Group #27214784 — subgroups

Subgroups — C₅₃×#7272 #27214784

Order 13568.

208 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 114 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×#364	C₂	1	2	yes	maximal
2	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₅₃	—	2	2	no	—
4	C₂×#108×C₅₃	—	2	2	no	—
5	C₅₃×#423	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#364	—	2	2	no	—
8	C₅₃×#364	C₄	1	4	yes	—
9	C₂×M₁₆×C₅₃	—	2	2	no	—
10	C₂²×C₈×C₅₃	D₄	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
12	M₃₂×C₅₃	—	2	4	no	—
13	M₃₂×C₅₃	—	4	2	no	—
14	C₂×C₁₆×C₅₃	—	4	2	no	—
15	#108×C₅₃	—	4	2	no	—
16	#108×C₅₃	—	4	2	no	—
17	C₄⋊C₈×C₅₃	—	4	2	no	—
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	M₁₆×C₅₃	—	4	4	no	—
24	C₂×C₈×C₅₃	—	4	2	no	—
25	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	D₈	1	2	yes	—
28	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
29	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
30	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
31	C₄²×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	C₄²	1	1	yes	characteristic
33	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₁₆×C₅₃	—	4	4	no	—
35	C₂³×C₅₃	#101	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	2	1	no	—
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#102	1	2	yes	—
42	C₂×C₄×C₅₃	#103	1	2	yes	—
43	C₈×C₅₃	—	8	2	no	—
44	C₈×C₅₃	#102	1	2	yes	—
45	C₈×C₅₃	#103	1	2	yes	—
46	C₈×C₅₃	#106	1	2	yes	—
47	C₈×C₅₃	#107	1	2	yes	—
48	#7272	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²×C₅₃	—	4	1	no	—
50	C₂²×C₅₃	#332	1	2	yes	—
51	C₂²×C₅₃	#343	1	1	yes	characteristic
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	—	4	1	no	—
54	C₄×C₅₃	#331	1	1	yes	characteristic
55	C₄×C₅₃	#333	1	1	yes	characteristic
56	C₄×C₅₃	#340	1	2	yes	—
57	C₂×#364	C₂×C₅₃	1	2	yes	—
58	#1848	C₂×C₅₃	1	1	yes	characteristic
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	#1296	1	1	yes	characteristic
62	C₂×M₃₂	—	2	2	no	—
63	C₂×#108	—	2	2	no	—
64	#423	C₄×C₅₃	1	2	yes	—
65	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
66	#364	—	2	2	no	—
67	#364	C₄×C₅₃	1	4	yes	—
68	C₅₃	#7272	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×M₁₆	—	2	2	no	—
70	C₂²×C₈	D₄×C₅₃	1	2	yes	—
71	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
72	M₃₂	—	2	4	no	—
73	M₃₂	—	4	2	no	—
74	C₂×C₁₆	—	4	2	no	—
75	#108	—	4	2	no	—
76	#108	—	4	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
79	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
80	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
83	M₁₆	—	4	4	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
86	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
87	C₂×C₈	D₈×C₅₃	1	2	yes	—
88	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
89	C₂×C₈	SD₁₆×C₅₃	1	2	yes	—
90	C₂×C₈	Q₁₆×C₅₃	1	2	yes	—
91	C₄²	—	2	1	no	—
92	C₄²	C₄²×C₅₃	1	1	yes	characteristic
93	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
94	C₁₆	—	4	4	no	—
95	C₂³	#101×C₅₃	1	1	yes	characteristic
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
100	C₂×C₄	#100×C₅₃	1	1	yes	characteristic
101	C₂×C₄	#102×C₅₃	1	2	yes	—
102	C₂×C₄	#103×C₅₃	1	2	yes	—
103	C₈	—	8	2	no	—
104	C₈	#102×C₅₃	1	2	yes	—
105	C₈	#103×C₅₃	1	2	yes	—
106	C₈	#106×C₅₃	1	2	yes	—
107	C₈	#107×C₅₃	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	C₅₃×#332	1	2	yes	—
110	C₂²	C₅₃×#343	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	4	1	no	—
113	C₄	C₅₃×#331	1	1	yes	characteristic
114	C₄	C₅₃×#333	1	1	yes	characteristic
115	C₄	C₅₃×#340	1	2	yes	—
116	C₂	—	2	1	no	—
117	C₂	—	2	2	no	—
118	C₂	C₅₃×#1296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#364

C₂

yes

maximal

C₅₃×#1848

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₃

—

C₂×#108×C₅₃

—

C₅₃×#423

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#364

—

C₅₃×#364

C₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

#108×C₅₃

—

#108×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

#101

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₈×C₅₃

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

#103

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#107

yes

—

#7272

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#332

yes

—

C₂²×C₅₃

#343

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#364

C₂×C₅₃

yes

—

#1848

C₂×C₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1296

yes

characteristic

C₂×M₃₂

—

C₂×#108

—

#423

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#364

—

#364

C₄×C₅₃

yes

—

C₅₃

#7272

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

—

C₂²×C₈

D₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

Q₈×C₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

D₈×C₅₃

yes

—

C₂×C₈

SD₁₆×C₅₃

yes

—

C₂×C₈

SD₁₆×C₅₃

yes

—

C₂×C₈

Q₁₆×C₅₃

yes

—

C₄²

—

C₄²

C₄²×C₅₃

yes

characteristic

C₄²

C₄⋊C₄×C₅₃

yes

characteristic

C₁₆

—

C₂³

#101×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₅₃

yes

characteristic

100

C₂×C₄

#100×C₅₃

yes

characteristic

101

C₂×C₄

#102×C₅₃

yes

—

102

C₂×C₄

#103×C₅₃

yes

—

103

C₈

—

104

C₈

#102×C₅₃

yes

—

105

C₈

#103×C₅₃

yes

—

106

C₈

#106×C₅₃

yes

—

107

C₈

#107×C₅₃

yes

—

108

C₂²

—

109

C₂²

C₅₃×#332

yes

—

110

C₂²

C₅₃×#343

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

C₅₃×#331

yes

characteristic

114

C₄

C₅₃×#333

yes

characteristic

115

C₄

C₅₃×#340

yes

—

116

C₂

—

117

C₂

—

118

C₂

C₅₃×#1296

yes

characteristic

	subgroup	G/H
center	C₄×C₅₃	#333
commutator	C₄²	C₄²×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂×C₅₃	#1296
prime radical	C₂³×C₅₃	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₄×C₅₃

#333

commutator

C₄²

C₄²×C₅₃

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₅₃

C₂×C₅₃

C₂³×C₅₃

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
48. #7272	C₅₃	direct (N × H)
68. C₅₃	#7272	direct (N × H)

H ≅ G/N

type

48. #7272

C₅₃

direct (N × H)

68. C₅₃

#7272

direct (N × H)

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

subgroup	index
C₂×C₅₃×#364	2
C₅₃×#1848	2
#7272	53

subgroup

index

C₂×C₅₃×#364

C₅₃×#1848

#7272

Subgroups — C₅₃×#7272 #27214784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7272 #27214784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7272 #27214784