirect
4. C₁₆×Dic₆₁	C₂	semidirect
12. D₁₆×C₆₁	C₄	semidirect
17. D₈×C₆₁	C₂×C₄	semidirect
23. C₁₆×C₆₁	C₂×C₄	semidirect
44. Dic₆₁	D₁₆	direct (N × H)
55. C₆₁	C₄×D₁₆	semidirect
58. D₁₆	Dic₆₁	direct (N × H)
67. D₈	C₂×Dic₆₁	semidirect
74. C₁₆	C₂×Dic₆₁	semidirect

H ≅ G/N

type

1. D₈×Dic₆₁

C₂

semidirect

2. #16839588

C₂

semidirect

3. #16839563

C₂

semidirect

4. C₁₆×Dic₆₁

C₂

semidirect

12. D₁₆×C₆₁

C₄

semidirect

17. D₈×C₆₁

C₂×C₄

semidirect

23. C₁₆×C₆₁

C₂×C₄

semidirect

44. Dic₆₁

D₁₆

direct (N × H)

55. C₆₁

C₄×D₁₆

semidirect

58. D₁₆

Dic₆₁

direct (N × H)

67. D₈

C₂×Dic₆₁

semidirect

74. C₁₆

C₂×Dic₆₁

semidirect

Subgroups — D₁₆×Dic₆₁ #21447666

Conjugacy classes of subgroups

Complements

Subgroups — D16×Dic61 #21447666

Conjugacy classes of subgroups

Complements

Subgroups — D₁₆×Dic₆₁ #21447666