Group #20282423 — subgroups

Subgroups — #20282423

Order 6784.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16810540	C₂	1	3	yes	maximal
2	#16810534	C₂	1	2	yes	maximal, complemented
3	#16810515	C₂	1	6	yes	maximal, complemented
4	#16810488	C₂	1	3	yes	maximal, complemented
5	C₅₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
8	C₂×#11719921	C₂²	1	6	yes	—
9	C₂×#11719921	C₂²	1	6	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
11	#13133124	C₂²	1	6	yes	—
12	#13133124	C₂²	1	3	yes	—
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
15	#95×C₅₃	C₂²	1	3	yes	—
16	C₂³×D₅₃	C₂³	1	2	yes	—
17	C₂×Q₈×C₅₃	—	4	3	no	—
18	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
21	C₂²×Dic₅₃	C₂³	1	6	yes	—
22	C₂×D₂₁₂	—	4	3	no	—
23	C₄⋊C₄×C₅₃	—	4	3	no	—
24	#11719921	—	2	12	no	—
25	#11719921	—	4	6	no	—
26	C₄⋊Dic₅₃	—	4	3	no	—
27	C₄×Dic₅₃	—	2	6	no	—
28	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
29	C₂²×D₅₃	—	2	6	no	—
30	C₂²×D₅₃	—	2	6	no	—
31	C₂²×D₅₃	—	2	2	no	—
32	Q₈×C₅₃	—	4	4	no	—
33	C₂×C₄×C₅₃	—	2	3	no	—
34	C₂×C₄×C₅₃	—	2	3	no	—
35	C₂×C₄×C₅₃	—	2	3	no	—
36	C₂×C₄×C₅₃	—	2	6	no	—
37	C₂×C₄×C₅₃	—	2	3	no	—
38	C₂×C₄×C₅₃	#41	1	6	yes	—
39	C₂×Dic₅₃	—	2	6	no	—
40	C₂×Dic₅₃	—	2	6	no	—
41	C₂×Dic₅₃	—	2	6	no	—
42	D₂₁₂	—	4	4	no	—
43	C₂²×C₅₃	C₂×#41	1	3	yes	—
44	C₂²×C₅₃	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
46	C₂×D₅₃	—	4	6	no	—
47	C₂×D₅₃	—	4	6	no	—
48	C₂×D₅₃	—	4	2	no	—
49	C₄×C₅₃	—	2	6	no	—
50	C₄×C₅₃	—	4	1	no	—
51	C₄×C₅₃	—	4	3	no	—
52	Dic₅₃	—	4	6	no	—
53	#2785	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	#539	1	3	yes	—
55	C₂×C₅₃	#555	1	3	yes	—
56	C₂×C₅₃	H(3,𝔽₄)	1	1	yes	characteristic
57	D₅₃	—	8	2	no	—
58	#395	D₅₃	1	1	yes	characteristic, complemented
59	#394	—	53	3	no	—
60	#393	—	53	2	no	—
61	#388	—	53	6	no	—
62	#382	—	53	3	no	—
63	C₅₃	#2785	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
65	C₂²×D₄	—	53	1	no	—
66	C₂×C₄⋊C₄	—	53	3	no	—
67	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
68	C₂×C₂²⋊C₄	—	53	6	no	—
69	C₂×C₂²⋊C₄	—	53	6	no	—
70	C₂×C₄²	—	53	3	no	—
71	#95	—	53	3	no	—
72	#95	—	53	6	no	—
73	#95	C₂×D₅₃	1	3	yes	—
74	C₂⁴	—	53	2	no	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	212	3	no	—
77	C₂²×C₄	—	53	6	no	—
78	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
80	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	212	3	no	—
83	C₂²⋊C₄	—	106	12	no	—
84	C₂²⋊C₄	—	212	6	no	—
85	C₄²	—	106	6	no	—
86	C₂³	—	106	6	no	—
87	C₂³	—	106	6	no	—
88	C₂³	—	106	2	no	—
89	C₂³	C₂³×D₅₃	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	212	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	106	6	no	—
98	C₂×C₄	—	106	6	no	—
99	C₂×C₄	—	106	6	no	—
100	C₂×C₄	#11719949	1	6	yes	—
101	C₂²	—	212	6	no	—
102	C₂²	—	212	2	no	—
103	C₂²	—	212	6	no	—
104	C₂²	C₂×#11719949	1	3	yes	—
105	C₂²	#13133262	1	3	yes	—
106	C₂²	#13133264	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	212	6	no	—
111	C₂	—	424	2	no	—
112	C₂	#16811222	1	3	yes	—
113	C₂	#16811339	1	3	yes	—
114	C₂	#16811366	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810540

C₂

yes

maximal

#16810534

C₂

yes

maximal, complemented

#16810515

C₂

yes

maximal, complemented

#16810488

C₂

yes

maximal, complemented

C₅₃×#395

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#95×C₅₃

C₂²

yes

—

C₂³×D₅₃

C₂³

yes

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

—

C₄⋊C₄×C₅₃

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2785

—

maximal, Sylow

C₂×C₅₃

#539

yes

—

C₂×C₅₃

#555

yes

—

C₂×C₅₃

H(3,𝔽₄)

yes

characteristic

D₅₃

—

#395

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#2785

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₅₃

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

212

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

212

—

C₄²

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

#11719949

yes

—

101

C₂²

—

212

—

102

C₂²

—

212

—

103

C₂²

—

212

—

104

C₂²

C₂×#11719949

yes

—

105

C₂²

#13133262

yes

—

106

C₂²

#13133264

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

212

—

111

C₂

—

424

—

112

C₂

#16811222

yes

—

113

C₂

#16811339

yes

—

114

C₂

#16811366

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#395	C₂
hypercenter	#395	D₅₃
hypocenter	C₅₃	#2785
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#395

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
2. #16810534	C₂	semidirect
3. #16810515	C₂	semidirect
4. #16810488	C₂	semidirect
5. C₅₃×#395	C₂	semidirect
58. #395	D₅₃	semidirect
63. C₅₃	#2785	semidirect

H ≅ G/N

type

2. #16810534

C₂

semidirect

3. #16810515

C₂

semidirect

4. #16810488

C₂

semidirect

5. C₅₃×#395

C₂

semidirect

58. #395

D₅₃

semidirect

63. C₅₃

#2785

semidirect

Subgroups — #20282423

Conjugacy classes of subgroups

Complements