Group #20275891 — subgroups

Subgroups — #20275891

Order 6784.

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

normal subgroups only

This group has 278 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×#13133131	C₂	1	3	yes	maximal
2	#16810389	C₂	1	3	yes	maximal, complemented
3	C₅₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
6	#13133131	—	2	6	no	—
7	C₂²⋊Q₈×C₅₃	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
9	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
10	C₂³×D₅₃	Q₈	1	1	yes	characteristic, complemented
11	C₂×Q₈×C₅₃	—	2	2	no	—
12	C₂²×C₄×C₅₃	—	2	3	no	—
13	C₂×C₂²⋊D₅₃	—	2	3	no	—
14	C₂×C₂²⋊D₅₃	—	2	3	no	—
15	C₂²×Dic₅₃	—	2	3	no	—
16	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic, complemented
17	C₄⋊C₄×C₅₃	—	2	6	no	—
18	C₂²⋊C₄×C₅₃	—	2	3	no	—
19	C₂²⋊C₄×C₅₃	D₄	1	6	yes	—
20	C₂²⋊Dic₅₃	—	2	3	no	—
21	C₂²⋊Dic₅₃	—	2	3	no	—
22	C₂³×C₅₃	—	8	1	no	—
23	C₂³×C₅₃	C₂×D₄	1	3	yes	—
24	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
25	C₂³×C₅₃	#41	1	3	yes	—
26	C₂²×D₅₃	—	2	3	no	—
27	C₂²×D₅₃	—	8	1	no	—
28	Q₈×C₅₃	—	8	2	no	—
29	C₂×C₄×C₅₃	—	2	6	no	—
30	C₂×C₄×C₅₃	—	8	3	no	—
31	C₂²⋊D₅₃	—	4	1	no	—
32	C₂²⋊D₅₃	—	4	3	no	—
33	C₂×Dic₅₃	—	2	3	no	—
34	C₂×Dic₅₃	—	4	3	no	—
35	C₂×Dic₅₃	—	8	3	no	—
36	C₂²×C₅₃	—	4	3	no	—
37	C₂²×C₅₃	—	4	3	no	—
38	C₂²×C₅₃	—	4	1	no	—
39	C₂²×C₅₃	#120	1	1	yes	characteristic
40	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
41	C₂²×C₅₃	#124	1	3	yes	—
42	C₂×D₅₃	—	4	1	no	—
43	C₂×D₅₃	—	8	3	no	—
44	C₄×C₅₃	—	4	6	no	—
45	Dic₅₃	—	4	1	no	—
46	Dic₅₃	—	8	3	no	—
47	#2032	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	—	2	3	no	—
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	—	2	3	no	—
51	C₂×C₅₃	#393	1	1	yes	characteristic
52	D₅₃	—	8	1	no	—
53	#543	D₅₃	1	1	yes	characteristic, complemented
54	C₂×#99	—	53	3	no	—
55	#342	—	53	3	no	—
56	C₅₃	#2032	1	1	yes	characteristic, complemented, Sylow
57	C₂²×D₄	—	53	1	no	—
58	C₂²⋊Q₈	—	2	6	no	—
59	C₂×C₂²⋊C₄	—	53	3	no	—
60	C₂×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
61	#99	—	106	6	no	—
62	C₂⁴	—	53	1	no	—
63	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
64	C₂×Q₈	—	2	2	no	—
65	C₂×D₄	—	106	3	no	—
66	C₂×D₄	—	106	3	no	—
67	C₂²×C₄	—	2	3	no	—
68	C₂²×C₄	—	53	1	no	—
69	C₂²×C₄	—	106	3	no	—
70	C₄⋊C₄	—	2	6	no	—
71	C₂²⋊C₄	—	2	3	no	—
72	C₂²⋊C₄	—	106	3	no	—
73	C₂²⋊C₄	—	106	3	no	—
74	C₂²⋊C₄	C₂²⋊D₅₃	1	6	yes	—
75	C₂³	—	8	1	no	—
76	C₂³	—	106	3	no	—
77	C₂³	—	424	1	no	—
78	C₂³	Q₈×D₅₃	1	1	yes	characteristic
79	C₂³	#11719949	1	3	yes	—
80	C₂³	C₂×C₂²⋊D₅₃	1	3	yes	—
81	Q₈	—	8	2	no	—
82	D₄	—	212	1	no	—
83	D₄	—	212	3	no	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	—	8	3	no	—
86	C₂×C₄	—	106	3	no	—
87	C₂×C₄	—	424	3	no	—
88	C₂×C₄	—	212	3	no	—
89	C₂²	—	4	3	no	—
90	C₂²	—	4	3	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	424	3	no	—
93	C₂²	—	212	1	no	—
94	C₂²	#13133186	1	1	yes	characteristic
95	C₂²	#13133193	1	3	yes	—
96	C₂²	#13133206	1	3	yes	—
97	C₄	—	4	6	no	—
98	C₄	—	424	3	no	—
99	C₄	—	212	1	no	—
100	C₂	—	2	1	no	—
101	C₂	—	2	3	no	—
102	C₂	—	2	3	no	—
103	C₂	—	424	1	no	—
104	C₂	#16810534	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133131

C₂

yes

maximal

#16810389

C₂

yes

maximal, complemented

C₅₃×#543

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₅₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

#13133131

—

C₂²⋊Q₈×C₅₃

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂³×D₅₃

Q₈

yes

characteristic, complemented

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂×C₂²⋊D₅₃

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

Q₈

yes

characteristic, complemented

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

D₄

yes

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

#41

yes

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#120

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

#124

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

#2032

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#393

yes

characteristic

D₅₃

—

#543

D₅₃

yes

characteristic, complemented

C₂×#99

—

#342

—

C₅₃

#2032

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

#99

—

106

—

C₂⁴

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

C₂²⋊D₅₃

yes

—

C₂³

—

C₂³

—

106

—

C₂³

—

424

—

C₂³

Q₈×D₅₃

yes

characteristic

C₂³

#11719949

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

—

Q₈

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

424

—

C₂²

—

212

—

C₂²

#13133186

yes

characteristic

C₂²

#13133193

yes

—

C₂²

#13133206

yes

—

C₄

—

C₄

—

424

—

C₄

—

212

—

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

—

424

—

104

C₂

#16810534

yes

characteristic

	subgroup	G/H
center	C₂	#16810534
commutator	C₂⁴×C₅₃	C₂³
Frattini	C₂⁴	C₂²×D₅₃
Fitting	C₅₃×#543	C₂
socle	C₂×C₅₃	#393
hypercenter	#543	D₅₃
hypocenter	C₅₃	#2032
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂

#16810534

commutator

C₂⁴×C₅₃

C₂³

Frattini

C₂⁴

C₂²×D₅₃

Fitting

C₅₃×#543

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #16810389	C₂	semidirect
3. C₅₃×#543	C₂	semidirect
10. C₂³×D₅₃	Q₈	semidirect
16. C₂²×Dic₅₃	Q₈	semidirect
53. #543	D₅₃	semidirect
56. C₅₃	#2032	semidirect

H ≅ G/N

type

2. #16810389

C₂

semidirect

3. C₅₃×#543

C₂

semidirect

10. C₂³×D₅₃

Q₈

semidirect

16. C₂²×Dic₅₃

Q₈

semidirect

53. #543

D₅₃

semidirect

56. C₅₃

#2032

semidirect

Subgroups — #20275891

Conjugacy classes of subgroups

Complements