Group #20272638 — subgroups

Subgroups — #20272638

Order 6784.

140 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 43 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×#13133129	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₅₃	C₄	1	1	yes	characteristic
5	C₂²×#88322	C₄	1	1	yes	characteristic
6	C₂×#11719908	—	2	1	no	—
7	#13133129	—	2	1	no	—
8	#13133129	—	2	1	no	—
9	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	—	2	1	no	—
11	C₂³×D₅₃	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
14	C₂×C₄×D₅₃	—	2	1	no	—
15	C₂×C₄×D₅₃	—	2	1	no	—
16	C₂×C₄×D₅₃	—	4	1	no	—
17	C₂×C₄×D₅₃	—	4	1	no	—
18	C₂×C₈×C₅₃	—	4	1	no	—
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×#88322	—	2	1	no	—
22	C₂×#88322	—	4	1	no	—
23	C₁₆×C₅₃	—	4	1	no	—
24	#11719908	—	4	1	no	—
25	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×D₅₃	—	2	1	no	—
27	C₂²×D₅₃	—	2	1	no	—
28	C₂²×D₅₃	—	4	1	no	—
29	C₂²×D₅₃	—	4	1	no	—
30	C₂²×D₅₃	C₁₆	1	2	yes	complemented
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
35	C₂×Dic₅₃	—	2	1	no	—
36	C₂×Dic₅₃	—	4	1	no	—
37	C₄×D₅₃	—	4	2	no	—
38	C₄×D₅₃	—	4	1	no	—
39	C₄×D₅₃	—	4	1	no	—
40	C₈×C₅₃	—	2	1	no	—
41	C₈×C₅₃	—	2	1	no	—
42	#88322	—	4	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	—	2	1	no	—
45	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂²×C₅₃	M₃₂	1	1	yes	characteristic
48	C₂×D₅₃	—	2	2	no	—
49	C₂×D₅₃	—	4	2	no	—
50	C₂×D₅₃	—	4	1	no	—
51	C₂×D₅₃	—	4	1	no	—
52	C₂×D₅₃	—	4	2	no	—
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	#99	1	1	yes	characteristic
55	C₄×C₅₃	#100	1	1	yes	characteristic
56	Dic₅₃	—	4	1	no	—
57	#1314	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	#323	1	1	yes	characteristic
61	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
62	C₂×C₅₃	#349	1	1	yes	characteristic
63	D₅₃	—	4	2	no	—
64	C₂×C₂²⋊C₈	—	53	1	no	—
65	C₂²⋊C₁₆	—	53	1	no	—
66	C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
67	C₅₃	#1314	1	1	yes	characteristic, complemented, Sylow
68	C₂³×C₄	—	53	1	no	—
69	C₂²×C₈	—	53	1	no	—
70	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	106	1	no	—
73	C₂²⋊C₈	—	106	1	no	—
74	C₂²⋊C₈	—	106	1	no	—
75	C₂⁴	—	53	1	no	—
76	C₂²×C₄	—	53	1	no	—
77	C₂²×C₄	—	106	1	no	—
78	C₂²×C₄	—	106	1	no	—
79	C₂²×C₄	—	212	1	no	—
80	C₂²×C₄	—	212	1	no	—
81	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	106	1	no	—
84	C₂×C₈	—	212	1	no	—
85	C₂×C₈	D₂₁₂	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
87	C₁₆	—	4	1	no	—
88	C₁₆	—	212	1	no	—
89	C₂³	—	53	2	no	—
90	C₂³	—	106	1	no	—
91	C₂³	—	106	1	no	—
92	C₂³	—	212	1	no	—
93	C₂³	—	212	1	no	—
94	C₂³	C₈×D₅₃	1	1	yes	characteristic
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	106	1	no	—
98	C₂×C₄	—	212	2	no	—
99	C₂×C₄	—	212	1	no	—
100	C₂×C₄	—	212	1	no	—
101	C₂×C₄	—	212	1	no	—
102	C₂×C₄	#11719912	1	1	yes	characteristic
103	C₂×C₄	#11719921	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	—	2	1	no	—
106	C₈	—	212	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	106	2	no	—
110	C₂²	—	212	2	no	—
111	C₂²	—	212	2	no	—
112	C₂²	—	212	1	no	—
113	C₂²	—	212	1	no	—
114	C₂²	#13133129	1	1	yes	characteristic
115	C₂²	C₁₆×D₅₃	1	1	yes	characteristic
116	C₂²	#13133156	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	212	1	no	—
119	C₄	#13133131	1	1	yes	characteristic
120	C₄	#13133133	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	—	212	2	no	—
124	C₂	#16810346	1	1	yes	characteristic
125	C₂	#16810402	1	1	yes	characteristic
126	C₂	#16810404	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133129

C₂

yes

maximal, characteristic

C₂²⋊#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₄

yes

characteristic

C₂²×#88322

C₄

yes

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂³×D₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

C₂×C₁₆

yes

characteristic

C₂²×C₅₃

M₃₂

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1314

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#323

yes

characteristic

C₂×C₅₃

C₂²⋊C₁₆

yes

characteristic

C₂×C₅₃

#349

yes

characteristic

D₅₃

—

C₂×C₂²⋊C₈

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₅₃

yes

characteristic, complemented

C₅₃

#1314

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

212

—

C₂³

—

C₂³

—

106

—

C₂³

—

106

—

C₂³

—

212

—

C₂³

—

212

—

C₂³

C₈×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

100

C₂×C₄

—

212

—

101

C₂×C₄

—

212

—

102

C₂×C₄

#11719912

yes

characteristic

103

C₂×C₄

#11719921

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

212

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

106

—

110

C₂²

—

212

—

111

C₂²

—

212

—

112

C₂²

—

212

—

113

C₂²

—

212

—

114

C₂²

#13133129

yes

characteristic

115

C₂²

C₁₆×D₅₃

yes

characteristic

116

C₂²

#13133156

yes

characteristic

117

C₄

—

118

C₄

—

212

—

119

C₄

#13133131

yes

characteristic

120

C₄

#13133133

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

212

—

124

C₂

#16810346

yes

characteristic

125

C₂

#16810402

yes

characteristic

126

C₂

#16810404

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11719921
commutator	C₂²×C₅₃	C₂×C₁₆
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₅₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#1314
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂×C₄

#11719921

commutator

C₂²×C₅₃

C₂×C₁₆

Frattini

C₂²×C₈

C₂×D₅₃

Fitting

C₅₃×C₂²⋊C₁₆

C₂

socle

C₂²×C₅₃

C₂²⋊C₈

hypercenter

C₂²⋊C₁₆

D₅₃

hypocenter

C₅₃

#1314

LCS

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂²⋊#11719908	C₂	semidirect
3. C₅₃×C₂²⋊C₁₆	C₂	semidirect
30. C₂²×D₅₃	C₁₆	semidirect
66. C₂²⋊C₁₆	D₅₃	semidirect
67. C₅₃	#1314	semidirect

H ≅ G/N

type

2. C₂²⋊#11719908

C₂

semidirect

3. C₅₃×C₂²⋊C₁₆

C₂

semidirect

30. C₂²×D₅₃

C₁₆

semidirect

66. C₂²⋊C₁₆

D₅₃

semidirect

67. C₅₃

#1314

semidirect

Subgroups — #20272638

Conjugacy classes of subgroups

Complements