Group #20270705 — subgroups

Subgroups — C₅₃×#1768 #20270705

Order 6784.

260 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 168 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×C₄⋊C₈×C₅₃	C₂	1	2	yes	maximal
2	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#336	C₂	1	2	yes	maximal
6	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₃	C₄	1	8	yes	complemented
12	#97×C₅₃	—	2	2	no	—
13	C₄×C₈×C₅₃	—	2	2	no	—
14	#95×C₅₃	C₄	1	4	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	C₂×C₄	1	4	yes	—
27	C₄²×C₅₃	D₄	1	2	yes	—
28	C₄²×C₅₃	Q₈	1	2	yes	—
29	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	4	no	—
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	C₄²	1	4	yes	—
35	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#41	1	2	yes	—
40	C₈×C₅₃	—	2	4	no	—
41	C₈×C₅₃	—	2	4	no	—
42	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₅₃	#117	1	1	yes	characteristic
45	C₂²×C₅₃	#131	1	4	yes	—
46	C₄×C₅₃	—	2	4	no	—
47	C₄×C₅₃	—	2	4	no	—
48	C₄×C₅₃	—	4	2	no	—
49	C₄×C₅₃	C₄×D₄	1	2	yes	—
50	C₄×C₅₃	C₄×Q₈	1	2	yes	—
51	#1768	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₅₃	#405	1	2	yes	—
54	C₂×C₅₃	#424	1	2	yes	—
55	C₂×C₅₃	#433	1	2	yes	—
56	C₂×C₄⋊C₈	C₂×C₅₃	1	2	yes	—
57	C₂×#97	C₂×C₅₃	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
59	#376	C₂×C₅₃	1	1	yes	characteristic
60	#336	C₂×C₅₃	1	2	yes	—
61	C₅₃	#1768	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
63	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
64	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
65	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
66	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
67	C₄⋊C₈	C₄×C₅₃	1	8	yes	complemented
68	#97	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	#95	C₄×C₅₃	1	4	yes	—
71	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
72	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
73	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
74	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	C₂×C₄×C₅₃	1	4	yes	—
80	C₂×C₈	C₂×C₄×C₅₃	1	4	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂×C₄×C₅₃	1	4	yes	—
83	C₄²	D₄×C₅₃	1	2	yes	—
84	C₄²	Q₈×C₅₃	1	2	yes	—
85	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄²×C₅₃	1	4	yes	—
91	C₂×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
92	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
93	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
95	C₂×C₄	#41×C₅₃	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	4	no	—
98	C₂²	C₂×C₄²×C₅₃	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
100	C₂²	#117×C₅₃	1	1	yes	characteristic
101	C₂²	#131×C₅₃	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₄×C₅₃	1	2	yes	—
106	C₄	C₄×Q₈×C₅₃	1	2	yes	—
107	C₂	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
108	C₂	C₅₃×#405	1	2	yes	—
109	C₂	C₅₃×#424	1	2	yes	—
110	C₂	C₅₃×#433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal

C₂×#97×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#376

C₂

yes

maximal, characteristic

C₅₃×#336

C₂

yes

maximal

C₂²×C₈×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

complemented

#97×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

#1768

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#405

yes

—

C₂×C₅₃

#424

yes

—

C₂×C₅₃

#433

yes

—

C₂×C₄⋊C₈

C₂×C₅₃

yes

—

C₂×#97

C₂×C₅₃

yes

characteristic

C₂×C₄×C₈

C₂×C₅₃

yes

characteristic

#376

C₂×C₅₃

yes

characteristic

#336

C₂×C₅₃

yes

—

C₅₃

#1768

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂²×C₅₃

yes

—

C₂²×C₈

C₂²×C₅₃

yes

—

C₂×C₄²

C₂²×C₅₃

yes

characteristic

C₂×C₄²

C₂²×C₅₃

yes

characteristic

C₂×C₄²

C₂²×C₅₃

yes

characteristic

C₄⋊C₈

C₄×C₅₃

yes

complemented

#97

—

C₄×C₈

—

#95

C₄×C₅₃

yes

—

C₂²×C₄

C₂×C₄×C₅₃

yes

—

C₂²×C₄

C₂×C₄×C₅₃

yes

—

C₂²×C₄

C₂×C₄×C₅₃

yes

—

C₂²×C₄

C₂³×C₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₈

C₂×C₄×C₅₃

yes

—

C₄²

—

C₄²

C₂×C₄×C₅₃

yes

—

C₄²

D₄×C₅₃

yes

—

C₄²

Q₈×C₅₃

yes

—

C₂³

C₂²×C₄×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄²×C₅₃

yes

—

C₂×C₄

C₄⋊C₄×C₅₃

yes

—

C₂×C₄

C₂²×C₄×C₅₃

yes

—

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂×C₄

C₂×Q₈×C₅₃

yes

characteristic

C₂×C₄

#41×C₅₃

yes

—

C₈

—

C₈

—

C₂²

C₂×C₄²×C₅₃

yes

characteristic

C₂²

C₂×C₄⋊C₄×C₅₃

yes

characteristic

100

C₂²

#117×C₅₃

yes

characteristic

101

C₂²

#131×C₅₃

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₄×C₅₃

yes

—

106

C₄

C₄×Q₈×C₅₃

yes

—

107

C₂

C₄×C₄⋊C₄×C₅₃

yes

characteristic

108

C₂

C₅₃×#405

yes

—

109

C₂

C₅₃×#424

yes

—

110

C₂

C₅₃×#433

yes

—

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

subgroup

G/H

center

C₂²×C₄×C₅₃

C₂³

commutator

C₂²

C₂×C₄²×C₅₃

Frattini, cosocle

C₂²×C₄

C₂³×C₅₃

socle, prime radical

C₂³×C₅₃

C₂²×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₅₃ ◁ G

N	H ≅ G/N	type
11. C₄⋊C₈×C₅₃	C₄	semidirect
51. #1768	C₅₃	direct (N × H)
61. C₅₃	#1768	direct (N × H)
67. C₄⋊C₈	C₄×C₅₃	semidirect

H ≅ G/N

type

11. C₄⋊C₈×C₅₃

C₄

semidirect

51. #1768

C₅₃

direct (N × H)

61. C₅₃

#1768

direct (N × H)

67. C₄⋊C₈

C₄×C₅₃

semidirect

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

subgroup	index
C₂×C₄⋊C₈×C₅₃	2
C₂×#97×C₅₃	2
C₂×C₄×C₈×C₅₃	2
C₅₃×#376	2
C₅₃×#336	2
#1768	53

subgroup

index

C₂×C₄⋊C₈×C₅₃

C₂×#97×C₅₃

C₂×C₄×C₈×C₅₃

C₅₃×#376

C₅₃×#336

#1768

Subgroups — C₅₃×#1768 #20270705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1768 #20270705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1768 #20270705