Group #20271101 — subgroups

Subgroups — C₅₃×#2164 #20271101

Order 6784.

138 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 96 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×#405	C₂	1	1	yes	maximal, characteristic
2	C₅₃×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₅₃×C₂²⋊C₁₆	C₂	1	2	yes	maximal
4	C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	#117×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
10	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
11	#97×C₅₃	C₄	1	1	yes	characteristic
12	#97×C₅₃	C₄	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	M₁₆×C₅₃	—	2	2	no	—
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	C₈	1	2	yes	—
23	C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
24	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₁₆×C₅₃	—	2	2	no	—
27	C₁₆×C₅₃	—	2	2	no	—
28	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₅₃	—	2	1	no	—
36	C₈×C₅₃	—	2	1	no	—
37	C₈×C₅₃	M₁₆	1	2	yes	—
38	C₈×C₅₃	#41	1	2	yes	—
39	C₂²×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	—	2	2	no	—
41	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₅₃	—	2	1	no	—
43	C₄×C₅₃	—	2	2	no	—
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	#117	1	1	yes	characteristic
46	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
47	#2164	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₅₃	—	4	1	no	—
49	C₂×C₅₃	#431	1	1	yes	characteristic
50	C₂×C₅₃	#504	1	2	yes	—
51	#405	C₂×C₅₃	1	1	yes	characteristic
52	C₄⋊C₁₆	C₂×C₅₃	1	2	yes	complemented
53	C₂²⋊C₁₆	C₂×C₅₃	1	2	yes	—
54	#346	C₂×C₅₃	1	1	yes	characteristic, complemented
55	C₄×C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
56	C₅₃	#2164	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×M₁₆	C₄×C₅₃	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
59	#117	C₄×C₅₃	1	1	yes	characteristic
60	C₂×C₁₆	C₂²×C₅₃	1	2	yes	—
61	C₂×C₁₆	C₂²×C₅₃	1	2	yes	—
62	#97	C₄×C₅₃	1	1	yes	characteristic
63	#97	C₄×C₅₃	1	1	yes	characteristic
64	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
65	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
66	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
67	M₁₆	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
70	C₂×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
71	C₂×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
72	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
73	C₄⋊C₄	C₈×C₅₃	1	2	yes	—
74	C₂²⋊C₄	C₈×C₅₃	1	2	yes	—
75	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
76	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	2	2	no	—
79	C₂³	C₂×C₈×C₅₃	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×C₈×C₅₃	1	2	yes	—
83	C₂×C₄	C₂×C₈×C₅₃	1	2	yes	—
84	C₂×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
85	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	M₁₆×C₅₃	1	2	yes	—
89	C₈	#41×C₅₃	1	2	yes	—
90	C₂²	—	2	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	C₂²×C₈×C₅₃	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	#117×C₅₃	1	1	yes	characteristic
97	C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	C₅₃×#431	1	1	yes	characteristic
100	C₂	C₅₃×#504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#405

C₂

yes

maximal, characteristic

C₅₃×C₄⋊C₁₆

C₂

yes

maximal, complemented

C₅₃×C₂²⋊C₁₆

C₂

yes

maximal

C₅₃×#346

C₂

yes

maximal, characteristic, complemented

C₄×C₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂×C₁₆×C₅₃

C₂²

yes

—

#97×C₅₃

C₄

yes

characteristic

#97×C₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₈

yes

—

C₂²⋊C₄×C₅₃

C₈

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

M₁₆

yes

—

C₈×C₅₃

#41

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

#2164

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

#431

yes

characteristic

C₂×C₅₃

#504

yes

—

#405

C₂×C₅₃

yes

characteristic

C₄⋊C₁₆

C₂×C₅₃

yes

complemented

C₂²⋊C₁₆

C₂×C₅₃

yes

—

#346

C₂×C₅₃

yes

characteristic, complemented

C₄×C₁₆

C₂×C₅₃

yes

characteristic, complemented

C₅₃

#2164

yes

characteristic, direct product factor, Sylow

C₂×M₁₆

C₄×C₅₃

yes

characteristic

C₂²×C₈

C₂²×C₅₃

yes

characteristic

#117

C₄×C₅₃

yes

characteristic

C₂×C₁₆

C₂²×C₅₃

yes

—

C₂×C₁₆

C₂²×C₅₃

yes

—

#97

C₄×C₅₃

yes

characteristic

#97

C₄×C₅₃

yes

characteristic

C₄×C₈

C₂²×C₅₃

yes

characteristic

C₄×C₈

C₂²×C₅₃

yes

characteristic

C₂²×C₄

C₂×C₄×C₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂×C₈

C₂³×C₅₃

yes

characteristic

C₄⋊C₄

C₈×C₅₃

yes

—

C₂²⋊C₄

C₈×C₅₃

yes

—

C₄²

C₂×C₄×C₅₃

yes

characteristic

C₄²

C₂×C₄×C₅₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂×C₈×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₈×C₅₃

yes

—

C₂×C₄

C₂×C₈×C₅₃

yes

—

C₂×C₄

C₂×C₈×C₅₃

yes

characteristic

C₂×C₄

C₂²×C₄×C₅₃

yes

characteristic

C₈

—

C₈

—

C₈

M₁₆×C₅₃

yes

—

C₈

#41×C₅₃

yes

—

C₂²

—

C₂²

—

C₂²

C₂²×C₈×C₅₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#117×C₅₃

yes

characteristic

C₄

C₂×M₁₆×C₅₃

yes

characteristic

C₂

—

C₂

C₅₃×#431

yes

characteristic

100

C₂

C₅₃×#504

yes

—

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

subgroup

G/H

center

C₂×C₈×C₅₃

C₂³

commutator

C₂²

C₂²×C₈×C₅₃

Frattini, cosocle

C₂×C₈

C₂³×C₅₃

socle

C₂²×C₅₃

C₂²×C₈

prime radical

C₂³×C₅₃

C₂×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×C₄⋊C₁₆	C₂	semidirect
4. C₅₃×#346	C₂	semidirect
5. C₄×C₁₆×C₅₃	C₂	semidirect
47. #2164	C₅₃	direct (N × H)
52. C₄⋊C₁₆	C₂×C₅₃	semidirect
54. #346	C₂×C₅₃	semidirect
55. C₄×C₁₆	C₂×C₅₃	semidirect
56. C₅₃	#2164	direct (N × H)

H ≅ G/N

type

2. C₅₃×C₄⋊C₁₆

C₂

semidirect

4. C₅₃×#346

C₂

semidirect

5. C₄×C₁₆×C₅₃

C₂

semidirect

47. #2164

C₅₃

direct (N × H)

52. C₄⋊C₁₆

C₂×C₅₃

semidirect

54. #346

C₂×C₅₃

semidirect

55. C₄×C₁₆

C₂×C₅₃

semidirect

56. C₅₃

#2164

direct (N × H)

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

subgroup	index
C₅₃×#405	2
C₅₃×C₄⋊C₁₆	2
C₅₃×C₂²⋊C₁₆	2
C₅₃×#346	2
C₄×C₁₆×C₅₃	2
#2164	53

subgroup

index

C₅₃×#405

C₅₃×C₄⋊C₁₆

C₅₃×C₂²⋊C₁₆

C₅₃×#346

C₄×C₁₆×C₅₃

#2164

Subgroups — C₅₃×#2164 #20271101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#2164 #20271101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2164 #20271101