Group #27214754 — subgroups

Subgroups — C₅₃×#7242 #27214754

Order 13568.

190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1568	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
4	C₅₃×#432	—	2	2	no	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₅₃×C₄⋊C₁₆	—	2	2	no	—
7	C₅₃×#335	—	2	2	no	—
8	C₅₃×#334	C₄	1	4	yes	—
9	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
11	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	—	2	2	no	—
13	C₂×C₁₆×C₅₃	—	4	2	no	—
14	C₄⋊C₈×C₅₃	—	2	4	no	—
15	C₂²⋊C₈×C₅₃	—	4	2	no	—
16	#97×C₅₃	—	4	2	no	—
17	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	—	4	1	no	—
25	C₂×C₈×C₅₃	—	4	4	no	—
26	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₅₃	SD₁₆	1	2	yes	—
31	C₁₆×C₅₃	—	4	2	no	—
32	C₂³×C₅₃	#107	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#102	1	2	yes	—
39	C₂×C₄×C₅₃	#103	1	2	yes	—
40	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
41	C₈×C₅₃	—	2	2	no	—
42	C₈×C₅₃	—	2	1	no	—
43	C₈×C₅₃	—	8	4	no	—
44	C₈×C₅₃	#100	1	1	yes	characteristic
45	C₈×C₅₃	#101	1	1	yes	characteristic
46	#7242	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	#340	1	1	yes	characteristic
50	C₂²×C₅₃	#368	1	2	yes	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	#338	1	1	yes	characteristic
55	C₄×C₅₃	#343	1	1	yes	characteristic
56	C₂×C₄⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic
57	#1568	C₂×C₅₃	1	2	yes	—
58	C₂×C₅₃	—	2	2	no	—
59	C₂×C₅₃	#1292	1	1	yes	characteristic
60	C₂×C₅₃	#1381	1	1	yes	characteristic
61	C₂×C₅₃	#1384	1	1	yes	characteristic
62	C₂²×C₁₆	C₄×C₅₃	1	2	yes	—
63	#432	—	2	2	no	—
64	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	2	2	no	—
67	#334	C₄×C₅₃	1	4	yes	—
68	C₅₃	#7242	1	1	yes	characteristic, direct product factor, Sylow
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	—	4	2	no	—
74	C₄⋊C₈	—	2	4	no	—
75	C₂²⋊C₈	—	4	2	no	—
76	#97	—	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	D₈×C₅₃	1	1	yes	characteristic
81	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
88	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
89	C₄²	C₂²⋊C₄×C₅₃	1	2	yes	—
90	C₄²	SD₁₆×C₅₃	1	2	yes	—
91	C₁₆	—	4	2	no	—
92	C₂³	#107×C₅₃	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
98	C₂×C₄	#102×C₅₃	1	2	yes	—
99	C₂×C₄	#103×C₅₃	1	2	yes	—
100	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	2	1	no	—
103	C₈	—	8	4	no	—
104	C₈	#100×C₅₃	1	1	yes	characteristic
105	C₈	#101×C₅₃	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₅₃×#340	1	1	yes	characteristic
109	C₂²	C₅₃×#368	1	2	yes	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	2	no	—
113	C₄	C₅₃×#338	1	1	yes	characteristic
114	C₄	C₅₃×#343	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	C₅₃×#1292	1	1	yes	characteristic
117	C₂	C₅₃×#1381	1	1	yes	characteristic
118	C₂	C₅₃×#1384	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₅₃×#1568

C₂

yes

maximal

C₂²×C₁₆×C₅₃

C₄

yes

—

C₅₃×#432

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₅₃×#335

—

C₅₃×#334

C₄

yes

—

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

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×C₈×C₅₃

—

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

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₁₆×C₅₃

—

C₂³×C₅₃

#107

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₂²×C₅₃

#368

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₄⋊C₁₆

C₂×C₅₃

yes

characteristic

#1568

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

C₄×C₅₃

yes

—

#432

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

C₄⋊C₁₆

—

#335

—

#334

C₄×C₅₃

yes

—

C₅₃

#7242

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

Q₈×C₅₃

yes

characteristic

C₂×C₄²

D₄×C₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

—

C₂²×C₄

C₄⋊C₄×C₅₃

yes

characteristic

C₂²×C₄

D₈×C₅₃

yes

characteristic

C₂²×C₄

Q₁₆×C₅₃

yes

characteristic

C₂×C₈

—

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

—

C₄²

C₂²⋊C₄×C₅₃

yes

—

C₄²

SD₁₆×C₅₃

yes

—

C₁₆

—

C₂³

#107×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₅₃

yes

characteristic

C₂×C₄

#102×C₅₃

yes

—

C₂×C₄

#103×C₅₃

yes

—

100

C₂×C₄

#106×C₅₃

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

#100×C₅₃

yes

characteristic

105

C₈

#101×C₅₃

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

C₅₃×#340

yes

characteristic

109

C₂²

C₅₃×#368

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₅₃×#338

yes

characteristic

114

C₄

C₅₃×#343

yes

characteristic

115

C₂

—

116

C₂

C₅₃×#1292

yes

characteristic

117

C₂

C₅₃×#1381

yes

characteristic

118

C₂

C₅₃×#1384

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#340
commutator	C₂×C₈	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
prime radical	C₂³×C₅₃	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#340

commutator

C₂×C₈

C₄²×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

prime radical

C₂³×C₅₃

#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
46. #7242	C₅₃	direct (N × H)
68. C₅₃	#7242	direct (N × H)

H ≅ G/N

type

46. #7242

C₅₃

direct (N × H)

68. C₅₃

#7242

direct (N × H)

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

subgroup	index
C₂×C₅₃×C₄⋊C₁₆	2
C₅₃×#1568	2
#7242	53

subgroup

index

C₂×C₅₃×C₄⋊C₁₆

C₅₃×#1568

#7242

Subgroups — C₅₃×#7242 #27214754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7242 #27214754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7242 #27214754