Group #20277005 — subgroups

Subgroups — M₆₄×D₅₃ #20277005

Order 6784.

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

normal subgroups only

This group has 112 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	C₂×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810457	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810456	C₂	1	2	yes	maximal, complemented
4	C₃₂×D₅₃	C₂	1	2	yes	maximal, complemented
5	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂×#11719908	C₄	1	1	yes	characteristic
8	C₁₆×D₅₃	C₄	1	1	yes	characteristic
9	C₁₆×D₅₃	C₄	1	1	yes	characteristic
10	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
11	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
12	#13133123	C₂²	1	2	yes	complemented
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₃₂×C₅₃	C₂²	1	2	yes	complemented
15	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×#88322	C₈	1	1	yes	characteristic
18	C₈×D₅₃	C₈	1	1	yes	characteristic
19	C₈×D₅₃	C₈	1	1	yes	characteristic
20	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
24	#11719908	C₂×C₄	1	1	yes	characteristic
25	#11719908	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₅₃	C₁₆	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×Dic₅₃	C₁₆	1	1	yes	characteristic
29	C₄×D₅₃	C₁₆	1	1	yes	characteristic
30	C₄×D₅₃	C₁₆	1	1	yes	characteristic
31	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
32	C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
33	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	#88322	C₂×C₈	1	1	yes	characteristic
36	#88322	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
38	C₂×D₅₃	—	2	2	no	—
39	C₂×D₅₃	C₂×C₁₆	1	1	yes	characteristic
40	C₂×D₅₃	C₂×C₁₆	1	1	yes	characteristic
41	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
42	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
43	Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
44	Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×M₆₄	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	1	no	—
47	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
48	D₅₃	—	2	1	no	—
49	D₅₃	M₆₄	1	2	yes	direct product factor
50	C₂²×C₁₆	—	53	1	no	—
51	M₆₄	—	53	1	no	—
52	M₆₄	—	53	2	no	—
53	M₆₄	D₅₃	1	1	yes	characteristic, direct product factor
54	C₂×C₃₂	—	53	2	no	—
55	C₅₃	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	53	1	no	—
57	C₂×C₁₆	—	53	1	no	—
58	C₂×C₁₆	—	53	1	no	—
59	C₂×C₁₆	—	53	1	no	—
60	C₂×C₁₆	—	53	1	no	—
61	C₂×C₁₆	—	53	1	no	—
62	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
63	C₃₂	—	53	2	no	—
64	C₃₂	C₂×D₅₃	1	2	yes	complemented
65	C₂²×C₄	—	53	1	no	—
66	C₂×C₈	—	53	1	no	—
67	C₂×C₈	—	53	1	no	—
68	C₂×C₈	—	53	1	no	—
69	C₂×C₈	—	53	1	no	—
70	C₂×C₈	—	53	1	no	—
71	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
72	C₁₆	—	53	1	no	—
73	C₁₆	—	53	1	no	—
74	C₁₆	C₄×D₅₃	1	1	yes	characteristic
75	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
76	C₂³	—	53	1	no	—
77	C₂×C₄	—	53	1	no	—
78	C₂×C₄	—	53	1	no	—
79	C₂×C₄	—	53	1	no	—
80	C₂×C₄	—	53	1	no	—
81	C₂×C₄	—	53	1	no	—
82	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
83	C₈	—	53	1	no	—
84	C₈	—	53	1	no	—
85	C₈	C₈×D₅₃	1	1	yes	characteristic
86	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
87	C₂²	—	53	1	no	—
88	C₂²	—	53	1	no	—
89	C₂²	—	106	2	no	—
90	C₂²	C₁₆×D₅₃	1	1	yes	characteristic
91	C₄	—	53	1	no	—
92	C₄	—	53	1	no	—
93	C₄	C₁₆×D₅₃	1	1	yes	characteristic
94	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
95	C₂	—	2	1	no	—
96	C₂	—	53	2	no	—
97	C₂	—	106	1	no	—
98	C₂	C₂×C₁₆×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×D₅₃

