Group #20283082 — subgroups

Subgroups — #20283082

Order 6784.

516 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13133211	C₂	1	2	yes	maximal, complemented
2	#16810525	C₂	1	3	yes	maximal
3	#16810521	C₂	1	3	yes	maximal, complemented
4	#16810514	C₂	1	6	yes	maximal, complemented
5	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
7	#13133211	—	2	8	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
9	C₂×#11719919	C₂²	1	6	yes	—
10	C₂×#11719921	C₂²	1	6	yes	—
11	C₂×#11719921	C₂²	1	6	yes	—
12	#13133124	C₂²	1	6	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×D₅₃	C₂³	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂³	1	6	yes	—
21	C₂×D₂₁₂	—	2	6	no	—
22	C₄²×C₅₃	—	2	6	no	—
23	C₄²×C₅₃	—	2	6	no	—
24	C₄²×C₅₃	D₄	1	4	yes	complemented
25	#11719921	—	2	12	no	—
26	#11719921	—	2	12	no	—
27	C₄⋊Dic₅₃	—	2	6	no	—
28	#11719919	—	2	12	no	—
29	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂²×D₅₃	—	2	2	no	—
31	C₂²×D₅₃	—	2	6	no	—
32	C₂²×D₅₃	—	2	6	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	3	no	—
37	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
38	C₂×C₄×C₅₃	#41	1	6	yes	—
39	C₂×C₄×C₅₃	#41	1	6	yes	—
40	C₂×Dic₅₃	—	2	6	no	—
41	C₂×Dic₅₃	—	2	6	no	—
42	C₂×Dic₅₃	—	2	6	no	—
43	D₂₁₂	—	4	4	no	—
44	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
45	C₂²×C₅₃	C₂×#41	1	3	yes	—
46	C₂²×C₅₃	C₂×#41	1	3	yes	—
47	C₂×D₅₃	—	4	2	no	—
48	C₂×D₅₃	—	4	6	no	—
49	C₂×D₅₃	—	4	6	no	—
50	C₄×C₅₃	—	2	6	no	—
51	C₄×C₅₃	—	2	6	no	—
52	C₄×C₅₃	#126	1	4	yes	complemented
53	Dic₅₃	—	4	6	no	—
54	#2860	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	C₂×#126	1	1	yes	characteristic
56	C₂×C₅₃	#529	1	3	yes	—
57	C₂×C₅₃	#532	1	3	yes	—
58	D₅₃	—	8	2	no	—
59	C₂×#126	—	53	2	no	—
60	#390	—	53	3	no	—
61	#389	—	53	3	no	—
62	#388	—	53	6	no	—
63	C₄³	D₅₃	1	1	yes	characteristic, complemented
64	C₅₃	#2860	1	1	yes	characteristic, complemented, Sylow
65	C₂²×D₄	—	53	1	no	—
66	#126	—	106	8	no	—
67	C₂×C₄⋊C₄	—	53	6	no	—
68	C₂×C₄⋊C₄	—	53	3	no	—
69	C₂×C₂²⋊C₄	—	53	6	no	—
70	C₂×C₂²⋊C₄	—	53	6	no	—
71	C₂×C₄²	C₂×D₅₃	1	3	yes	—
72	C₂×C₄²	C₂×D₅₃	1	3	yes	—
73	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
74	#95	—	53	6	no	—
75	C₂⁴	—	53	2	no	—
76	C₂×D₄	—	106	6	no	—
77	C₂²×C₄	—	53	6	no	—
78	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
79	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
80	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
81	C₄⋊C₄	—	106	12	no	—
82	C₄⋊C₄	—	106	6	no	—
83	C₂²⋊C₄	—	106	12	no	—
84	C₂²⋊C₄	—	106	12	no	—
85	C₄²	—	2	6	no	—
86	C₄²	—	2	6	no	—
87	C₄²	D₂₁₂	1	4	yes	complemented
88	C₂³	—	106	2	no	—
89	C₂³	—	106	6	no	—
90	C₂³	—	106	6	no	—
91	C₂³	C₂³×D₅₃	1	1	yes	characteristic
92	D₄	—	212	4	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	3	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	C₂×D₂₁₂	1	6	yes	—
101	C₂×C₄	#11719945	1	6	yes	—
102	C₂×C₄	#11719945	1	6	yes	—
103	C₂²	—	212	2	no	—
104	C₂²	—	212	6	no	—
105	C₂²	—	212	6	no	—
106	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
107	C₂²	C₂×#11719945	1	3	yes	—
108	C₂²	C₂×#11719945	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	2	6	no	—
111	C₄	—	212	6	no	—
112	C₄	#13133211	1	4	yes	complemented
113	C₂	—	424	2	no	—
114	C₂	C₂×#13133211	1	1	yes	characteristic
115	C₂	#16811157	1	3	yes	—
116	C₂	#16811173	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133211

C₂

yes

maximal, complemented

#16810525

C₂

yes

maximal

#16810521

C₂

yes

maximal, complemented

#16810514

C₂

yes

maximal, complemented

C₄³×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

#13133211

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

C₂×#11719921

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

C₂×D₂₁₂

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

complemented

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

#11719919

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#126

yes

complemented

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#532

yes

—

D₅₃

—

—

—

—

—

C₄³

D₅₃

yes

characteristic, complemented

C₅₃

#2860

yes

characteristic, complemented, Sylow

C₂²×D₄

—

#126

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

C₂⁴

—

C₂×D₄

—

106

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

D₂₁₂

yes

complemented

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

C₂³×D₅₃

yes

characteristic

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

C₂×D₂₁₂

yes

—

101

C₂×C₄

#11719945

yes

—

102

C₂×C₄

#11719945

yes

—

103

C₂²

—

212

—

104

C₂²

—

212

—

105

C₂²

—

212

—

106

C₂²

C₂²×D₂₁₂

yes

characteristic

107

C₂²

C₂×#11719945

yes

—

108

C₂²

C₂×#11719945

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

212

—

112

C₄

#13133211

yes

complemented

113

C₂

—

424

—

114

C₂

C₂×#13133211

yes

characteristic

115

C₂

#16811157

yes

—

116

C₂

#16811173

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₄³×C₅₃	C₂
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#2860
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₄³×C₅₃

C₂

hypercenter

C₄³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
1. C₂×#13133211	C₂	semidirect
3. #16810521	C₂	semidirect
4. #16810514	C₂	semidirect
5. C₄³×C₅₃	C₂	semidirect
24. C₄²×C₅₃	D₄	semidirect
52. C₄×C₅₃	#126	semidirect
63. C₄³	D₅₃	semidirect
64. C₅₃	#2860	semidirect
87. C₄²	D₂₁₂	semidirect
112. C₄	#13133211	semidirect

H ≅ G/N

type

1. C₂×#13133211

C₂

semidirect

3. #16810521

C₂

semidirect

4. #16810514

C₂

semidirect

5. C₄³×C₅₃

C₂

semidirect

24. C₄²×C₅₃

D₄

semidirect

52. C₄×C₅₃

#126

semidirect

63. C₄³

D₅₃

semidirect

64. C₅₃

#2860

semidirect

87. C₄²

D₂₁₂

semidirect

112. C₄

#13133211

semidirect

Subgroups — #20283082

Conjugacy classes of subgroups

Complements