Group #27270813 — subgroups

Subgroups — #27270813

Order 13568.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20272779	C₂	1	2	yes	maximal, complemented
2	C₅₃×#1709	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	—	2	2	no	—
4	C₅₃×#494	C₄	1	1	yes	characteristic
5	C₅₃×#493	C₄	1	1	yes	characteristic
6	C₅₃×#486	—	2	2	no	—
7	C₈²×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#11719908	—	4	2	no	—
9	C₂×Q₁₆×C₅₃	—	4	1	no	—
10	C₂×D₈×C₅₃	—	4	1	no	—
11	C₄⋊Q₈×C₅₃	—	2	1	no	—
12	#127×C₅₃	—	2	1	no	—
13	#102×C₅₃	—	4	2	no	—
14	#102×C₅₃	—	4	2	no	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄×C₈×C₅₃	D₄	1	2	yes	—
17	C₂×Q₈×C₅₃	—	4	1	no	—
18	C₂×D₄×C₅₃	—	4	1	no	—
19	C₂×D₄×C₅₃	—	4	2	no	—
20	Q₁₆×C₅₃	—	8	2	no	—
21	D₈×C₅₃	—	8	2	no	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	D₈	1	2	yes	—
24	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
25	C₄⋊C₄×C₅₃	—	4	2	no	—
26	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	#11719908	—	8	2	no	—
28	C₂³×C₅₃	—	8	1	no	—
29	Q₈×C₅₃	—	8	2	no	—
30	D₄×C₅₃	—	8	2	no	—
31	D₄×C₅₃	—	8	2	no	—
32	D₄×C₅₃	—	8	2	no	—
33	C₂×C₄×C₅₃	—	8	1	no	—
34	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
35	C₂×C₄×C₅₃	#102	1	2	yes	—
36	C₈×C₅₃	—	2	2	no	—
37	C₈×C₅₃	—	2	2	no	—
38	C₈×C₅₃	—	2	2	no	—
39	#7092	—	53	1	no	maximal, Sylow
40	C₂²×C₅₃	—	16	1	no	—
41	C₂²×C₅₃	—	16	2	no	—
42	C₂²×C₅₃	#331	1	1	yes	characteristic
43	C₄×C₅₃	—	2	1	no	—
44	C₄×C₅₃	—	16	1	no	—
45	C₄×C₅₃	#359	1	2	yes	—
46	C₄×C₅₃	#360	1	2	yes	—
47	#1709	D₅₃	1	1	yes	characteristic
48	#1367	—	53	2	no	—
49	C₂×C₅₃	—	32	1	no	—
50	C₂×C₅₃	#1357	1	1	yes	characteristic
51	C₂×C₅₃	#1360	1	2	yes	—
52	#494	Dic₅₃	1	1	yes	characteristic
53	#493	Dic₅₃	1	1	yes	characteristic
54	#486	—	2	2	no	—
55	C₄⋊C₁₆	—	106	2	no	—
56	C₈²	C₂×D₅₃	1	1	yes	characteristic
57	C₅₃	#7092	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	—	4	1	no	—
59	C₂×D₈	—	4	1	no	—
60	C₄⋊Q₈	—	2	1	no	—
61	#127	—	2	1	no	—
62	C₂×C₁₆	—	212	2	no	—
63	#102	—	4	2	no	—
64	#102	—	4	2	no	—
65	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
66	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
67	C₂×Q₈	—	4	1	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	2	no	—
70	Q₁₆	—	8	2	no	—
71	D₈	—	8	2	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	#11719922	1	2	yes	—
74	C₂×C₈	#11719923	1	2	yes	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
77	C₁₆	—	424	2	no	—
78	C₂³	—	8	1	no	—
79	Q₈	—	8	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	2	no	—
82	D₄	—	8	2	no	—
83	C₂×C₄	—	8	1	no	—
84	C₂×C₄	#13133134	1	1	yes	characteristic
85	C₂×C₄	#13133138	1	2	yes	—
86	C₈	—	2	2	no	—
87	C₈	—	2	2	no	—
88	C₈	—	2	2	no	—
89	C₂²	—	16	1	no	—
90	C₂²	—	16	2	no	—
91	C₂²	#16810367	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	16	1	no	—
94	C₄	#16810430	1	2	yes	—
95	C₄	#16810433	1	2	yes	—
96	C₂	—	32	1	no	—
97	C₂	#20272752	1	1	yes	characteristic
98	C₂	#20272758	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20272779

C₂

yes

maximal, complemented

C₅₃×#1709

C₂

yes

maximal, characteristic

C₄⋊#11719908

—

C₅₃×#494

C₄

yes

characteristic

C₅₃×#493

C₄

yes

characteristic

C₅₃×#486

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×#11719908

—

C₂×Q₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#127×C₅₃

—

#102×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

Q₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

#11719908

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#7092

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#331

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#359

yes

—

C₄×C₅₃

#360

yes

—

#1709

D₅₃

yes

characteristic

#1367

—

C₂×C₅₃

—

C₂×C₅₃

#1357

yes

characteristic

C₂×C₅₃

#1360

yes

—

#494

Dic₅₃

yes

characteristic

#493

Dic₅₃

yes

characteristic

#486

—

C₄⋊C₁₆

—

106

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#7092

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

212

—

#102

—

#102

—

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂²⋊D₅₃

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

#11719922

yes

—

C₂×C₈

#11719923

yes

—

C₄⋊C₄

—

C₄²

C₂²⋊Dic₅₃

yes

characteristic

C₁₆

—

424

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#13133134

yes

characteristic

C₂×C₄

#13133138

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#16810367

yes

characteristic

C₄

—

C₄

—

C₄

#16810430

yes

—

C₄

#16810433

yes

—

C₂

—

C₂

#20272752

yes

characteristic

C₂

#20272758

yes

—

	subgroup	G/H
center	C₂²	#16810367
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1709	C₂
socle	C₂²×C₅₃	#331
prime radical	C₅₃×#493	C₄
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1709	D₅₃
hypocenter	C₅₃	#7092
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16810367

commutator

C₄×C₈×C₅₃

C₂×C₄

Frattini

C₈²

C₂×D₅₃

Fitting

C₅₃×#1709

C₂

socle

C₂²×C₅₃

#331

prime radical

C₅₃×#493

C₄

cosocle

C₈²×C₅₃

C₂²

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #20272779	C₂	semidirect
57. C₅₃	#7092	semidirect

H ≅ G/N

type

1. #20272779

C₂

semidirect

57. C₅₃

#7092

semidirect

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

subgroup	index
#20272779	2
C₅₃×#1709	2
#7092	53

subgroup

index

#20272779

C₅₃×#1709

#7092

Subgroups — #27270813

Conjugacy classes of subgroups

Complements

Maximal subgroups