Group #27271950 — subgroups

Subgroups — #27271950

Order 13568.

186 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 27 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	#20272878	C₂	1	2	yes	maximal, complemented
2	C₅₃×#1712	C₂	1	1	yes	maximal, characteristic
3	#16810416	—	2	2	no	—
4	C₅₃×#493	—	2	2	no	—
5	C₅₃×#493	C₄	1	1	yes	characteristic, complemented
6	C₅₃×#493	C₂²	1	1	yes	characteristic
7	C₈²×C₅₃	C₄	1	1	yes	characteristic, complemented
8	#13133132	—	4	2	no	—
9	C₂×D₈×C₅₃	—	4	1	no	—
10	C₂×D₈×C₅₃	—	4	4	no	—
11	C₂×D₈×C₅₃	—	4	1	no	—
12	#127×C₅₃	—	2	1	no	—
13	#127×C₅₃	D₄	1	2	yes	—
14	C₄×C₈×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×D₄×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	4	2	no	—
18	C₂×D₄×C₅₃	—	4	2	no	—
19	C₂×D₄×C₅₃	—	4	1	no	—
20	D₈×C₅₃	—	8	2	no	—
21	D₈×C₅₃	—	8	8	no	—
22	D₈×C₅₃	—	8	2	no	—
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²⋊Dic₅₃	—	8	2	no	—
27	C₂³×C₅₃	—	4	2	no	—
28	C₂³×C₅₃	—	8	1	no	—
29	D₄×C₅₃	—	8	2	no	—
30	D₄×C₅₃	—	8	4	no	—
31	D₄×C₅₃	—	8	4	no	—
32	D₄×C₅₃	—	8	2	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
35	C₂×Dic₅₃	—	16	2	no	—
36	C₈×C₅₃	—	2	4	no	—
37	C₈×C₅₃	—	2	2	no	—
38	#7529	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	—	8	2	no	—
40	C₂²×C₅₃	—	16	2	no	—
41	C₂²×C₅₃	—	16	2	no	—
42	C₂²×C₅₃	—	16	1	no	—
43	C₂²×C₅₃	#353	1	1	yes	characteristic
44	C₄×C₅₃	—	2	2	no	—
45	C₄×C₅₃	—	2	1	no	—
46	Dic₅₃	—	32	2	no	—
47	#1712	D₅₃	1	1	yes	characteristic
48	#1408	—	53	2	no	—
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	—	16	2	no	—
51	C₂×C₅₃	—	32	1	no	—
52	C₂×C₅₃	#1408	1	1	yes	characteristic
53	#493	—	2	2	no	—
54	#493	Dic₅₃	1	1	yes	characteristic, complemented
55	#493	C₂×D₅₃	1	1	yes	characteristic
56	#353	—	106	2	no	—
57	C₈²	Dic₅₃	1	1	yes	characteristic, complemented
58	C₅₃	#7529	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	—	4	1	no	—
61	C₂×D₈	—	4	4	no	—
62	#127	—	2	1	no	—
63	#127	C₂²⋊D₅₃	1	2	yes	—
64	#99	—	212	2	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
67	C₂×D₄	—	2	2	no	—
68	C₂×D₄	—	4	2	no	—
69	C₂×D₄	—	4	2	no	—
70	C₂×D₄	—	4	1	no	—
71	D₈	—	8	2	no	—
72	D₈	—	8	2	no	—
73	D₈	—	8	8	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂²⋊C₄	—	424	2	no	—
77	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
78	C₂³	—	4	2	no	—
79	C₂³	—	8	1	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	8	4	no	—
83	D₄	—	8	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	848	2	no	—
86	C₂×C₄	#13133132	1	1	yes	characteristic
87	C₈	—	2	4	no	—
88	C₈	—	2	2	no	—
89	C₂²	—	8	2	no	—
90	C₂²	—	16	2	no	—
91	C₂²	—	16	2	no	—
92	C₂²	—	16	1	no	—
93	C₂²	#16810416	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	1696	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	16	2	no	—
99	C₂	—	32	1	no	—
100	C₂	#20272878	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272878

C₂

yes

maximal, complemented

C₅₃×#1712

C₂

yes

maximal, characteristic

#16810416

—

C₅₃×#493

—

C₅₃×#493

C₄

yes

characteristic, complemented

C₅₃×#493

C₂²

yes

characteristic

C₈²×C₅₃

C₄

yes

characteristic, complemented

#13133132

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

#127×C₅₃

—

#127×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#99

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#7529

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#353

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1712

D₅₃

yes

characteristic

#1408

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

Dic₅₃

yes

characteristic, complemented

#493

C₂×D₅₃

yes

characteristic

#353

—

106

—

C₈²

Dic₅₃

yes

characteristic, complemented

C₅₃

#7529

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

C₂²⋊D₅₃

yes

—

#99

—

212

—

C₄×C₈

—

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

424

—

C₄²

C₂²⋊Dic₅₃

yes

characteristic

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

848

—

C₂×C₄

#13133132

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#16810416

yes

characteristic

C₄

—

C₄

—

C₄

—

1696

—

C₂

—

C₂

—

C₂

—

100

C₂

#20272878

yes

characteristic

	subgroup	G/H
center	C₂	#20272878
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	#493	C₂×D₅₃
Fitting, prime radical	C₅₃×#1712	C₂
socle	C₂×C₅₃	#1408
cosocle	C₅₃×#493	C₂²
hypercenter	#1712	D₅₃
hypocenter	C₅₃	#7529
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1712

subgroup

G/H

center

C₂

#20272878

commutator

C₄×C₈×C₅₃

C₂×C₄

Frattini

#493

C₂×D₅₃

Fitting, prime radical

C₅₃×#1712

C₂

socle

C₂×C₅₃

#1408

cosocle

C₅₃×#493

C₂²

D₅₃

C₅₃

G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1712

N	H ≅ G/N	type
1. #20272878	C₂	semidirect
5. C₅₃×#493	C₄	semidirect
7. C₈²×C₅₃	C₄	semidirect
54. #493	Dic₅₃	semidirect
57. C₈²	Dic₅₃	semidirect
58. C₅₃	#7529	semidirect

H ≅ G/N

type

1. #20272878

C₂

semidirect

5. C₅₃×#493

C₄

semidirect

7. C₈²×C₅₃

C₄

semidirect

54. #493

Dic₅₃

semidirect

57. C₈²

Dic₅₃

semidirect

58. C₅₃

#7529

semidirect

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

subgroup	index
#20272878	2
C₅₃×#1712	2
#7529	53

subgroup

index

#20272878

C₅₃×#1712

#7529

Subgroups — #27271950

Conjugacy classes of subgroups

Complements

Maximal subgroups