Group #27215364 — subgroups

Subgroups — C₅₃×#7852 #27215364

Order 13568.

802 conjugacy classes of subgroups; only the 334 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 334 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×M₁₆×C₅₃	C₂	1	3	yes	maximal
2	C₅₃×#1273	C₂	1	4	yes	maximal, complemented
3	C₂²×M₁₆×C₅₃	C₄	1	6	yes	—
4	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
5	C₄×M₁₆×C₅₃	—	2	12	no	—
6	C₂×#97×C₅₃	C₄	1	6	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
8	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
9	C₂×M₁₆×C₅₃	—	2	24	no	—
10	C₂×M₁₆×C₅₃	—	2	12	no	—
11	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
12	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
13	C₂×C₄²×C₅₃	D₄	1	6	yes	—
14	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	#97×C₅₃	—	2	12	no	—
17	C₄×C₈×C₅₃	—	2	12	no	—
18	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	—	2	12	no	—
20	C₂²×C₄×C₅₃	C₄²	1	3	yes	—
21	C₂²×C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
22	C₂²×C₄×C₅₃	C₄⋊C₄	1	6	yes	—
23	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
24	M₁₆×C₅₃	—	2	48	no	—
25	C₂×C₈×C₅₃	—	2	12	no	—
26	C₂×C₈×C₅₃	—	2	12	no	—
27	C₂×C₈×C₅₃	M₁₆	1	24	yes	complemented
28	C₄²×C₅₃	C₂²⋊C₄	1	6	yes	—
29	C₄²×C₅₃	C₄⋊C₄	1	6	yes	—
30	C₄²×C₅₃	C₂×D₄	1	3	yes	—
31	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₂³×C₅₃	—	2	6	no	—
33	C₂³×C₅₃	#95	1	2	yes	—
34	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	6	no	—
36	C₂×C₄×C₅₃	—	2	12	no	—
37	C₂×C₄×C₅₃	—	2	12	no	—
38	C₂×C₄×C₅₃	#95	1	6	yes	—
39	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
40	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
41	C₂×C₄×C₅₃	C₂×M₁₆	1	12	yes	—
42	C₈×C₅₃	—	2	24	no	—
43	#7852	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₅₃	—	2	6	no	—
45	C₂²×C₅₃	—	4	4	no	—
46	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
47	C₂²×C₅₃	C₄×M₁₆	1	6	yes	—
48	C₄×C₅₃	—	2	6	no	—
49	C₄×C₅₃	#407	1	6	yes	—
50	C₄×C₅₃	#423	1	6	yes	—
51	C₂×C₄×M₁₆	C₂×C₅₃	1	3	yes	—
52	#1273	C₂×C₅₃	1	4	yes	complemented
53	C₂×C₅₃	—	4	2	no	—
54	C₂×C₅₃	#1455	1	4	yes	—
55	C₂×C₅₃	#1728	1	3	yes	—
56	C₂²×M₁₆	C₄×C₅₃	1	6	yes	—
57	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
58	C₄×M₁₆	—	2	12	no	—
59	C₂×#97	C₄×C₅₃	1	6	yes	—
60	C₂×C₄×C₈	C₂²×C₅₃	1	6	yes	—
61	C₅₃	#7852	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
63	C₂×M₁₆	—	2	12	no	—
64	C₂×M₁₆	—	2	24	no	—
65	C₂²×C₈	C₂×C₄×C₅₃	1	12	yes	—
66	C₂×C₄²	C₂×C₄×C₅₃	1	3	yes	—
67	C₂×C₄²	D₄×C₅₃	1	6	yes	—
68	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
69	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
70	#97	—	2	12	no	—
71	C₄×C₈	—	2	12	no	—
72	C₂⁴	C₄²×C₅₃	1	1	yes	characteristic
73	C₂²×C₄	—	2	12	no	—
74	C₂²×C₄	C₄²×C₅₃	1	3	yes	—
75	C₂²×C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
76	C₂²×C₄	C₄⋊C₄×C₅₃	1	6	yes	—
77	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
78	M₁₆	—	2	48	no	—
79	C₂×C₈	—	2	12	no	—
80	C₂×C₈	—	2	12	no	—
81	C₂×C₈	M₁₆×C₅₃	1	24	yes	complemented
82	C₄²	C₂²⋊C₄×C₅₃	1	6	yes	—
83	C₄²	C₄⋊C₄×C₅₃	1	6	yes	—
84	C₄²	C₂×D₄×C₅₃	1	3	yes	—
85	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
86	C₂³	—	2	6	no	—
87	C₂³	#95×C₅₃	1	2	yes	—
88	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	2	12	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	#95×C₅₃	1	6	yes	—
93	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
94	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
95	C₂×C₄	C₂×M₁₆×C₅₃	1	12	yes	—
96	C₈	—	2	24	no	—
97	C₂²	—	2	6	no	—
98	C₂²	—	4	4	no	—
99	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
100	C₂²	C₄×M₁₆×C₅₃	1	6	yes	—
101	C₄	—	2	6	no	—
102	C₄	C₅₃×#407	1	6	yes	—
103	C₄	C₅₃×#423	1	6	yes	—
104	C₂	—	4	2	no	—
105	C₂	C₅₃×#1455	1	4	yes	—
106	C₂	C₅₃×#1728	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×M₁₆×C₅₃

