Group #20271153 — subgroups

Subgroups — C₅₃×#2216 #20271153

Order 6784.

154 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 68 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×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#467	C₂	1	2	yes	maximal
4	C₅₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#358	C₂	1	2	yes	maximal, complemented
6	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²⋊Q₈×C₅₃	—	2	2	no	—
9	Q₃₂×C₅₃	—	2	2	no	—
10	C₂×C₁₆×C₅₃	—	2	2	no	—
11	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	#107×C₅₃	C₂²	1	2	yes	—
14	#103×C₅₃	—	2	2	no	—
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	Q₁₆×C₅₃	—	2	2	no	—
18	Q₁₆×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	—	2	1	no	—
20	C₂×C₈×C₅₃	—	2	1	no	—
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	4	2	no	—
25	C₂²⋊C₄×C₅₃	—	4	2	no	—
26	C₁₆×C₅₃	—	2	1	no	—
27	C₁₆×C₅₃	D₄	1	2	yes	—
28	C₂³×C₅₃	D₈	1	1	yes	characteristic
29	Q₈×C₅₃	—	4	2	no	—
30	Q₈×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₅₃	—	2	1	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	Q₃₂	1	2	yes	complemented
43	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
44	C₄×C₅₃	—	2	1	no	—
45	C₄×C₅₃	—	8	2	no	—
46	C₄×C₅₃	—	8	2	no	—
47	C₄×C₅₃	#121	1	1	yes	characteristic
48	C₄×C₅₃	#135	1	1	yes	characteristic
49	#2216	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₅₃	—	2	2	no	—
51	C₂×C₅₃	#466	1	1	yes	characteristic
52	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
53	C₂×C₅₃	#508	1	1	yes	characteristic
54	C₂×Q₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
55	C₂²×C₁₆	C₂×C₅₃	1	1	yes	characteristic
56	#467	C₂×C₅₃	1	2	yes	—
57	#366	C₂×C₅₃	1	1	yes	characteristic, complemented
58	#358	C₂×C₅₃	1	2	yes	complemented
59	C₅₃	#2216	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×Q₁₆	C₂²×C₅₃	1	2	yes	—
61	C₂²×C₈	C₂²×C₅₃	1	1	yes	characteristic
62	C₂²⋊Q₈	—	2	2	no	—
63	Q₃₂	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
66	C₂×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
67	#107	C₂²×C₅₃	1	2	yes	—
68	#103	—	2	2	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂²×C₄	D₄×C₅₃	1	1	yes	characteristic
71	Q₁₆	—	2	2	no	—
72	Q₁₆	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	D₄×C₅₃	1	1	yes	characteristic
76	C₂×C₈	C₂³×C₅₃	1	1	yes	characteristic
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	4	2	no	—
79	C₂²⋊C₄	—	4	2	no	—
80	C₁₆	—	2	1	no	—
81	C₁₆	D₄×C₅₃	1	2	yes	—
82	C₂³	D₈×C₅₃	1	1	yes	characteristic
83	Q₈	—	4	2	no	—
84	Q₈	—	4	2	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	D₈×C₅₃	1	1	yes	characteristic
90	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
93	C₈	#41×C₅₃	1	1	yes	characteristic
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	Q₃₂×C₅₃	1	2	yes	complemented
97	C₂²	C₂×D₈×C₅₃	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	8	2	no	—
100	C₄	—	8	2	no	—
101	C₄	#121×C₅₃	1	1	yes	characteristic
102	C₄	#135×C₅₃	1	1	yes	characteristic
103	C₂	—	2	2	no	—
104	C₂	C₅₃×#466	1	1	yes	characteristic
105	C₂	C₂×Q₃₂×C₅₃	1	1	yes	characteristic
106	C₂	C₅₃×#508	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₃₂×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#467

C₂

yes

maximal

C₅₃×#366

C₂

yes

maximal, characteristic, complemented

C₅₃×#358

C₂

yes

maximal, complemented

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₅₃

—

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

—

#103×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

D₄

yes

—

C₂³×C₅₃

D₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

Q₃₂

yes

complemented

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

#466

yes

characteristic

C₂×C₅₃

C₂×Q₃₂

yes

characteristic

C₂×C₅₃

#508

yes

characteristic

C₂×Q₃₂

C₂×C₅₃

yes

characteristic, complemented

C₂²×C₁₆

C₂×C₅₃

yes

characteristic

#467

C₂×C₅₃

yes

—

#366

C₂×C₅₃

yes

characteristic, complemented

#358

C₂×C₅₃

yes

complemented

C₅₃

#2216

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

C₂²×C₅₃

yes

—

C₂²×C₈

C₂²×C₅₃

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₅₃

yes

characteristic

C₂×C₁₆

C₂²×C₅₃

yes

characteristic

#107

C₂²×C₅₃

yes

—

#103

—

C₂×Q₈

—

C₂²×C₄

D₄×C₅₃

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₅₃

yes

characteristic

C₂×C₈

C₂³×C₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

C₁₆

D₄×C₅₃

yes

—

C₂³

D₈×C₅₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×C₅₃

yes

characteristic

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

C₈

—

C₈

C₂×D₄×C₅₃

yes

characteristic

C₈

#41×C₅₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

Q₃₂×C₅₃

yes

complemented

C₂²

C₂×D₈×C₅₃

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

#121×C₅₃

yes

characteristic

102

C₄

#135×C₅₃

yes

characteristic

103

C₂

—

104

C₂

C₅₃×#466

yes

characteristic

105

C₂

C₂×Q₃₂×C₅₃

yes

characteristic

106

C₂

C₅₃×#508

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₅₃

prime radical

C₂³×C₅₃

D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. C₂×Q₃₂×C₅₃	C₂	semidirect
4. C₅₃×#366	C₂	semidirect
5. C₅₃×#358	C₂	semidirect
42. C₂²×C₅₃	Q₃₂	semidirect
49. #2216	C₅₃	direct (N × H)
54. C₂×Q₃₂	C₂×C₅₃	semidirect
57. #366	C₂×C₅₃	semidirect
58. #358	C₂×C₅₃	semidirect
59. C₅₃	#2216	direct (N × H)
96. C₂²	Q₃₂×C₅₃	semidirect

H ≅ G/N

type

1. C₂×Q₃₂×C₅₃

C₂

semidirect

4. C₅₃×#366

C₂

semidirect

5. C₅₃×#358

C₂

semidirect

42. C₂²×C₅₃

Q₃₂

semidirect

49. #2216

C₅₃

direct (N × H)

54. C₂×Q₃₂

C₂×C₅₃

semidirect

57. #366

C₂×C₅₃

semidirect

58. #358

C₂×C₅₃

semidirect

59. C₅₃

#2216

direct (N × H)

96. C₂²

Q₃₂×C₅₃

semidirect

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

subgroup	index
C₂×Q₃₂×C₅₃	2
C₂²×C₁₆×C₅₃	2
C₅₃×#467	2
C₅₃×#366	2
C₅₃×#358	2
#2216	53

subgroup

index

C₂×Q₃₂×C₅₃

C₂²×C₁₆×C₅₃

C₅₃×#467

C₅₃×#366

C₅₃×#358

#2216

Subgroups — C₅₃×#2216 #20271153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#2216 #20271153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2216 #20271153