Group #20287600 — subgroups

Subgroups — Q₈×Dic₂₁₂ #20287600

Order 6784.

336 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 149 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	Q₈×Dic₁₀₆	C₂	1	2	yes	maximal
2	#16811028	C₂	1	3	yes	maximal
3	#16810919	C₂	1	6	yes	maximal
4	C₄×Dic₂₁₂	C₂	1	3	yes	maximal
5	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	#13133217	C₂²	1	6	yes	—
8	#13133216	—	2	6	no	—
9	Q₈×Dic₅₃	—	2	2	no	—
10	C₄×Dic₁₀₆	C₂²	1	6	yes	—
11	#13133151	C₂²	1	3	yes	—
12	#13133151	C₂²	1	6	yes	—
13	#13133142	C₂²	1	6	yes	—
14	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
16	C₄×C₈×C₅₃	C₂²	1	3	yes	—
17	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×Dic₁₀₆	—	2	6	no	—
19	C₂×Dic₁₀₆	C₂³	1	2	yes	—
20	C₂×C₈×C₅₃	C₂³	1	3	yes	—
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	D₄	1	3	yes	—
23	C₄²×C₅₃	C₂³	1	3	yes	—
24	C₄⋊Dic₅₃	—	2	6	no	—
25	C₄⋊Dic₅₃	C₂³	1	6	yes	—
26	#11719919	—	2	6	no	—
27	C₄×Dic₅₃	—	2	6	no	—
28	Dic₂₁₂	Q₈	1	4	yes	direct product factor
29	Q₈×C₅₃	Q₁₆	1	4	yes	complemented
30	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
31	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
32	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	2	no	—
34	C₂×Dic₅₃	—	2	6	no	—
35	Dic₁₀₆	—	2	2	no	—
36	Dic₁₀₆	—	8	6	no	—
37	Dic₁₀₆	C₂×Q₈	1	4	yes	—
38	C₈×C₅₃	—	2	3	no	—
39	C₈×C₅₃	C₂×Q₈	1	2	yes	—
40	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₅₃	—	2	3	no	—
42	C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
43	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
44	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
45	Dic₅₃	—	2	4	no	—
46	Dic₅₃	—	4	6	no	—
47	Q₈×Q₁₆	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
49	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
50	C₂×C₅₃	#576	1	1	yes	characteristic
51	Q₈²	—	53	2	no	—
52	#500	—	53	3	no	—
53	#477	—	53	6	no	—
54	C₈×Q₈	D₅₃	1	1	yes	characteristic
55	C₄×Q₁₆	—	53	3	no	—
56	C₅₃	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₁₆	—	53	1	no	—
58	C₄⋊Q₈	—	53	6	no	—
59	C₄⋊Q₈	—	106	6	no	—
60	C₄×Q₈	—	53	6	no	—
61	C₄×Q₈	—	106	2	no	—
62	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
63	#107	—	53	3	no	—
64	#107	—	53	6	no	—
65	C₄⋊C₈	C₂×D₅₃	1	3	yes	—
66	#103	—	53	6	no	—
67	C₄×C₈	C₂×D₅₃	1	3	yes	—
68	C₂×Q₈	—	53	2	no	—
69	C₂×Q₈	—	106	6	no	—
70	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
71	Q₁₆	—	53	4	no	—
72	C₂×C₈	C₂²×D₅₃	1	3	yes	—
73	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
74	C₄⋊C₄	—	53	6	no	—
75	C₄⋊C₄	—	106	6	no	—
76	C₄⋊C₄	—	106	6	no	—
77	C₄⋊C₄	D₂₁₂	1	3	yes	—
78	C₄²	—	106	6	no	—
79	C₄²	C₂²×D₅₃	1	3	yes	—
80	Q₈	—	53	4	no	—
81	Q₈	—	106	2	no	—
82	Q₈	—	424	6	no	—
83	Q₈	Dic₂₁₂	1	4	yes	direct product factor
84	C₂×C₄	—	106	2	no	—
85	C₂×C₄	—	106	6	no	—
86	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
87	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
88	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	Q₈×D₅₃	1	2	yes	—
91	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	106	4	no	—
94	C₄	—	212	6	no	—
95	C₄	C₂×Dic₂₁₂	1	6	yes	—
96	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
97	C₄	#13133264	1	1	yes	characteristic
98	C₂	Q₈×D₂₁₂	1	1	yes	characteristic
99	C₂	C₂²×Dic₂₁₂	1	1	yes	characteristic
100	C₂	#16811428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal

#16811028

C₂

yes

maximal

#16810919

C₂

yes

maximal

C₄×Dic₂₁₂

C₂

yes

maximal

C₈×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

#13133217

C₂²

yes

—

#13133216

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133151

C₂²

yes

—

#13133151

C₂²

yes

—

#13133142

C₂²

yes

—

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

Q₈

yes

direct product factor

Q₈×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂×Q₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Q₈×Q₁₆

—

maximal, Sylow

C₂×C₅₃

D₄×Q₈

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

#576

yes

characteristic

Q₈²

—

#500

—

#477

—

C₈×Q₈

D₅₃

yes

characteristic

C₄×Q₁₆

—

C₅₃

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

#107

—

#107

—

C₄⋊C₈

C₂×D₅₃

yes

—

#103

—

C₄×C₈

C₂×D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

D₂₁₂

yes

characteristic

Q₁₆

—

C₂×C₈

C₂²×D₅₃

yes

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

D₂₁₂

yes

—

C₄²

—

106

—

C₄²

C₂²×D₅₃

yes

—

Q₈

—

Q₈

—

106

—

Q₈

—

424

—

Q₈

Dic₂₁₂

yes

direct product factor

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂×C₄

C₂³×D₅₃

yes

characteristic

C₈

—

C₈

Q₈×D₅₃

yes

—

C₂²

C₂²×D₂₁₂

yes

characteristic

C₄

—

C₄

—

106

—

C₄

—

212

—

C₄

C₂×Dic₂₁₂

yes

—

C₄

C₂×Q₈×D₅₃

yes

characteristic

C₄

#13133264

yes

characteristic

C₂

Q₈×D₂₁₂

yes

characteristic

C₂

C₂²×Dic₂₁₂

yes

characteristic

100

C₂

#16811428

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₈×Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₈×Q₈

D₅₃

hypocenter

C₅₃

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
28. Dic₂₁₂	Q₈	direct (N × H)
29. Q₈×C₅₃	Q₁₆	semidirect
56. C₅₃	Q₈×Q₁₆	semidirect
83. Q₈	Dic₂₁₂	direct (N × H)

H ≅ G/N

type

28. Dic₂₁₂

Q₈

direct (N × H)

29. Q₈×C₅₃

Q₁₆

semidirect

56. C₅₃

Q₈×Q₁₆

semidirect

83. Q₈

Dic₂₁₂

direct (N × H)

Subgroups — Q₈×Dic₂₁₂ #20287600

Conjugacy classes of subgroups

Complements

Subgroups — Q8×Dic212 #20287600

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×Dic₂₁₂ #20287600