Group #27288601 — subgroups

Subgroups — #27288601

Order 13568.

350 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20274848	C₂	1	3	yes	maximal
2	#20272596	C₂	1	3	yes	maximal
3	C₅₃×#2845	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133128	C₂²	1	3	yes	—
5	#16810493	C₂²	1	1	yes	characteristic
6	#16810385	—	2	6	no	—
7	C₅₃×#398	—	2	6	no	—
8	C₅₃×#382	C₂²	1	3	yes	—
9	C₂²×#88322	—	2	3	no	—
10	C₂×C₄⋊Dic₅₃	—	2	3	no	—
11	#13133128	—	2	3	no	—
12	#13133128	Q₈	1	6	yes	—
13	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
14	C₂×C₄⋊C₄×C₅₃	D₄	1	6	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	#95×C₅₃	—	2	2	no	—
17	#95×C₅₃	—	2	6	no	—
18	C₂²×C₄×C₅₃	—	2	6	no	—
19	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
20	C₂²×Dic₅₃	—	4	1	no	—
21	C₄⋊C₄×C₅₃	—	4	6	no	—
22	C₄⋊C₄×C₅₃	—	8	6	no	—
23	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
24	C₄²×C₅₃	#41	1	1	yes	characteristic
25	C₄⋊Dic₅₃	—	4	6	no	—
26	C₂×#88322	—	2	6	no	—
27	C₂×#88322	—	4	3	no	—
28	C₂×#88322	—	4	3	no	—
29	C₂³×C₅₃	#120	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	3	no	—
31	C₂×C₄×C₅₃	—	2	3	no	—
32	C₂×C₄×C₅₃	—	4	6	no	—
33	C₂×C₄×C₅₃	—	4	6	no	—
34	C₂×C₄×C₅₃	—	4	6	no	—
35	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
36	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
37	C₂×Dic₅₃	—	8	3	no	—
38	#88322	—	4	6	no	—
39	#10586	—	53	1	no	maximal, Sylow
40	C₂²×C₅₃	#395	1	1	yes	characteristic
41	C₂²×C₅₃	#454	1	3	yes	—
42	C₂²×C₅₃	#455	1	3	yes	—
43	C₄×C₅₃	—	2	3	no	—
44	C₄×C₅₃	—	2	3	no	—
45	C₄×C₅₃	—	8	6	no	—
46	Dic₅₃	—	16	1	no	—
47	#2845	D₅₃	1	1	yes	characteristic
48	#1836	—	53	3	no	—
49	#1295	—	53	3	no	—
50	C₂×C₅₃	#1658	1	1	yes	characteristic
51	C₂×C₅₃	#1662	1	3	yes	—
52	C₂×C₅₃	#2061	1	3	yes	—
53	C₂×#97	—	53	3	no	—
54	#398	—	2	6	no	—
55	#384	—	53	1	no	—
56	#382	C₂×D₅₃	1	3	yes	—
57	#340	—	106	6	no	—
58	C₅₃	#10586	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₈	—	106	3	no	—
60	C₂×C₄⋊C₄	—	2	6	no	—
61	C₂×C₄⋊C₄	—	106	3	no	—
62	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	6	yes	—
63	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
64	#97	—	53	6	no	—
65	#97	—	106	3	no	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	212	1	no	—
70	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
71	C₂×C₈	—	106	6	no	—
72	C₂×C₈	—	212	3	no	—
73	C₂×C₈	—	212	3	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	8	6	no	—
76	C₄⋊C₄	—	212	6	no	—
77	C₄²	Q₈×D₅₃	1	3	yes	—
78	C₄²	#11719949	1	1	yes	characteristic
79	C₂³	#13133186	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	424	3	no	—
86	C₂×C₄	#13133193	1	3	yes	—
87	C₂×C₄	#13133215	1	3	yes	—
88	C₈	—	212	6	no	—
89	C₂²	#16810542	1	1	yes	characteristic
90	C₂²	#16810796	1	3	yes	—
91	C₂²	#16810801	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	848	1	no	—
96	C₂	#20274017	1	1	yes	characteristic
97	C₂	#20274038	1	3	yes	—
98	C₂	#20276062	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274848

C₂

yes

maximal

#20272596

C₂

yes

maximal

C₅₃×#2845

C₂

yes

maximal, characteristic

C₂×#13133128

C₂²

yes

—

#16810493

C₂²

yes

characteristic

#16810385

—

C₅₃×#398

—

C₅₃×#382

C₂²

yes

—

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

#13133128

—

#13133128

Q₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×Dic₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#455

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2845

D₅₃

yes

characteristic

#1836

—

#1295

—

C₂×C₅₃

#1658

yes

characteristic

C₂×C₅₃

#1662

yes

—

C₂×C₅₃

#2061

yes

—

C₂×#97

—

#398

—

#384

—

#382

C₂×D₅₃

yes

—

#340

—

106

—

C₅₃

#10586

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#97

—

#97

—

106

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄²

Q₈×D₅₃

yes

—

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

#13133193

yes

—

C₂×C₄

#13133215

yes

—

C₈

—

212

—

C₂²

#16810542

yes

characteristic

C₂²

#16810796

yes

—

C₂²

#16810801

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

848

—

C₂

#20274017

yes

characteristic

C₂

#20274038

yes

—

C₂

#20276062

yes

—

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2845	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#2845	D₅₃
hypocenter	C₅₃	#10586
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#13133186

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2845

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
58. C₅₃	#10586	semidirect

H ≅ G/N

type

58. C₅₃

#10586

semidirect

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

subgroup	index
#20274848	2
#20272596	2
C₅₃×#2845	2
#10586	53

subgroup

index

#20274848

#20272596

C₅₃×#2845

#10586

Subgroups — #27288601

Conjugacy classes of subgroups

Complements

Maximal subgroups