Group #20288863 — subgroups

Subgroups — #20288863

Order 6784.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#16811377	C₂	1	10	yes	maximal, complemented
2	#16811371	C₂	1	5	yes	maximal, complemented
3	#16811179	C₂	1	5	yes	maximal, complemented
4	#16811158	C₂	1	10	yes	maximal, complemented
5	C₅₃×#564	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133216	C₂²	1	10	yes	—
7	#13133212	C₂²	1	20	yes	—
8	#13133209	C₂²	1	5	yes	—
9	#13133205	C₂²	1	10	yes	—
10	#13133202	C₂²	1	20	yes	—
11	#13133197	C₂²	1	10	yes	—
12	#13133195	C₂²	1	20	yes	—
13	#13133192	C₂²	1	10	yes	—
14	#13133182	C₂²	1	10	yes	—
15	#13133179	C₂²	1	10	yes	—
16	#13133175	C₂²	1	10	yes	—
17	C₄²×D₅₃	C₂²	1	5	yes	—
18	#125×C₅₃	C₂²	1	5	yes	—
19	#125×C₅₃	C₂²	1	10	yes	—
20	C₂×D₂₁₂	C₂³	1	10	yes	—
21	C₂×C₄×D₅₃	C₂³	1	5	yes	—
22	C₂×C₄×D₅₃	C₂³	1	10	yes	—
23	C₂×Dic₁₀₆	C₂³	1	10	yes	—
24	C₄⋊C₄×C₅₃	C₂³	1	10	yes	—
25	C₄⋊C₄×C₅₃	C₂³	1	10	yes	—
26	C₄⋊C₄×C₅₃	C₂³	1	10	yes	—
27	C₄²×C₅₃	C₂³	1	5	yes	—
28	#11719921	C₂³	1	20	yes	—
29	#11719921	C₂³	1	20	yes	—
30	C₄⋊Dic₅₃	C₂³	1	10	yes	—
31	#11719919	C₂³	1	20	yes	—
32	C₄×Dic₅₃	C₂³	1	10	yes	—
33	C₄×Dic₅₃	C₂³	1	5	yes	—
34	C₂²×D₅₃	C₂⁴	1	5	yes	—
35	C₂×C₄×C₅₃	C₂⁴	1	10	yes	—
36	C₂×C₄×C₅₃	C₂⁴	1	5	yes	—
37	C₂×Dic₅₃	C₂⁴	1	10	yes	—
38	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
39	D₂₁₂	—	4	10	no	—
40	C₄×D₅₃	—	2	10	no	—
41	C₄×D₅₃	—	4	10	no	—
42	Dic₁₀₆	—	4	10	no	—
43	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
44	C₂×D₅₃	—	2	5	no	—
45	C₂×D₅₃	—	2	10	no	—
46	C₄×C₅₃	—	2	5	no	—
47	C₄×C₅₃	—	2	10	no	—
48	Dic₅₃	—	2	10	no	—
49	Dic₅₃	2₋¹⁺⁴	1	2	yes	—
50	#3531	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
52	C₂×C₅₃	#585	1	2	yes	—
53	D₅₃	—	4	5	no	—
54	#564	D₅₃	1	1	yes	characteristic, complemented
55	#563	—	53	10	no	—
56	#562	—	53	5	no	—
57	#533	—	53	5	no	—
58	#529	—	53	10	no	—
59	C₅₃	#3531	1	1	yes	characteristic, complemented, Sylow
60	C₄⋊Q₈	—	53	10	no	—
61	#126	—	53	20	no	—
62	#125	—	53	5	no	—
63	#125	C₂×D₅₃	1	10	yes	—
64	#125	C₂×D₅₃	1	5	yes	—
65	#124	—	53	10	no	—
66	#123	—	53	20	no	—
67	C₂²⋊Q₈	—	53	10	no	—
68	C₂²⋊Q₈	—	53	20	no	—
69	#121	—	53	10	no	—
70	C₄×Q₈	—	53	10	no	—
71	C₄×D₄	—	53	10	no	—
72	#117	—	53	10	no	—
73	C₂×C₄²	—	53	5	no	—
74	C₂×Q₈	—	53	10	no	—
75	C₂×D₄	—	53	10	no	—
76	C₂²×C₄	—	53	5	no	—
77	C₂²×C₄	—	53	10	no	—
78	C₄⋊C₄	—	53	10	no	—
79	C₄⋊C₄	—	53	20	no	—
80	C₄⋊C₄	C₂²×D₅₃	1	10	yes	—
81	C₄⋊C₄	C₂²×D₅₃	1	10	yes	—
82	C₄⋊C₄	C₂²×D₅₃	1	10	yes	—
83	C₂²⋊C₄	—	53	20	no	—
84	C₂²⋊C₄	—	53	20	no	—
85	C₄²	—	53	10	no	—
86	C₄²	—	53	5	no	—
87	C₄²	C₂²×D₅₃	1	5	yes	—
88	C₂³	—	53	5	no	—
89	Q₈	—	212	10	no	—
90	D₄	—	212	10	no	—
91	C₂×C₄	—	53	1	no	—
92	C₂×C₄	—	53	10	no	—
93	C₂×C₄	—	106	10	no	—
94	C₂×C₄	—	212	10	no	—
95	C₂×C₄	C₂³×D₅₃	1	5	yes	—
96	C₂×C₄	C₂³×D₅₃	1	10	yes	—
97	C₂²	—	106	5	no	—
98	C₂²	—	106	10	no	—
99	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	53	2	no	—
103	C₄	—	106	10	no	—
104	C₂	—	212	5	no	—
105	C₂	2₋¹⁺⁴×D₅₃	1	1	yes	characteristic
106	C₂	#16811460	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811377

