Group #20283800 — subgroups

Subgroups — #20283800

Order 6784.

356 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 217 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	#16810754	C₂	1	8	yes	maximal, complemented
2	#16810680	C₂	1	4	yes	maximal
3	#16810676	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#88322	C₂²	1	4	yes	—
6	C₄⋊#88322	C₂²	1	8	yes	—
7	C₂²⋊#88322	C₂²	1	4	yes	—
8	#13133128	C₂²	1	8	yes	—
9	C₄×#88322	C₂²	1	4	yes	—
10	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₅₃	C₄	1	4	yes	—
12	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
17	C₂×Q₈×C₅₃	—	2	4	no	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
25	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₄²×C₅₃	C₂×C₄	1	4	yes	—
27	C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₄²×C₅₃	C₂³	1	2	yes	—
29	C₄²×C₅₃	C₂³	1	4	yes	—
30	C₂×#88322	C₂³	1	8	yes	—
31	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	Q₈×C₅₃	—	2	8	no	—
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
44	#88322	—	2	8	no	—
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
48	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	4	no	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	—	2	4	no	—
53	C₄×C₅₃	C₂×M₁₆	1	4	yes	—
54	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
55	#2983	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	2	no	—
57	C₂×C₅₃	#519	1	1	yes	characteristic
58	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
59	C₂×C₅₃	#568	1	1	yes	characteristic
60	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
61	#446	—	53	8	no	—
62	#432	—	53	4	no	—
63	#431	—	53	2	no	—
64	C₅₃	#2983	1	1	yes	characteristic, complemented, Sylow
65	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
66	C₄×Q₈	Dic₅₃	1	4	yes	—
67	C₄×Q₈	C₂×D₅₃	1	4	yes	—
68	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	—
69	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
71	C₂×C₄²	C₂×D₅₃	1	2	yes	—
72	C₄⋊C₈	—	53	4	no	—
73	C₄⋊C₈	—	53	8	no	—
74	C₂²⋊C₈	—	53	4	no	—
75	#97	—	53	8	no	—
76	C₄×C₈	—	53	4	no	—
77	C₂×Q₈	—	2	4	no	—
78	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
79	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
80	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
82	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	C₂×C₈	—	53	8	no	—
85	C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
86	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
87	C₄²	C₂×Dic₅₃	1	4	yes	—
88	C₄²	C₂×Dic₅₃	1	2	yes	—
89	C₄²	C₂²×D₅₃	1	2	yes	—
90	C₄²	C₂²×D₅₃	1	4	yes	—
91	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	#11719917	1	4	yes	—
98	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
99	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
100	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
101	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
102	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
103	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
104	C₈	—	106	8	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×#11719917	1	2	yes	—
108	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	4	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	4	no	—
113	C₄	C₂×#11719917	1	4	yes	—
114	C₄	#13133265	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	#16811101	1	1	yes	characteristic
117	C₂	C₂²×#11719917	1	1	yes	characteristic
118	C₂	#16811390	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810754

C₂

yes

maximal, complemented

#16810680

C₂

yes

maximal

#16810676

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

C₄

yes

—

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

2₋¹⁺⁴

yes

—

#2983

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#519

yes

characteristic

C₂×C₅₃

C₂²×M₁₆

yes

characteristic

C₂×C₅₃

#568

yes

characteristic

C₂×C₄×Q₈

D₅₃

yes

—

—

—

C₅₃

#2983

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₄×Q₈

Dic₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₄⋊C₄

Dic₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11719917

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

—

100

C₂×C₄

C₂²×Dic₅₃

yes

—

101

C₂×C₄

C₂²×Dic₅₃

yes

—

102

C₂×C₄

C₂²×Dic₅₃

yes

—

103

C₂×C₄

C₂³×D₅₃

yes

characteristic

104

C₈

—

106

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×#11719917

yes

—

108

C₂²

C₂³×Dic₅₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₂×#11719917

yes

—

114

C₄

#13133265

yes

—

115

C₂

—

116

C₂

#16811101

yes

characteristic

117

C₂

C₂²×#11719917

yes

characteristic

118

C₂

#16811390

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₅₃	C₂
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#2983
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₅₃

C₂

hypercenter

C₂×C₄×Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
64. C₅₃	#2983	semidirect

H ≅ G/N

type

1. #16810754

C₂

semidirect

64. C₅₃

#2983

semidirect

Subgroups — #20283800

Conjugacy classes of subgroups

Complements