Group #20272669 — subgroups

Subgroups — #20272669

Order 6784.

110 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 47 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	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₅₃	—	2	1	no	—
5	C₄×D₂₁₂	C₄	1	1	yes	characteristic
6	C₂×#11719908	—	2	1	no	—
7	C₄⋊#88322	C₄	1	1	yes	characteristic
8	#13133129	—	2	1	no	—
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×D₂₁₂	C₈	1	1	yes	characteristic
12	C₂×C₄×D₅₃	—	2	1	no	—
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	#11719921	—	2	1	no	—
17	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
18	C₂×#88322	—	2	1	no	—
19	C₈×D₅₃	—	4	1	no	—
20	C₈×D₅₃	—	4	1	no	—
21	C₁₆×C₅₃	—	4	1	no	—
22	#11719908	—	2	2	no	—
23	C₂²×D₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×Dic₅₃	—	2	1	no	—
28	D₂₁₂	—	2	1	no	—
29	D₂₁₂	C₁₆	1	2	yes	complemented
30	C₄×D₅₃	—	4	1	no	—
31	C₄×D₅₃	—	4	1	no	—
32	C₈×C₅₃	—	2	1	no	—
33	C₈×C₅₃	D₈	1	1	yes	characteristic
34	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
35	#88322	—	4	1	no	—
36	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×D₅₃	—	2	2	no	—
38	C₂×D₅₃	—	4	1	no	—
39	C₂×D₅₃	—	4	1	no	—
40	C₄×C₅₃	—	2	1	no	—
41	C₄×C₅₃	#102	1	1	yes	characteristic
42	C₄×C₅₃	#104	1	1	yes	characteristic
43	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
44	C₄×C₅₃	M₃₂	1	1	yes	characteristic
45	Dic₅₃	—	4	1	no	—
46	D₄⋊C₁₆	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
48	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
49	C₂×C₅₃	#350	1	1	yes	characteristic
50	D₅₃	—	4	2	no	—
51	C₈×D₄	—	53	1	no	—
52	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
53	C₄×C₁₆	—	53	1	no	—
54	C₅₃	D₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₈	—	106	1	no	—
56	C₄×D₄	—	53	1	no	—
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	—	106	1	no	—
59	C₄⋊C₈	—	53	1	no	—
60	C₂²⋊C₈	—	106	1	no	—
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₂×D₄	—	53	1	no	—
63	C₂²×C₄	—	106	1	no	—
64	C₂×C₈	—	106	1	no	—
65	C₂×C₈	—	212	1	no	—
66	C₂×C₈	—	212	1	no	—
67	C₂×C₈	D₂₁₂	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
69	C₄⋊C₄	—	53	1	no	—
70	C₂²⋊C₄	—	106	1	no	—
71	C₄²	C₄×D₅₃	1	1	yes	characteristic
72	C₁₆	—	4	1	no	—
73	C₁₆	—	106	2	no	—
74	C₂³	—	106	1	no	—
75	D₄	—	53	2	no	—
76	D₄	—	106	1	no	—
77	C₂×C₄	—	106	1	no	—
78	C₂×C₄	—	212	1	no	—
79	C₂×C₄	—	212	1	no	—
80	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
81	C₂×C₄	#11719912	1	1	yes	characteristic
82	C₂×C₄	#11719921	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	212	1	no	—
85	C₈	#11719922	1	1	yes	characteristic
86	C₈	#11719924	1	1	yes	characteristic
87	C₂²	—	106	2	no	—
88	C₂²	—	212	1	no	—
89	C₂²	—	212	1	no	—
90	C₂²	#13133129	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	212	1	no	—
93	C₄	#13133137	1	1	yes	characteristic
94	C₄	#13133143	1	1	yes	characteristic
95	C₄	C₁₆×D₅₃	1	1	yes	characteristic
96	C₄	#13133156	1	1	yes	characteristic
97	C₂	—	212	2	no	—
98	C₂	#16810350	1	1	yes	characteristic
99	C₂	#16810402	1	1	yes	characteristic
100	C₂	#16810408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×D₂₁₂

C₂

yes

maximal, characteristic

C₄×#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

—

C₄×D₂₁₂

C₄

yes

characteristic

C₂×#11719908

—

C₄⋊#88322

C₄

yes

characteristic

#13133129

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

C₈

yes

characteristic

C₂×#88322

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

C₁₆

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

D₈

yes

characteristic

C₈×C₅₃

SD₁₆

yes

characteristic

#88322

—

C₂²×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₄×C₅₃

M₃₂

yes

characteristic

Dic₅₃

—

D₄⋊C₁₆

—

maximal, Sylow

C₂×C₅₃

D₄⋊C₈

yes

characteristic

C₂×C₅₃

C₂²⋊C₁₆

yes

characteristic

C₂×C₅₃

#350

yes

characteristic

D₅₃

—

C₈×D₄

—

C₄⋊C₁₆

D₅₃

yes

characteristic, complemented

C₄×C₁₆

—

C₅₃

D₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

C₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₂³

—

106

—

D₄

—

D₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₈×D₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

212

—

C₈

yes

C₈

yes

C₂²

—

106

—

C₂²

—

212

—

C₂²

—

212

—

C₂²

#13133129

yes

characteristic

C₄

—

C₄

—

212

—

C₄

yes

C₄

yes

C₄

C₁₆×D₅₃

yes

characteristic

C₄

#13133156

yes

characteristic

C₂

—

212

—

C₂

yes

C₂

yes

100

C₂

#16810408

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	D₄⋊C₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂×C₈

C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂×C₁₆

Frattini

C₄×C₈

C₂×D₅₃

Fitting

C₅₃×C₄⋊C₁₆

C₂

socle

C₂²×C₅₃

C₂²⋊C₈

hypercenter

C₄⋊C₁₆

D₅₃

hypocenter

C₅₃

D₄⋊C₁₆

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. C₄×#11719908	C₂	semidirect
3. C₅₃×C₄⋊C₁₆	C₂	semidirect
29. D₂₁₂	C₁₆	semidirect
52. C₄⋊C₁₆	D₅₃	semidirect
54. C₅₃	D₄⋊C₁₆	semidirect

H ≅ G/N

type

2. C₄×#11719908

C₂

semidirect

3. C₅₃×C₄⋊C₁₆

C₂

semidirect

29. D₂₁₂

C₁₆

semidirect

52. C₄⋊C₁₆

D₅₃

semidirect

54. C₅₃

D₄⋊C₁₆

semidirect

Subgroups — #20272669

Conjugacy classes of subgroups

Complements