C₂

yes

maximal, complemented

#16811371

C₂

yes

maximal, complemented

#16811179

C₂

yes

maximal, complemented

#16811158

C₂

yes

maximal, complemented

C₅₃×#564

C₂

yes

maximal, characteristic, complemented

#13133216

C₂²

yes

—

#13133212

C₂²

yes

—

#13133209

C₂²

yes

—

#13133205

C₂²

yes

—

#13133202

C₂²

yes

—

#13133197

C₂²

yes

—

#13133195

C₂²

yes

—

#13133192

C₂²

yes

—

#13133182

C₂²

yes

—

#13133179

C₂²

yes

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

—

#125×C₅₃

C₂²

yes

—

#125×C₅₃

C₂²

yes

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

C₂³

yes

—

#11719921

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂²×D₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

2₋¹⁺⁴

yes

—

#3531

—

maximal, Sylow

C₂×C₅₃

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₅₃

#585

yes

—

D₅₃

—

#564

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#3531

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#126

—

#125

—

#125

C₂×D₅₃

yes

—

#125

C₂×D₅₃

yes

—

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅₃

yes

—

C₂³

—

Q₈

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂³×D₅₃

yes

—

C₂×C₄

C₂³×D₅₃

yes

—

C₂²

—

106

—

C₂²

—

106

—

C₂²

C₂⁴×D₅₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

106

—

104

C₂

—

212

—

105

C₂

2₋¹⁺⁴×D₅₃

yes

characteristic

106

C₂

#16811460

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#564	C₂
hypercenter	#564	D₅₃
hypocenter	C₅₃	#3531
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₅₃

commutator, socle

C₂²×C₅₃

C₂⁵

Fitting

C₅₃×#564

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #16811377	C₂	semidirect
2. #16811371	C₂	semidirect
3. #16811179	C₂	semidirect
4. #16811158	C₂	semidirect
5. C₅₃×#564	C₂	semidirect
54. #564	D₅₃	semidirect
59. C₅₃	#3531	semidirect

H ≅ G/N

type

1. #16811377

C₂

semidirect

2. #16811371

C₂

semidirect

3. #16811179

C₂

semidirect

4. #16811158

C₂

semidirect

5. C₅₃×#564

C₂

semidirect

54. #564

D₅₃

semidirect

59. C₅₃

#3531

semidirect

Subgroups — #20288863

Conjugacy classes of subgroups

Complements