Group #20272792 — subgroups

Subgroups — #20272792

Order 6784.

160 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
6	C₂×#11719908	—	2	1	no	—
7	C₂×#11719908	—	2	1	no	—
8	C₂×#11719908	C₄	1	2	yes	complemented
9	C₈×Dic₅₃	—	2	1	no	—
10	C₈×Dic₅₃	—	2	1	no	—
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	—	2	1	no	—
13	C₂×C₁₆×C₅₃	—	2	1	no	—
14	C₂×C₁₆×C₅₃	C₄	1	2	yes	complemented
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₅₃	C₈	1	2	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₄×Dic₅₃	—	2	1	no	—
24	C₄×Dic₅₃	—	2	1	no	—
25	C₂×#88322	—	2	1	no	—
26	C₂×#88322	—	2	1	no	—
27	C₂×#88322	C₈	1	2	yes	—
28	C₁₆×C₅₃	—	2	2	no	—
29	#11719908	—	2	2	no	—
30	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
36	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×Dic₅₃	—	2	2	no	—
38	C₂×Dic₅₃	—	2	2	no	—
39	C₂×Dic₅₃	C₁₆	1	4	yes	complemented
40	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	#88322	—	2	2	no	—
45	C₂²×C₅₃	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
47	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
48	C₂²×C₅₃	C₂×C₁₆	1	2	yes	—
49	C₂²×C₅₃	M₃₂	1	2	yes	—
50	C₄×C₅₃	#95	1	1	yes	characteristic
51	C₄×C₅₃	#97	1	1	yes	characteristic
52	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
54	Dic₅₃	—	2	4	no	—
55	#1374	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	#336	1	1	yes	characteristic
57	C₂×C₅₃	C₄×C₁₆	1	1	yes	characteristic
58	C₂×C₅₃	#346	1	1	yes	characteristic
59	C₂×C₅₃	C₂²⋊C₁₆	1	2	yes	—
60	C₂×C₅₃	C₄⋊C₁₆	1	2	yes	—
61	C₂²×C₁₆	—	53	1	no	—
62	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
63	C₂×C₄×C₈	—	53	1	no	—
64	C₅₃	#1374	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	—	53	1	no	—
66	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
67	C₂×C₄²	—	53	1	no	—
68	C₂×C₁₆	—	2	1	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	53	2	no	—
71	C₂×C₁₆	—	106	1	no	—
72	C₂×C₁₆	—	106	1	no	—
73	C₂×C₁₆	Dic₅₃	1	2	yes	complemented
74	C₄×C₈	—	106	1	no	—
75	C₄×C₈	—	106	1	no	—
76	C₂²×C₄	—	53	2	no	—
77	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
78	C₂×C₈	—	53	2	no	—
79	C₂×C₈	—	106	1	no	—
80	C₂×C₈	—	106	1	no	—
81	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
82	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
83	C₂×C₈	D₂₁₂	1	1	yes	characteristic
84	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
87	C₄²	—	106	1	no	—
88	C₄²	—	106	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	106	2	no	—
91	C₂³	C₈×D₅₃	1	1	yes	characteristic
92	C₂×C₄	—	53	4	no	—
93	C₂×C₄	—	106	2	no	—
94	C₂×C₄	—	106	2	no	—
95	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
96	C₂×C₄	#11719912	1	1	yes	characteristic
97	C₂×C₄	#11719912	1	1	yes	characteristic
98	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
99	C₂×C₄	#11719919	1	1	yes	characteristic
100	C₂×C₄	#11719921	1	1	yes	characteristic
101	C₈	—	106	2	no	—
102	C₈	#11719919	1	1	yes	characteristic
103	C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
104	C₈	#11719921	1	1	yes	characteristic
105	C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
106	C₂²	C₈×Dic₅₃	1	1	yes	characteristic
107	C₂²	#13133129	1	1	yes	characteristic
108	C₂²	#13133146	1	1	yes	characteristic
109	C₂²	C₁₆×D₅₃	1	2	yes	—
110	C₂²	#13133156	1	2	yes	—
111	C₄	—	106	4	no	—
112	C₄	#13133124	1	1	yes	characteristic
113	C₄	#13133127	1	1	yes	characteristic
114	C₄	#13133129	1	1	yes	characteristic
115	C₄	#13133146	1	1	yes	characteristic
116	C₂	#16810377	1	1	yes	characteristic
117	C₂	C₁₆×Dic₅₃	1	1	yes	characteristic
118	C₂	#16810398	1	1	yes	characteristic
119	C₂	#16810402	1	2	yes	—
120	C₂	#16810441	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719908

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₄

yes

—

—

C₄

yes

complemented

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₈

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₈

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₁₆

yes

complemented

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

#88322

—

C₂²×C₅₃

C₄×C₈

yes

characteristic

C₂²×C₅₃

C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₂×C₁₆

yes

—

C₂²×C₅₃

M₃₂

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×C₁₆

yes

characteristic

C₂×C₅₃

#346

yes

characteristic

C₂×C₅₃

C₂²⋊C₁₆

yes

—

C₂×C₅₃

C₄⋊C₁₆

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

D₅₃

yes

characteristic

C₂×C₄×C₈

—

C₅₃

#1374

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

Dic₅₃

yes

complemented

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄²

—

106

—

C₄²

—

106

—

C₁₆

—

C₁₆

—

106

—

C₂³

C₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₈×D₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

#11719919

yes

characteristic

100

C₂×C₄

#11719921

yes

characteristic

101

C₈

—

106

—

102

C₈

#11719919

yes

characteristic

103

C₈

C₄⋊Dic₅₃

yes

characteristic

104

C₈

#11719921

yes

characteristic

105

C₈

C₂²⋊Dic₅₃

yes

characteristic

106

C₂²

C₈×Dic₅₃

yes

characteristic

107

C₂²

#13133129

yes

characteristic

108

C₂²

#13133146

yes

characteristic

109

C₂²

C₁₆×D₅₃

yes

—

110

C₂²

#13133156

yes

—

111

C₄

—

106

—

112

C₄

#13133124

yes

characteristic

113

C₄

#13133127

yes

characteristic

114

C₄

#13133129

yes

characteristic

115

C₄

#13133146

yes

characteristic

116

C₂

#16810377

yes

characteristic

117

C₂

C₁₆×Dic₅₃

yes

characteristic

118

C₂

#16810398

yes

characteristic

119

C₂

#16810402

yes

—

120

C₂

#16810441

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₄×C₁₆
Fitting	C₂²×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₈
hypercenter	C₂²×C₁₆	D₅₃
hypocenter	C₅₃	#1374
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂×D₅₃

commutator

C₂×C₅₃

C₄×C₁₆

Fitting

C₂²×C₁₆×C₅₃

C₂

socle

C₂³×C₅₃

C₂×C₈

hypercenter

C₂²×C₁₆

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
8. C₂×#11719908	C₄	semidirect
14. C₂×C₁₆×C₅₃	C₄	semidirect
39. C₂×Dic₅₃	C₁₆	semidirect
64. C₅₃	#1374	semidirect
73. C₂×C₁₆	Dic₅₃	semidirect

H ≅ G/N

type

8. C₂×#11719908

C₄

semidirect

14. C₂×C₁₆×C₅₃

C₄

semidirect

39. C₂×Dic₅₃

C₁₆

semidirect

64. C₅₃

#1374

semidirect

73. C₂×C₁₆

Dic₅₃

semidirect

Subgroups — #20272792

Conjugacy classes of subgroups

Complements