Group #20279957 — subgroups

Subgroups — #20279957

Order 6784.

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

normal subgroups only

This group has 380 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	#16810562	C₂	1	4	yes	maximal
2	#16810522	C₂	1	4	yes	maximal
3	#16810488	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
5	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
10	#13133124	C₂²	1	8	yes	—
11	#13133124	C₂²	1	4	yes	—
12	#13133124	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
19	C₂²×Dic₅₃	C₂³	1	4	yes	—
20	C₂²×Dic₅₃	C₂³	1	4	yes	—
21	C₄⋊C₄×C₅₃	—	2	8	no	—
22	C₄⋊C₄×C₅₃	—	4	2	no	—
23	C₄²×C₅₃	Q₈	1	4	yes	—
24	C₄⋊Dic₅₃	—	2	4	no	—
25	C₄⋊Dic₅₃	—	2	4	no	—
26	C₄⋊Dic₅₃	—	4	4	no	—
27	C₄×Dic₅₃	—	2	8	no	—
28	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	2	8	no	—
32	C₂×C₄×C₅₃	—	2	4	no	—
33	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
35	C₂×C₄×C₅₃	#41	1	4	yes	—
36	C₂×C₄×C₅₃	#41	1	4	yes	—
37	C₂×Dic₅₃	—	2	4	no	—
38	C₂×Dic₅₃	—	2	4	no	—
39	C₂×Dic₅₃	—	2	4	no	—
40	C₂×Dic₅₃	—	2	4	no	—
41	C₂×Dic₅₃	—	2	4	no	—
42	C₂×Dic₅₃	—	2	4	no	—
43	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
44	C₂²×C₅₃	C₂×#41	1	2	yes	—
45	C₂²×C₅₃	C₂×#41	1	2	yes	—
46	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	—	4	4	no	—
50	C₄×C₅₃	#125	1	8	yes	—
51	Dic₅₃	—	4	4	no	—
52	Dic₅₃	—	4	4	no	—
53	#2571	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	C₂×#125	1	2	yes	—
55	C₂×C₅₃	#544	1	1	yes	characteristic
56	C₂×C₅₃	#555	1	1	yes	characteristic
57	C₂×C₅₃	#556	1	2	yes	—
58	C₂×C₅₃	#559	1	1	yes	characteristic
59	#400	—	53	4	no	—
60	#389	—	53	4	no	—
61	#384	D₅₃	1	1	yes	characteristic
62	#382	—	53	4	no	—
63	C₄×C₄⋊C₄	—	53	2	no	—
64	C₅₃	#2571	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄⋊C₄	—	53	2	no	—
66	C₂×C₄⋊C₄	—	53	2	no	—
67	C₂×C₄⋊C₄	—	53	4	no	—
68	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
69	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
70	C₂×C₄²	—	53	4	no	—
71	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
72	#95	—	53	4	no	—
73	#95	—	53	4	no	—
74	#95	—	53	8	no	—
75	C₂²×C₄	—	53	4	no	—
76	C₂²×C₄	—	53	4	no	—
77	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
78	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
80	C₄⋊C₄	—	2	8	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	106	4	no	—
83	C₄⋊C₄	—	106	4	no	—
84	C₄⋊C₄	—	212	4	no	—
85	C₄²	—	106	8	no	—
86	C₄²	Dic₁₀₆	1	4	yes	—
87	C₂³	C₂³×D₅₃	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	106	4	no	—
93	C₂×C₄	—	106	4	no	—
94	C₂×C₄	—	106	4	no	—
95	C₂×C₄	—	106	4	no	—
96	C₂×C₄	—	106	4	no	—
97	C₂×C₄	—	106	4	no	—
98	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
99	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
100	C₂×C₄	#11719947	1	4	yes	—
101	C₂×C₄	#11719949	1	4	yes	—
102	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
103	C₂²	C₂×#11719947	1	2	yes	—
104	C₂²	C₂×#11719949	1	2	yes	—
105	C₂²	#13133262	1	1	yes	characteristic
106	C₂²	#13133265	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	212	4	no	—
110	C₄	—	212	4	no	—
111	C₄	#13133208	1	8	yes	—
112	C₂	C₂×#13133208	1	2	yes	—
113	C₂	#16811254	1	1	yes	characteristic
114	C₂	#16811339	1	1	yes	characteristic
115	C₂	#16811347	1	2	yes	—
116	C₂	#16811359	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810562

C₂

yes

maximal

#16810522

C₂

yes

maximal

#16810488

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal

C₅₃×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₅₃

2₋¹⁺⁴

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#125

yes

—

Dic₅₃

—

Dic₅₃

—

#2571

—

maximal, Sylow

C₂×C₅₃

C₂×#125

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#382

—

C₄×C₄⋊C₄

—

C₅₃

#2571

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

106

—

C₄²

Dic₁₀₆

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

100

C₂×C₄

#11719947

yes

—

101

C₂×C₄

#11719949

yes

—

102

C₂²

C₂²×Dic₁₀₆

yes

characteristic

103

C₂²

C₂×#11719947

yes

—

104

C₂²

C₂×#11719949

yes

—

105

C₂²

#13133262

yes

characteristic

106

C₂²

#13133265

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

212

—

110

C₄

—

212

—

111

C₄

#13133208

yes

—

112

C₂

C₂×#13133208

yes

—

113

C₂

#16811254

yes

characteristic

114

C₂

#16811339

yes

characteristic

115

C₂

#16811347

yes

—

116

C₂

#16811359

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#384	C₂
hypercenter	#384	D₅₃
hypocenter	C₅₃	#2571
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₅₃×#384

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
64. C₅₃	#2571	semidirect

H ≅ G/N

type

64. C₅₃

#2571

semidirect

Subgroups — #20279957

Conjugacy classes of subgroups

Complements