Group #20276515 — subgroups

Subgroups — #20276515

Order 6784.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810441	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
4	#16810398	C₂	1	2	yes	maximal, complemented
5	C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂×#11719908	C₂²	1	2	yes	—
9	C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
17	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄×Dic₅₃	C₈	1	1	yes	characteristic
22	C₄×Dic₅₃	C₈	1	1	yes	characteristic
23	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂×#88322	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₅₃	—	2	2	no	—
29	#11719908	—	2	2	no	—
30	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	—	2	2	no	—
36	C₂×Dic₅₃	—	2	2	no	—
37	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
41	C₈×C₅₃	—	2	1	no	—
42	C₈×C₅₃	#41	1	2	yes	—
43	#88322	—	2	1	no	—
44	#88322	M₁₆	1	2	yes	—
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	M₃₂	1	2	yes	—
48	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	#117	1	1	yes	characteristic
51	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
52	Dic₅₃	—	2	1	no	—
53	Dic₅₃	—	2	1	no	—
54	Dic₅₃	—	2	1	no	—
55	Dic₅₃	M₃₂	1	2	yes	—
56	#2163	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	2	2	no	—
58	C₂×C₅₃	#431	1	1	yes	characteristic
59	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
60	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
61	C₂×C₄×C₈	—	53	1	no	—
62	C₄⋊C₁₆	—	53	2	no	—
63	C₂²⋊C₁₆	—	53	1	no	—
64	C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
65	#346	—	53	2	no	—
66	C₅₃	#2163	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	—	53	1	no	—
68	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
69	C₂×C₄²	—	53	1	no	—
70	C₂×C₁₆	—	53	2	no	—
71	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
72	C₄×C₈	—	53	1	no	—
73	C₄×C₈	—	53	1	no	—
74	C₄×C₈	—	53	1	no	—
75	C₄×C₈	—	53	1	no	—
76	C₂²×C₄	—	53	1	no	—
77	C₂²×C₄	—	53	1	no	—
78	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	53	1	no	—
81	C₂×C₈	—	53	1	no	—
82	C₂×C₈	—	106	2	no	—
83	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
85	C₄²	—	53	1	no	—
86	C₄²	—	53	1	no	—
87	C₄²	—	53	1	no	—
88	C₄²	—	53	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	106	2	no	—
91	C₂³	C₈×D₅₃	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	53	1	no	—
95	C₂×C₄	—	53	1	no	—
96	C₂×C₄	—	53	1	no	—
97	C₂×C₄	—	53	1	no	—
98	C₂×C₄	—	106	2	no	—
99	C₂×C₄	—	106	2	no	—
100	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	53	2	no	—
104	C₈	—	106	1	no	—
105	C₈	#11719947	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#13133156	1	2	yes	—
109	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	53	2	no	—
112	C₄	—	106	1	no	—
113	C₄	—	106	1	no	—
114	C₄	—	106	1	no	—
115	C₄	#13133173	1	1	yes	characteristic
116	C₄	M₁₆×D₅₃	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#16810673	1	1	yes	characteristic
119	C₂	M₃₂×D₅₃	1	1	yes	characteristic
120	C₂	C₂×#13133156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810441

C₂

yes

maximal, complemented

C₂²⋊#11719908

C₂

yes

maximal, characteristic

#16810398

C₂

yes

maximal, complemented

C₅₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×#88322

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

characteristic

C₂×#11719908

C₂²

yes

—

C₈×Dic₅₃

C₄

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₈

yes

characteristic

C₂²×Dic₅₃

C₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₈

yes

characteristic

C₄×Dic₅₃

C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

#88322

—

#88322

M₁₆

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

M₃₂

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

M₃₂

yes

—

#2163

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#431

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₅₃

yes

characteristic

#346

—

C₅₃

#2163

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

106

—

C₂³

C₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

C₈×D₅₃

yes

characteristic

101

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

106

—

105

C₈

#11719947

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

#13133156

yes

—

109

C₂²

C₂×C₈×D₅₃

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

106

—

113

C₄

—

106

—

114

C₄

—

106

—

115

C₄

#13133173

yes

characteristic

116

C₄

M₁₆×D₅₃

yes

characteristic

117

C₂

—

118

C₂

#16810673

yes

characteristic

119

C₂

M₃₂×D₅₃

yes

characteristic

120

C₂

C₂×#13133156

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₅₃×C₂²⋊C₁₆	C₂
hypercenter	C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#2163
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₅₃×C₂²⋊C₁₆

C₂

hypercenter

C₂²⋊C₁₆

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. #16810441	C₂	semidirect
4. #16810398	C₂	semidirect
66. C₅₃	#2163	semidirect

H ≅ G/N

type

2. #16810441

C₂

semidirect

4. #16810398

C₂

semidirect

66. C₅₃

#2163

semidirect

Subgroups — #20276515

Conjugacy classes of subgroups

Complements