Group #27214668 — subgroups

Subgroups — C₅₃×#7156 #27214668

Order 13568.

234 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 142 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×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#334	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
4	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×#346	—	2	2	no	—
7	C₅₃×#334	—	2	2	no	—
8	C₅₃×#334	C₄	1	4	yes	—
9	C₂²×C₈×C₅₃	—	4	2	no	—
10	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	—	2	2	no	—
14	C₂×C₁₆×C₅₃	—	2	2	no	—
15	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
16	C₄⋊C₈×C₅₃	—	2	4	no	—
17	C₄⋊C₈×C₅₃	—	4	2	no	—
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	—	4	4	no	—
24	C₂×C₈×C₅₃	—	8	4	no	—
25	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
30	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
31	C₄²×C₅₃	D₈	1	1	yes	characteristic
32	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
33	C₁₆×C₅₃	—	2	4	no	—
34	C₂³×C₅₃	#106	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#102	1	2	yes	—
38	C₂×C₄×C₅₃	#103	1	2	yes	—
39	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
40	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#108	1	2	yes	—
42	C₈×C₅₃	—	2	2	no	—
43	C₈×C₅₃	—	8	4	no	—
44	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
45	C₈×C₅₃	#97	1	1	yes	characteristic
46	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
47	#7156	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₅₃	#334	1	2	yes	—
49	C₂²×C₅₃	#340	1	1	yes	characteristic
50	C₂²×C₅₃	#365	1	4	yes	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	#335	1	2	yes	—
54	C₄×C₅₃	#336	1	1	yes	characteristic
55	C₄×C₅₃	#341	1	1	yes	characteristic
56	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
57	C₂×#334	C₂×C₅₃	1	2	yes	—
58	C₂×C₅₃	#1277	1	1	yes	characteristic
59	C₂×C₅₃	#1368	1	4	yes	—
60	C₂×C₅₃	#1383	1	2	yes	—
61	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
62	C₂×C₄⋊C₈	—	2	2	no	—
63	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
64	#346	—	2	2	no	—
65	#334	—	2	2	no	—
66	#334	C₄×C₅₃	1	4	yes	—
67	C₅₃	#7156	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₈	—	4	2	no	—
69	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
70	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
71	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	C₈×C₅₃	1	4	yes	complemented
75	C₄⋊C₈	—	2	4	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
78	C₄×C₈	D₄×C₅₃	1	2	yes	—
79	C₂²×C₄	C₄⋊C₄×C₅₃	1	1	yes	characteristic
80	C₂²×C₄	SD₁₆×C₅₃	1	2	yes	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	4	4	no	—
83	C₂×C₈	—	8	4	no	—
84	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
85	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
86	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
87	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
88	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
89	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
90	C₄²	D₈×C₅₃	1	1	yes	characteristic
91	C₄²	Q₁₆×C₅₃	1	1	yes	characteristic
92	C₁₆	—	2	4	no	—
93	C₂³	#106×C₅₃	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
96	C₂×C₄	#102×C₅₃	1	2	yes	—
97	C₂×C₄	#103×C₅₃	1	2	yes	—
98	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
99	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
100	C₂×C₄	#108×C₅₃	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	8	4	no	—
103	C₈	C₄×C₈×C₅₃	1	1	yes	characteristic
104	C₈	#97×C₅₃	1	1	yes	characteristic
105	C₈	C₂²⋊C₈×C₅₃	1	2	yes	—
106	C₂²	C₅₃×#334	1	2	yes	—
107	C₂²	C₅₃×#340	1	1	yes	characteristic
108	C₂²	C₅₃×#365	1	4	yes	—
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	C₅₃×#335	1	2	yes	—
112	C₄	C₅₃×#336	1	1	yes	characteristic
113	C₄	C₅₃×#341	1	1	yes	characteristic
114	C₂	C₅₃×#1277	1	1	yes	characteristic
115	C₂	C₅₃×#1368	1	4	yes	—
116	C₂	C₅₃×#1383	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂×C₅₃×#334

C₂

yes

maximal

C₂²×C₁₆×C₅₃

C₄

yes

—

C₂×C₄⋊C₈×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

—

C₅₃×#334

—

C₅₃×#334

C₄

yes

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₈

yes

complemented

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

Q₁₆

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

#107

yes

characteristic

C₂×C₄×C₅₃

#108

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

characteristic

C₈×C₅₃

#97

yes

characteristic

C₈×C₅₃

C₂²⋊C₈

yes

—

#7156

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

#334

yes

—

C₂²×C₅₃

#340

yes

characteristic

C₂²×C₅₃

#365

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#335

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

characteristic

C₂×#334

C₂×C₅₃

yes

—

C₂×C₅₃

#1277

yes

characteristic

C₂×C₅₃

#1368

yes

—

C₂×C₅₃

#1383

yes

—

C₂²×C₁₆

C₄×C₅₃

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂²×C₅₃

yes

—

—

C₄×C₅₃

yes

—

C₅₃

#7156

yes

characteristic, direct product factor, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

Q₈×C₅₃

yes

characteristic

C₂×C₄²

D₄×C₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₈×C₅₃

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

—

C₂²×C₄

C₄⋊C₄×C₅₃

yes

characteristic

C₂²×C₄

SD₁₆×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₅₃

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

C₂×C₈×C₅₃

yes

—

C₂×C₈

M₁₆×C₅₃

yes

—

C₄²

C₂²⋊C₄×C₅₃

yes

—

C₄²

D₈×C₅₃

yes

characteristic

C₄²

Q₁₆×C₅₃

yes

characteristic

C₁₆

—

C₂³

#106×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

#95×C₅₃

yes

characteristic

C₂×C₄

#102×C₅₃

yes

—

C₂×C₄

#103×C₅₃

yes

—

C₂×C₄

C₄⋊C₈×C₅₃

yes

—

C₂×C₄

#107×C₅₃

yes

characteristic

100

C₂×C₄

#108×C₅₃

yes

—

101

C₈

—

102

C₈

—

103

C₈

C₄×C₈×C₅₃

yes

characteristic

104

C₈

#97×C₅₃

yes

characteristic

105

C₈

C₂²⋊C₈×C₅₃

yes

—

106

C₂²

C₅₃×#334

yes

—

107

C₂²

C₅₃×#340

yes

characteristic

108

C₂²

C₅₃×#365

yes

—

109

C₄

—

110

C₄

—

111

C₄

C₅₃×#335

yes

—

112

C₄

C₅₃×#336

yes

characteristic

113

C₄

C₅₃×#341

yes

characteristic

114

C₂

C₅₃×#1277

yes

characteristic

115

C₂

C₅₃×#1368

yes

—

116

C₂

C₅₃×#1383

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#106
commutator	C₈	C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅₃

#106

commutator

C₈

C₄×C₈×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
15. C₂×C₁₆×C₅₃	C₈	semidirect
47. #7156	C₅₃	direct (N × H)
67. C₅₃	#7156	direct (N × H)
74. C₂×C₁₆	C₈×C₅₃	semidirect

H ≅ G/N

type

15. C₂×C₁₆×C₅₃

C₈

semidirect

47. #7156

C₅₃

direct (N × H)

67. C₅₃

#7156

direct (N × H)

74. C₂×C₁₆

C₈×C₅₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅₃×#346	2
C₂×C₅₃×#334	2
#7156	53

subgroup

index

C₂×C₅₃×#346

C₂×C₅₃×#334

#7156

Subgroups — C₅₃×#7156 #27214668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7156 #27214668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7156 #27214668