Group #20272583 — subgroups

Subgroups — #20272583

Order 6784.

228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16810483	C₂	1	2	yes	maximal
2	C₂×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₅₃	—	2	2	no	—
4	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
5	#13133124	C₄	1	4	yes	complemented
6	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₄	1	2	yes	complemented
8	C₂²⋊C₈×C₅₃	—	2	2	no	—
9	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	—	4	2	no	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₅₃	D₄	1	2	yes	—
14	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
15	C₂²×Dic₅₃	—	2	4	no	—
16	C₂×C₈×C₅₃	—	2	2	no	—
17	C₂×C₈×C₅₃	—	4	2	no	—
18	C₂²⋊Dic₅₃	—	4	2	no	—
19	C₂²⋊Dic₅₃	—	4	2	no	—
20	C₄⋊Dic₅₃	—	4	4	no	—
21	C₂³×C₅₃	—	4	1	no	—
22	C₂³×C₅₃	—	4	1	no	—
23	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
24	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂³×C₅₃	D₈	1	1	yes	characteristic
26	C₂³×C₅₃	SD₁₆	1	2	yes	—
27	C₂³×C₅₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	4	no	—
30	C₂×C₄×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₅₃	—	4	4	no	—
36	C₂×Dic₅₃	—	4	4	no	—
37	C₂×Dic₅₃	—	4	4	no	—
38	C₈×C₅₃	—	4	2	no	—
39	C₂²×C₅₃	—	2	2	no	—
40	C₂²×C₅₃	—	2	1	no	—
41	C₂²×C₅₃	—	2	2	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	4	1	no	—
44	C₂²×C₅₃	—	4	2	no	—
45	C₂²×C₅₃	—	4	1	no	—
46	C₂²×C₅₃	#95	1	1	yes	characteristic
47	C₂²×C₅₃	#102	1	2	yes	—
48	C₂²×C₅₃	#103	1	2	yes	—
49	C₂²×C₅₃	#106	1	1	yes	characteristic
50	C₂²×C₅₃	#107	1	1	yes	characteristic
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	#99	1	2	yes	—
54	Dic₅₃	—	8	4	no	—
55	#1289	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	2	no	—
57	C₂×C₅₃	—	2	2	no	—
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
61	C₂×C₅₃	#340	1	1	yes	characteristic
62	C₂×C₅₃	#342	1	1	yes	characteristic
63	C₂×C₂²⋊C₈	D₅₃	1	1	yes	characteristic
64	#380	—	53	2	no	—
65	C₅₃	#1289	1	1	yes	characteristic, complemented, Sylow
66	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
67	C₂²×C₈	Dic₅₃	1	2	yes	complemented
68	C₂×C₄⋊C₄	—	106	2	no	—
69	C₂×C₂²⋊C₄	—	106	2	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	#95	—	53	4	no	—
72	C₂⁴	D₂₁₂	1	1	yes	characteristic
73	C₂²×C₄	—	4	2	no	—
74	C₂²×C₄	—	106	4	no	—
75	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₅₃	1	2	yes	—
77	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₄⋊C₄	—	212	4	no	—
82	C₂²⋊C₄	—	212	2	no	—
83	C₂²⋊C₄	—	212	2	no	—
84	C₂³	—	4	1	no	—
85	C₂³	—	4	1	no	—
86	C₂³	#11719913	1	2	yes	—
87	C₂³	D₄₂₄	1	1	yes	characteristic
88	C₂³	Dic₂₁₂	1	1	yes	characteristic
89	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
90	C₂³	#11719921	1	2	yes	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	212	4	no	—
96	C₂×C₄	—	212	4	no	—
97	C₂×C₄	—	212	4	no	—
98	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
99	C₂×C₄	#11719919	1	2	yes	—
100	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	4	2	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	1	no	—
109	C₂²	#13133124	1	1	yes	characteristic
110	C₂²	#13133139	1	2	yes	—
111	C₂²	#13133142	1	2	yes	—
112	C₂²	#13133149	1	1	yes	characteristic
113	C₂²	#13133151	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	424	4	no	—
117	C₄	#13133132	1	2	yes	—
118	C₂	—	2	2	no	—
119	C₂	—	2	2	no	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#16810380	1	1	yes	characteristic
123	C₂	#16810386	1	1	yes	characteristic
124	C₂	#16810390	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810483

C₂

yes

maximal

C₂×C₂²⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

—

C₂×C₂²⋊Dic₅₃

—

#13133124

C₄

yes

complemented

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₄

yes

complemented

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂³×C₅₃

D₈

yes

characteristic

C₂³×C₅₃

SD₁₆

yes

—

C₂³×C₅₃

Q₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

#102

yes

—

C₂²×C₅₃

#103

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#99

yes

—

Dic₅₃

—

#1289

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₂²⋊C₈

D₅₃

yes

characteristic

#380

—

C₅₃

#1289

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

106

—

C₂×C₂²⋊C₄

—

106

—

C₂²⋊C₈

—

#95

—

C₂⁴

D₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₂³

—

C₂³

—

C₂³

#11719913

yes

—

C₂³

D₄₂₄

yes

characteristic

C₂³

Dic₂₁₂

yes

characteristic

C₂³

C₄⋊Dic₅₃

yes

characteristic

C₂³

#11719921

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₄

#11719919

yes

—

100

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

101

C₈

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#13133124

yes

characteristic

110

C₂²

#13133139

yes

—

111

C₂²

#13133142

yes

—

112

C₂²

#13133149

yes

characteristic

113

C₂²

#13133151

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

424

—

117

C₄

#13133132

yes

—

118

C₂

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#16810380

yes

characteristic

123

C₂

#16810386

yes

characteristic

124

C₂

#16810390

yes

characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂³×C₄	C₂×D₅₃
Fitting	C₂×C₂²⋊C₈×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₂×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	#1289
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂²

#13133124

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂³×C₄

C₂×D₅₃

Fitting

C₂×C₂²⋊C₈×C₅₃

C₂

socle

C₂²×C₅₃

#95

hypercenter

C₂×C₂²⋊C₈

D₅₃

hypocenter

C₅₃

#1289

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
5. #13133124	C₄	semidirect
7. C₂²×C₈×C₅₃	C₄	semidirect
65. C₅₃	#1289	semidirect
67. C₂²×C₈	Dic₅₃	semidirect

H ≅ G/N

type

5. #13133124

C₄

semidirect

7. C₂²×C₈×C₅₃

C₄

semidirect

65. C₅₃

#1289

semidirect

67. C₂²×C₈

Dic₅₃

semidirect

Subgroups — #20272583

Conjugacy classes of subgroups

Complements