Group #27274526 — subgroups

Subgroups — #27274526

Order 13568.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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	#20274460	C₂	1	3	yes	maximal
2	#20272542	C₂	1	3	yes	maximal
3	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₂²	1	3	yes	—
5	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
6	#16810377	C₄	1	6	yes	—
7	#16810377	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
9	C₂²×#88322	C₂×C₄	1	6	yes	—
10	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
11	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
12	C₄×#88322	—	2	3	no	—
13	C₄×#88322	—	2	3	no	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₃	—	2	6	no	—
17	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
18	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₄²	1	3	yes	—
20	C₂×C₈×C₅₃	—	2	6	no	—
21	C₂×C₈×C₅₃	—	2	6	no	—
22	C₂×C₈×C₅₃	—	2	6	no	—
23	C₄²×C₅₃	#41	1	3	yes	—
24	C₄²×C₅₃	#41	1	1	yes	characteristic
25	C₄×Dic₅₃	—	2	6	no	—
26	C₄×Dic₅₃	—	2	6	no	—
27	C₂×#88322	—	2	6	no	—
28	C₂×#88322	—	2	6	no	—
29	C₂×#88322	—	2	6	no	—
30	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₅₃	#117	1	3	yes	—
32	C₂×C₄×C₅₃	#117	1	3	yes	—
33	C₂×C₄×C₅₃	#131	1	6	yes	—
34	C₂×C₄×C₅₃	#131	1	6	yes	—
35	C₂×Dic₅₃	—	2	3	no	—
36	C₂×Dic₅₃	—	2	3	no	—
37	C₂×Dic₅₃	—	2	6	no	—
38	C₈×C₅₃	—	4	6	no	—
39	#88322	—	4	6	no	—
40	#8135	—	53	1	no	maximal, Sylow
41	C₂²×C₅₃	#376	1	1	yes	characteristic
42	C₂²×C₅₃	#405	1	3	yes	—
43	C₂²×C₅₃	#405	1	3	yes	—
44	C₄×C₅₃	#433	1	6	yes	—
45	C₄×C₅₃	#433	1	6	yes	—
46	Dic₅₃	—	4	1	no	—
47	Dic₅₃	—	4	3	no	—
48	#1744	—	53	3	no	—
49	#1273	—	53	3	no	—
50	#1273	D₅₃	1	1	yes	characteristic
51	C₂×C₅₃	#1454	1	1	yes	characteristic
52	C₂×C₅₃	#1454	1	3	yes	—
53	C₂×C₅₃	#1746	1	3	yes	—
54	C₂×C₄×C₈	—	53	3	no	—
55	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
56	C₄³	—	53	1	no	—
57	#336	—	53	6	no	—
58	#336	—	53	6	no	—
59	C₅₃	#8135	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	53	6	no	—
61	C₂²×C₈	C₄×D₅₃	1	6	yes	—
62	C₂×C₄²	—	53	3	no	—
63	C₂×C₄²	—	53	3	no	—
64	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	—	106	3	no	—
67	C₄×C₈	—	106	3	no	—
68	C₂²×C₄	—	53	1	no	—
69	C₂²×C₄	—	53	3	no	—
70	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	2	6	no	—
73	C₂×C₈	—	2	6	no	—
74	C₂×C₈	—	106	6	no	—
75	C₂×C₈	—	106	6	no	—
76	C₂×C₈	—	106	6	no	—
77	C₄²	—	106	6	no	—
78	C₄²	—	106	6	no	—
79	C₄²	#11719947	1	3	yes	—
80	C₄²	#11719949	1	1	yes	characteristic
81	C₂³	C₄²×D₅₃	1	1	yes	characteristic
82	C₂×C₄	—	106	3	no	—
83	C₂×C₄	—	106	3	no	—
84	C₂×C₄	—	106	6	no	—
85	C₂×C₄	#13133173	1	3	yes	—
86	C₂×C₄	#13133175	1	3	yes	—
87	C₂×C₄	#13133223	1	6	yes	—
88	C₂×C₄	#13133225	1	6	yes	—
89	C₈	—	4	6	no	—
90	C₈	—	212	6	no	—
91	C₂²	#16810468	1	1	yes	characteristic
92	C₂²	#16810577	1	3	yes	—
93	C₂²	#16810580	1	3	yes	—
94	C₄	—	212	1	no	—
95	C₄	—	212	3	no	—
96	C₄	#16810681	1	6	yes	—
97	C₄	#16810683	1	6	yes	—
98	C₂	#20273014	1	1	yes	characteristic
99	C₂	#20273016	1	3	yes	—
100	C₂	#20274465	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274460

C₂

yes

maximal

#20272542

C₂

yes

maximal

C₅₃×#1273

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810377

C₄

yes

—

#16810377

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₄×#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₄²

yes

characteristic

C₂²×Dic₅₃

C₄²

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#433

yes

—

Dic₅₃

—

Dic₅₃

—

#1744

—

#1273

—

#1273

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1746

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

—

#336

—

#336

—

C₅₃

#8135

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

#11719947

yes

—

C₄²

#11719949

yes

characteristic

C₂³

C₄²×D₅₃

yes

characteristic

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

#13133173

yes

—

C₂×C₄

#13133175

yes

—

C₂×C₄

#13133223

yes

—

C₂×C₄

#13133225

yes

—

C₈

—

C₈

—

212

—

C₂²

#16810468

yes

characteristic

C₂²

#16810577

yes

—

C₂²

#16810580

yes

—

C₄

—

212

—

C₄

—

212

—

C₄

#16810681

yes

—

C₄

#16810683

yes

—

C₂

#20273014

yes

characteristic

C₂

#20273016

yes

—

100

C₂

#20274465

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂×C₄²
Fitting	C₅₃×#1273	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1273	D₅₃
hypocenter	C₅₃	#8135
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₅₃

commutator, socle, prime radical

C₂³×C₅₃

C₂×C₄²

Fitting

C₅₃×#1273

C₂

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
59. C₅₃	#8135	semidirect

H ≅ G/N

type

59. C₅₃

#8135

semidirect

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

subgroup	index
#20274460	2
#20272542	2
C₅₃×#1273	2
#8135	53

subgroup

index

#20274460

#20272542

C₅₃×#1273

#8135

Subgroups — #27274526

Conjugacy classes of subgroups

Complements

Maximal subgroups