Group #27216295 — subgroups

Subgroups — C₅₃×#8783 #27216295

Order 13568.

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₅₃×#1746	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1564	C₂	1	2	yes	maximal
3	C₅₃×#1281	C₂	1	2	yes	maximal
4	C₅₃×#1274	C₂	1	2	yes	maximal
5	C₅₃×#431	C₂²	1	4	yes	—
6	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#376	C₂²	1	1	yes	characteristic
9	C₅₃×#336	C₄	1	4	yes	—
10	C₅₃×#335	—	2	2	no	—
11	C₅₃×#334	—	2	2	no	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	—	2	4	no	—
17	C₂²⋊C₈×C₅₃	—	2	4	no	—
18	#97×C₅₃	—	2	2	no	—
19	C₄×C₈×C₅₃	—	2	4	no	—
20	C₄×C₈×C₅₃	D₄	1	2	yes	—
21	C₄×C₈×C₅₃	Q₈	1	2	yes	—
22	#95×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	#41	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	4	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	—	4	4	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
31	C₄²×C₅₃	—	2	4	no	—
32	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₅₃	#41	1	2	yes	—
35	C₂³×C₅₃	#117	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	—	4	4	no	—
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#124	1	2	yes	—
42	C₂×C₄×C₅₃	#125	1	2	yes	—
43	C₂×C₄×C₅₃	#131	1	4	yes	—
44	C₈×C₅₃	—	2	4	no	—
45	C₈×C₅₃	—	4	2	no	—
46	C₈×C₅₃	—	8	4	no	—
47	#8783	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	#382	1	1	yes	characteristic
50	C₂²×C₅₃	#433	1	2	yes	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	—	8	2	no	—
54	C₄×C₅₃	#424	1	2	yes	—
55	C₄×C₅₃	#433	1	2	yes	—
56	#1746	C₂×C₅₃	1	1	yes	characteristic
57	#1564	C₂×C₅₃	1	2	yes	—
58	#1281	C₂×C₅₃	1	2	yes	—
59	#1274	C₂×C₅₃	1	2	yes	—
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	#1832	1	1	yes	characteristic
62	C₂×C₅₃	#1833	1	2	yes	—
63	#431	C₂²×C₅₃	1	4	yes	—
64	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
66	#376	C₂²×C₅₃	1	1	yes	characteristic
67	#336	C₄×C₅₃	1	4	yes	—
68	#335	—	2	2	no	—
69	#334	—	2	2	no	—
70	C₅₃	#8783	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
72	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
73	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
74	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
75	C₄⋊C₈	—	2	4	no	—
76	C₂²⋊C₈	—	2	4	no	—
77	#97	—	2	2	no	—
78	C₄×C₈	—	2	4	no	—
79	C₄×C₈	D₄×C₅₃	1	2	yes	—
80	C₄×C₈	Q₈×C₅₃	1	2	yes	—
81	#95	—	2	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
84	C₂²×C₄	#41×C₅₃	1	2	yes	—
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	4	4	no	—
89	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
90	C₄²	—	2	4	no	—
91	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
92	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
93	C₄²	#41×C₅₃	1	2	yes	—
94	C₂³	#117×C₅₃	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
99	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
100	C₂×C₄	#124×C₅₃	1	2	yes	—
101	C₂×C₄	#125×C₅₃	1	2	yes	—
102	C₂×C₄	#131×C₅₃	1	4	yes	—
103	C₈	—	2	4	no	—
104	C₈	—	4	2	no	—
105	C₈	—	8	4	no	—
106	C₂²	—	2	2	no	—
107	C₂²	C₅₃×#382	1	1	yes	characteristic
108	C₂²	C₅₃×#433	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	—	8	2	no	—
112	C₄	C₅₃×#424	1	2	yes	—
113	C₄	C₅₃×#433	1	2	yes	—
114	C₂	—	2	2	no	—
115	C₂	C₅₃×#1832	1	1	yes	characteristic
116	C₂	C₅₃×#1833	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1746

C₂

yes

maximal, characteristic

C₅₃×#1564

C₂

yes

maximal

C₅₃×#1281

C₂

yes

maximal

C₅₃×#1274

C₂

yes

maximal

C₅₃×#431

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#376

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₅₃×#335

—

C₅₃×#334

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#8783

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

#433

yes

—

#1746

C₂×C₅₃

yes

characteristic

#1564

C₂×C₅₃

yes

—

#1281

C₂×C₅₃

yes

—

#1274

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1832

yes

characteristic

C₂×C₅₃

#1833

yes

—

#431

C₂²×C₅₃

yes

—

C₂×#97

C₂²×C₅₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#376

C₂²×C₅₃

yes

characteristic

#336

C₄×C₅₃

yes

—

#335

—

#334

—

C₅₃

#8783

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

D₄×C₅₃

yes

—

C₄×C₈

Q₈×C₅₃

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

C₂²×C₄

#41×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₄²

—

C₄²

C₂×D₄×C₅₃

yes

characteristic

C₄²

C₂×Q₈×C₅₃

yes

characteristic

C₄²

#41×C₅₃

yes

—

C₂³

#117×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

C₂×C₄

#117×C₅₃

yes

characteristic

100

C₂×C₄

#124×C₅₃

yes

—

101

C₂×C₄

#125×C₅₃

yes

—

102

C₂×C₄

#131×C₅₃

yes

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₂²

—

107

C₂²

C₅₃×#382

yes

characteristic

108

C₂²

C₅₃×#433

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₅₃×#424

yes

—

113

C₄

C₅₃×#433

yes

—

114

C₂

—

115

C₂

C₅₃×#1832

yes

characteristic

116

C₂

C₅₃×#1833

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#376 ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#117

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂²×C₅₃

C₂³×C₅₃

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₅₃ ◁ C₅₃×#376 ◁ G

N	H ≅ G/N	type
47. #8783	C₅₃	direct (N × H)
70. C₅₃	#8783	direct (N × H)

H ≅ G/N

type

47. #8783

C₅₃

direct (N × H)

70. C₅₃

#8783

direct (N × H)

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

subgroup	index
C₅₃×#1746	2
C₅₃×#1564	2
C₅₃×#1281	2
C₅₃×#1274	2
#8783	53

subgroup

index

C₅₃×#1746

C₅₃×#1564

C₅₃×#1281

C₅₃×#1274

#8783

Subgroups — C₅₃×#8783 #27216295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8783 #27216295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8783 #27216295