Group #20276596 — subgroups

Subgroups — #20276596

Order 6784.

146 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810740	C₂	1	2	yes	maximal, complemented
2	#16810453	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810430	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#508	C₂	1	1	yes	maximal, characteristic
6	#13133224	—	2	2	no	—
7	C₂×#11719908	C₂²	1	1	yes	characteristic
8	#13133153	C₂²	1	2	yes	—
9	#13133144	—	2	2	no	—
10	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	#135×C₅₃	C₂²	1	2	yes	—
12	Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
13	SD₃₂×C₅₃	C₄	1	2	yes	complemented
14	D₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#41×C₅₃	—	2	2	no	—
17	Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
18	SD₁₆×C₅₃	—	2	2	no	—
19	D₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄×Dic₅₃	D₄	1	1	yes	characteristic
22	#11719917	—	2	2	no	—
23	#11719917	—	4	2	no	—
24	C₂×#88322	—	4	2	no	—
25	C₂×#88322	D₄	1	1	yes	characteristic
26	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#11719908	—	2	1	no	—
29	Q₈×C₅₃	—	2	2	no	—
30	D₄×C₅₃	—	2	2	no	—
31	D₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₅₃	—	2	1	no	—
35	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₅₃	#41	1	1	yes	characteristic
37	#88322	—	4	2	no	—
38	#88322	D₈	1	1	yes	characteristic
39	#88322	D₈	1	1	yes	characteristic
40	C₂²×C₅₃	—	4	2	no	—
41	C₂²×C₅₃	#135	1	1	yes	characteristic
42	C₄×C₅₃	—	4	2	no	—
43	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
44	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
45	Dic₅₃	—	2	2	no	—
46	#2178	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	2	1	no	—
48	C₂×C₅₃	—	8	2	no	—
49	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
50	#508	D₅₃	1	1	yes	characteristic
51	#443	—	53	2	no	—
52	#368	—	53	1	no	—
53	#359	—	53	2	no	—
54	C₄×C₁₆	—	53	1	no	—
55	C₅₃	#2178	1	1	yes	characteristic, complemented, Sylow
56	#135	C₂×D₅₃	1	2	yes	—
57	#131	—	106	2	no	—
58	Q₃₂	Dic₅₃	1	1	yes	characteristic, complemented
59	SD₃₂	Dic₅₃	1	2	yes	complemented
60	D₁₆	Dic₅₃	1	1	yes	characteristic, complemented
61	C₂×C₁₆	—	53	1	no	—
62	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
63	#108	—	53	2	no	—
64	#104	—	106	2	no	—
65	C₄×C₈	—	53	1	no	—
66	#41	—	2	2	no	—
67	Q₁₆	C₂×Dic₅₃	1	2	yes	—
68	SD₁₆	—	2	2	no	—
69	D₈	C₂×Dic₅₃	1	2	yes	—
70	M₁₆	—	106	2	no	—
71	M₁₆	—	212	2	no	—
72	C₂×C₈	—	53	1	no	—
73	C₂×C₈	—	212	2	no	—
74	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
75	C₄²	—	53	1	no	—
76	C₁₆	—	106	1	no	—
77	C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
78	C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
79	Q₈	—	2	2	no	—
80	D₄	—	2	2	no	—
81	D₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	106	1	no	—
84	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
85	C₈	—	53	1	no	—
86	C₈	—	53	1	no	—
87	C₈	—	212	2	no	—
88	C₈	#11719947	1	1	yes	characteristic
89	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
90	C₂²	—	4	2	no	—
91	C₂²	#13133236	1	1	yes	characteristic
92	C₄	—	4	2	no	—
93	C₄	—	106	2	no	—
94	C₄	D₄×Dic₅₃	1	1	yes	characteristic
95	C₄	D₈×D₅₃	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	8	2	no	—
98	C₂	D₈×Dic₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810740

C₂

yes

maximal, complemented

#16810453

C₂

yes

maximal, characteristic, complemented

#16810430

C₂

yes

maximal, complemented

C₁₆×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₅₃×#508

C₂

yes

maximal, characteristic

#13133224

—

C₂×#11719908

C₂²

yes

characteristic

#13133153

C₂²

yes

—

#13133144

—

C₈×Dic₅₃

C₂²

yes

characteristic

#135×C₅₃

C₂²

yes

—

Q₃₂×C₅₃

C₄

yes

characteristic, complemented

SD₃₂×C₅₃

C₄

yes

complemented

D₁₆×C₅₃

C₄

yes

characteristic, complemented

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

Q₁₆×C₅₃

C₂×C₄

yes

—

SD₁₆×C₅₃

—

D₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

—

—

D₄

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

characteristic

#11719908

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

C₂²×C₄

yes

C₈×C₅₃

yes

—

D₈

yes

characteristic

#88322

D₈

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

#135

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

#2178

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₄×D₈

yes

characteristic

#508

D₅₃

yes

—

—

—

C₄×C₁₆

—

C₅₃

#2178

yes

characteristic, complemented, Sylow

#135

C₂×D₅₃

yes

—

#131

—

106

—

Q₃₂

Dic₅₃

yes

characteristic, complemented

SD₃₂

Dic₅₃

yes

complemented

D₁₆

Dic₅₃

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#108

—

#104

—

106

—

C₄×C₈

—

#41

—

Q₁₆

C₂×Dic₅₃

yes

—

SD₁₆

—

D₈

C₂×Dic₅₃

yes

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

—

C₁₆

—

106

—

C₁₆

C₂×Dic₅₃

yes

characteristic

C₁₆

C₂×Dic₅₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

D₄×D₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

212

—

C₈

#11719947

yes

characteristic

C₈

C₂²×Dic₅₃

yes

characteristic

C₂²

—

C₂²

#13133236

yes

characteristic

C₄

—

C₄

—

106

—

C₄

D₄×Dic₅₃

yes

characteristic

C₄

D₈×D₅₃

yes

characteristic

C₂

—

C₂

—

C₂

D₈×Dic₅₃

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#508	C₂
socle	C₂×C₅₃	C₄×D₈
hypercenter	#508	D₅₃
hypocenter	C₅₃	#2178
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

subgroup

G/H

center

C₄

D₈×D₅₃

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#508

C₂

socle

C₂×C₅₃

C₄×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

N	H ≅ G/N	type
1. #16810740	C₂	semidirect
2. #16810453	C₂	semidirect
3. #16810430	C₂	semidirect
4. C₁₆×Dic₅₃	C₂	semidirect
12. Q₃₂×C₅₃	C₄	semidirect
13. SD₃₂×C₅₃	C₄	semidirect
14. D₁₆×C₅₃	C₄	semidirect
55. C₅₃	#2178	semidirect
58. Q₃₂	Dic₅₃	semidirect
59. SD₃₂	Dic₅₃	semidirect
60. D₁₆	Dic₅₃	semidirect

H ≅ G/N

type

1. #16810740

C₂

semidirect

2. #16810453

C₂

semidirect

3. #16810430

C₂

semidirect

4. C₁₆×Dic₅₃

C₂

semidirect

12. Q₃₂×C₅₃

C₄

semidirect

13. SD₃₂×C₅₃

C₄

semidirect

14. D₁₆×C₅₃

C₄

semidirect

55. C₅₃

#2178

semidirect

58. Q₃₂

Dic₅₃

semidirect

59. SD₃₂

Dic₅₃

semidirect

60. D₁₆

Dic₅₃

semidirect

Subgroups — #20276596

Conjugacy classes of subgroups

Complements