Group #20276529 — subgroups

Subgroups — #20276529

Order 6784.

122 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16810580	C₂	1	1	yes	maximal, characteristic
2	#16810443	C₂	1	2	yes	maximal, complemented
3	#16810404	C₂	1	2	yes	maximal
4	#16810401	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719912	C₂²	1	1	yes	characteristic
7	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
8	#13133174	C₄	1	1	yes	characteristic
9	#13133157	C₂²	1	2	yes	—
10	#13133127	C₄	1	1	yes	characteristic
11	C₄×#88322	C₄	1	1	yes	characteristic
12	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	M₃₂×C₅₃	C₂²	1	2	yes	—
14	#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#11719921	—	2	1	no	—
20	#11719919	—	2	1	no	—
21	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂×C₄	1	1	yes	characteristic
24	#11719912	—	2	2	no	—
25	C₈×D₅₃	—	2	2	no	—
26	C₁₆×C₅₃	—	2	2	no	—
27	#11719908	—	2	2	no	—
28	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	2	1	no	—
32	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
33	C₄×D₅₃	—	2	1	no	—
34	C₄×D₅₃	—	2	1	no	—
35	C₈×C₅₃	—	2	1	no	—
36	C₈×C₅₃	#41	1	2	yes	—
37	#88322	—	2	1	no	—
38	#88322	M₁₆	1	2	yes	—
39	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
40	C₂×D₅₃	—	2	1	no	—
41	C₂×D₅₃	—	4	1	no	—
42	C₄×C₅₃	—	4	1	no	—
43	C₄×C₅₃	#117	1	1	yes	characteristic
44	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
45	Dic₅₃	—	2	1	no	—
46	Dic₅₃	—	4	1	no	—
47	#2166	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	1	no	—
49	C₂×C₅₃	#432	1	1	yes	characteristic
50	D₅₃	—	4	1	no	—
51	#405	—	53	1	no	—
52	#364	—	53	2	no	—
53	#349	—	53	2	no	—
54	#347	—	53	1	no	—
55	#347	D₅₃	1	1	yes	characteristic, complemented
56	C₅₃	#2166	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	53	1	no	—
58	C₂²×C₈	—	53	1	no	—
59	#117	—	53	1	no	—
60	M₃₂	—	53	2	no	—
61	M₃₂	C₂×D₅₃	1	2	yes	—
62	#97	—	53	1	no	—
63	#97	C₂×D₅₃	1	1	yes	characteristic
64	C₄×C₈	—	53	1	no	—
65	C₄×C₈	—	53	1	no	—
66	C₂²×C₄	—	53	1	no	—
67	M₁₆	—	106	2	no	—
68	C₂×C₈	—	53	1	no	—
69	C₂×C₈	—	53	1	no	—
70	C₂×C₈	—	106	2	no	—
71	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
73	C₄⋊C₄	—	106	1	no	—
74	C₂²⋊C₄	—	106	1	no	—
75	C₄²	—	53	1	no	—
76	C₄²	C₄×D₅₃	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	106	2	no	—
79	C₂³	—	53	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	53	1	no	—
82	C₂×C₄	—	106	1	no	—
83	C₂×C₄	—	106	1	no	—
84	C₂×C₄	—	106	1	no	—
85	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	53	2	no	—
88	C₈	—	106	1	no	—
89	C₈	#11719945	1	2	yes	—
90	C₂²	—	106	1	no	—
91	C₂²	—	212	1	no	—
92	C₂²	M₁₆×D₅₃	1	1	yes	characteristic
93	C₄	—	4	1	no	—
94	C₄	—	106	1	no	—
95	C₄	—	212	1	no	—
96	C₄	#13133174	1	1	yes	characteristic
97	C₄	M₁₆×D₅₃	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	212	1	no	—
100	C₂	#16810678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810580

C₂

yes

maximal, characteristic

#16810443

C₂

yes

maximal, complemented

#16810404

C₂

yes

maximal

#16810401

C₂

yes

maximal, characteristic, complemented

C₅₃×#347

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

characteristic

#13133174

C₄

yes

characteristic

#13133157

C₂²

yes

—

#13133127

C₄

yes

characteristic

C₄×#88322

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

—

#97×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

M₁₆

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

#88322

—

#88322

M₁₆

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2166

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#432

yes

characteristic

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2166

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

—

#97

—

#97

C₂×D₅₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

C₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₈

#11719945

yes

—

C₂²

—

106

—

C₂²

—

212

—

C₂²

M₁₆×D₅₃

yes

characteristic

C₄

—

C₄

—

106

—

C₄

—

212

—

C₄

#13133174

yes

characteristic

C₄

M₁₆×D₅₃

yes

characteristic

C₂

—

C₂

—

212

—

100

C₂

#16810678

yes

characteristic

	subgroup	G/H
center	C₄	#13133174
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#347	C₂
socle	C₂×C₅₃	#432
hypercenter	#347	D₅₃
hypocenter	C₅₃	#2166
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #97 ◁ #347

subgroup

G/H

center

C₄

#13133174

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#347

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ #97 ◁ #347

N	H ≅ G/N	type
2. #16810443	C₂	semidirect
4. #16810401	C₂	semidirect
5. C₅₃×#347	C₂	semidirect
55. #347	D₅₃	semidirect
56. C₅₃	#2166	semidirect

H ≅ G/N

type

2. #16810443

C₂

semidirect

4. #16810401

C₂

semidirect

5. C₅₃×#347

C₂

semidirect

55. #347

D₅₃

semidirect

56. C₅₃

#2166

semidirect

Subgroups — #20276529

Conjugacy classes of subgroups

Complements