C₂

yes

maximal, characteristic

#16810457

C₂

yes

maximal, characteristic, complemented

#16810456

C₂

yes

maximal, complemented

C₃₂×D₅₃

C₂

yes

maximal, complemented

C₅₃×M₆₄

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×#11719908

C₄

yes

characteristic

C₁₆×D₅₃

C₄

yes

characteristic

C₁₆×D₅₃

C₄

yes

characteristic

C₁₆×D₅₃

C₂²

yes

characteristic

C₁₆×D₅₃

C₂²

yes

characteristic

#13133123

C₂²

yes

complemented

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₃₂×C₅₃

C₂²

yes

complemented

C₂×C₄×D₅₃

C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₈

yes

characteristic

C₈×D₅₃

C₈

yes

characteristic

C₈×D₅₃

C₈

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₁₆×C₅₃

C₂³

yes

characteristic

#11719908

C₂×C₄

yes

characteristic

#11719908

C₂×C₄

yes

characteristic

C₂²×D₅₃

C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₁₆

yes

characteristic

C₄×D₅₃

C₁₆

yes

characteristic

C₄×D₅₃

C₁₆

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

C₂×C₈

yes

characteristic

C₈×C₅₃

C₂×C₈

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₂×C₈

yes

characteristic

#88322

C₂×C₈

yes

characteristic

C₂²×C₅₃

C₂×C₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₂×C₁₆

yes

characteristic

C₂×D₅₃

C₂×C₁₆

yes

characteristic

C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

Dic₅₃

C₂×C₁₆

yes

characteristic

Dic₅₃

C₂×C₁₆

yes

characteristic

C₂×M₆₄

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₁₆

yes

characteristic

D₅₃

—

D₅₃

M₆₄

yes

direct product factor

C₂²×C₁₆

—

M₆₄

—

M₆₄

—

M₆₄

D₅₃

yes

characteristic, direct product factor

C₂×C₃₂

—

C₅₃

C₂×M₆₄

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₃₂

—

C₃₂

C₂×D₅₃

yes

complemented

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₄×D₅₃

yes

characteristic

C₁₆

C₂²×D₅₃

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₅₃

yes

characteristic

C₈

—

C₈

—

C₈

C₈×D₅₃

yes

characteristic

C₈

C₂×C₄×D₅₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

106

—

C₂²

C₁₆×D₅₃

yes

characteristic

C₄

—

C₄

—

C₄

C₁₆×D₅₃

yes

characteristic

C₄

C₂×C₈×D₅₃

yes

characteristic

C₂

—

C₂

—

C₂

—

106

—

C₂

C₂×C₁₆×D₅₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₅₃
commutator, socle	C₂×C₅₃	C₂²×C₁₆
Fitting	C₅₃×M₆₄	C₂
hypercenter	M₆₄	D₅₃
hypocenter	C₅₃	C₂×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ M₆₄

subgroup

G/H

center, Frattini

C₁₆

C₂²×D₅₃

commutator, socle

C₂×C₅₃

C₂²×C₁₆

Fitting

C₅₃×M₆₄

C₂

hypercenter

M₆₄

D₅₃

hypocenter

C₅₃

C₂×M₆₄

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₁₆ ◁ M₆₄

N	H ≅ G/N	type
2. #16810457	C₂	semidirect
3. #16810456	C₂	semidirect
4. C₃₂×D₅₃	C₂	semidirect
5. C₅₃×M₆₄	C₂	semidirect
12. #13133123	C₂²	semidirect
14. C₃₂×C₅₃	C₂²	semidirect
49. D₅₃	M₆₄	direct (N × H)
53. M₆₄	D₅₃	direct (N × H)
55. C₅₃	C₂×M₆₄	semidirect
64. C₃₂	C₂×D₅₃	semidirect

H ≅ G/N

type

2. #16810457

C₂

semidirect

3. #16810456

C₂

semidirect

4. C₃₂×D₅₃

C₂

semidirect

5. C₅₃×M₆₄

C₂

semidirect

12. #13133123

C₂²

semidirect

14. C₃₂×C₅₃

C₂²

semidirect

49. D₅₃

M₆₄

direct (N × H)

53. M₆₄

D₅₃

direct (N × H)

55. C₅₃

C₂×M₆₄

semidirect

64. C₃₂

C₂×D₅₃

semidirect

Subgroups — M₆₄×D₅₃ #20277005

Conjugacy classes of subgroups

Complements

Subgroups — M64×D53 #20277005

Conjugacy classes of subgroups

Complements

Subgroups — M₆₄×D₅₃ #20277005