C₂

yes

maximal

C₅₃×#1273

C₂

yes

maximal, complemented

C₂²×M₁₆×C₅₃

C₄

yes

—

C₂²×C₄²×C₅₃

C₂²

yes

characteristic

C₄×M₁₆×C₅₃

—

C₂×#97×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

M₁₆

yes

complemented

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

#95

yes

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₈×C₅₃

—

#7852

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#95

yes

characteristic

C₂²×C₅₃

C₄×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#407

yes

—

C₄×C₅₃

#423

yes

—

C₂×C₄×M₁₆

C₂×C₅₃

yes

—

#1273

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

#1455

yes

—

C₂×C₅₃

#1728

yes

—

C₂²×M₁₆

C₄×C₅₃

yes

—

C₂²×C₄²

C₂²×C₅₃

yes

characteristic

C₄×M₁₆

—

C₂×#97

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

—

C₅₃

#7852

yes

characteristic, direct product factor, Sylow

C₂³×C₄

C₂×C₄×C₅₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

—

C₂×C₄²

D₄×C₅₃

yes

—

C₂×C₄²

Q₈×C₅₃

yes

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

#97

—

C₄×C₈

—

C₂⁴

C₄²×C₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄²×C₅₃

yes

—

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

C₂²×C₄

C₄⋊C₄×C₅₃

yes

—

C₂²×C₄

C₂²×C₄×C₅₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

M₁₆×C₅₃

yes

complemented

C₄²

C₂²⋊C₄×C₅₃

yes

—

C₄²

C₄⋊C₄×C₅₃

yes

—

C₄²

C₂×D₄×C₅₃

yes

—

C₄²

C₂×Q₈×C₅₃

yes

characteristic

C₂³

—

C₂³

#95×C₅₃

yes

—

C₂³

C₂×C₄²×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₅₃

yes

—

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

—

C₂×C₄

C₂×M₁₆×C₅₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×#95×C₅₃

yes

characteristic

100

C₂²

C₄×M₁₆×C₅₃

yes

—

101

C₄

—

102

C₄

C₅₃×#407

yes

—

103

C₄

C₅₃×#423

yes

—

104

C₂

—

105

C₂

C₅₃×#1455

yes

—

106

C₂

C₅₃×#1728

yes

—

	subgroup	G/H
center	C₂×C₄²×C₅₃	C₂³
commutator	C₂³	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄²×C₅₃

C₂³

commutator

C₂³

C₂×C₄²×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

socle

C₂³×C₅₃

C₂×C₄²

prime radical

C₂⁴×C₅₃

C₄²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#1273	C₂	semidirect
27. C₂×C₈×C₅₃	M₁₆	semidirect
43. #7852	C₅₃	direct (N × H)
52. #1273	C₂×C₅₃	semidirect
61. C₅₃	#7852	direct (N × H)
81. C₂×C₈	M₁₆×C₅₃	semidirect

H ≅ G/N

type

2. C₅₃×#1273

C₂

semidirect

27. C₂×C₈×C₅₃

M₁₆

semidirect

43. #7852

C₅₃

direct (N × H)

52. #1273

C₂×C₅₃

semidirect

61. C₅₃

#7852

direct (N × H)

81. C₂×C₈

M₁₆×C₅₃

semidirect

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

subgroup	index
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#1273	2
#7852	53

subgroup

index

C₂×C₄×M₁₆×C₅₃

C₅₃×#1273

#7852

Subgroups — C₅₃×#7852 #27215364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#7852 #27215364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7852 #27215364