Group #20276525 — subgroups

Subgroups — #20276525

Order 6784.

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

normal subgroups only

This group has 128 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	#16810575	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	C₄	1	1	yes	characteristic
7	C₂×#11719912	C₂²	1	1	yes	characteristic
8	C₄²×D₅₃	C₄	1	1	yes	characteristic
9	#13133157	C₂²	1	2	yes	—
10	#13133128	C₄	1	1	yes	characteristic
11	C₈×Dic₅₃	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	—	2	1	no	—
16	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₅₃	—	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	#11719912	—	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	C₄×D₅₃	—	4	2	no	—
36	C₈×C₅₃	—	2	1	no	—
37	C₈×C₅₃	#41	1	2	yes	—
38	#88322	—	2	1	no	—
39	#88322	M₁₆	1	2	yes	—
40	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
41	C₂×D₅₃	—	2	1	no	—
42	C₂×D₅₃	—	4	1	no	—
43	C₄×C₅₃	—	4	1	no	—
44	C₄×C₅₃	#117	1	1	yes	characteristic
45	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
46	Dic₅₃	—	2	2	no	—
47	Dic₅₃	—	2	1	no	—
48	#2165	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	#432	1	1	yes	characteristic
51	D₅₃	—	4	1	no	—
52	C₄×M₁₆	—	53	1	no	—
53	#364	—	53	2	no	—
54	#349	—	53	2	no	—
55	#347	—	53	1	no	—
56	#347	D₅₃	1	1	yes	characteristic, complemented
57	C₅₃	#2165	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₁₆	—	53	1	no	—
59	C₂×M₁₆	—	53	1	no	—
60	C₂×C₄²	—	53	1	no	—
61	M₃₂	—	53	2	no	—
62	M₃₂	C₂×D₅₃	1	2	yes	—
63	#97	—	53	1	no	—
64	#97	C₂×D₅₃	1	1	yes	characteristic
65	C₄×C₈	—	53	1	no	—
66	C₄×C₈	—	53	1	no	—
67	C₂²×C₄	—	53	1	no	—
68	C₂²×C₄	—	106	1	no	—
69	M₁₆	—	106	2	no	—
70	M₁₆	—	106	2	no	—
71	C₂×C₈	—	53	1	no	—
72	C₂×C₈	—	53	1	no	—
73	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
75	C₄²	—	53	1	no	—
76	C₄²	—	106	1	no	—
77	C₄²	C₄×D₅₃	1	1	yes	characteristic
78	C₁₆	—	2	2	no	—
79	C₁₆	—	106	2	no	—
80	C₂³	—	53	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	53	1	no	—
83	C₂×C₄	—	106	1	no	—
84	C₂×C₄	—	106	1	no	—
85	C₂×C₄	—	106	1	no	—
86	C₂×C₄	—	212	2	no	—
87	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	53	2	no	—
90	C₈	—	106	1	no	—
91	C₈	#11719945	1	2	yes	—
92	C₂²	—	106	1	no	—
93	C₂²	—	212	1	no	—
94	C₂²	M₁₆×D₅₃	1	1	yes	characteristic
95	C₄	—	4	1	no	—
96	C₄	—	106	2	no	—
97	C₄	—	106	1	no	—
98	C₄	#13133174	1	1	yes	characteristic
99	C₄	M₁₆×D₅₃	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	212	1	no	—
102	C₂	#16810678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810575

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

C₄

yes

characteristic

C₂×#11719912

C₂²

yes

characteristic

C₄²×D₅₃

C₄

yes

characteristic

#13133157

C₂²

yes

—

#13133128

C₄

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

—

#97×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

—

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

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

—

#11719912

—

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

—

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

—

#2165

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#432

yes

characteristic

D₅₃

—

C₄×M₁₆

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2165

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

—

#97

—

#97

C₂×D₅₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

—

C₄²

—

106

—

C₄²

C₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

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

—

106

—

C₄

#13133174

yes

characteristic

C₄

M₁₆×D₅₃

yes

characteristic

100

C₂

—

101

C₂

—

212

—

102

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	#2165
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
56. #347	D₅₃	semidirect
57. C₅₃	#2165	semidirect

H ≅ G/N

type

2. #16810443

C₂

semidirect

4. #16810401

C₂

semidirect

5. C₅₃×#347

C₂

semidirect

56. #347

D₅₃

semidirect

57. C₅₃

#2165

semidirect

Subgroups — #20276525

Conjugacy classes of subgroups

Complements