Group #27274528 — subgroups

Subgroups — #27274528

Order 13568.

478 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 203 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	C₄×#13133127	C₂	1	3	yes	maximal
2	#20272542	C₂	1	3	yes	maximal
3	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133127	C₄	1	6	yes	—
5	C₂×#13133127	C₄	1	6	yes	—
6	C₂×C₄×#88322	C₂²	1	3	yes	—
7	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
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	#13133127	—	2	12	no	—
13	#13133127	—	2	12	no	—
14	C₄×#88322	—	2	3	no	—
15	C₄×#88322	—	2	3	no	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₅₃	—	2	6	no	—
19	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
20	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
21	C₂²×Dic₅₃	C₄²	1	3	yes	—
22	C₂×C₈×C₅₃	—	2	6	no	—
23	C₂×C₈×C₅₃	—	2	6	no	—
24	C₂×C₈×C₅₃	M₁₆	1	12	yes	—
25	C₄²×C₅₃	#41	1	3	yes	—
26	C₄²×C₅₃	#41	1	1	yes	characteristic
27	C₄×Dic₅₃	—	2	6	no	—
28	C₄×Dic₅₃	—	2	6	no	—
29	C₂×#88322	—	2	6	no	—
30	C₂×#88322	—	2	6	no	—
31	C₂×#88322	M₁₆	1	12	yes	—
32	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
33	C₂×C₄×C₅₃	#117	1	3	yes	—
34	C₂×C₄×C₅₃	#117	1	3	yes	—
35	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
36	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
37	C₂×Dic₅₃	—	2	3	no	—
38	C₂×Dic₅₃	—	2	3	no	—
39	C₂×Dic₅₃	—	2	6	no	—
40	C₈×C₅₃	—	2	12	no	—
41	#88322	—	2	12	no	—
42	#8136	—	53	1	no	maximal, Sylow
43	C₂²×C₅₃	#376	1	1	yes	characteristic
44	C₂²×C₅₃	C₄×M₁₆	1	3	yes	—
45	C₂²×C₅₃	C₄×M₁₆	1	3	yes	—
46	C₄×C₅₃	#432	1	6	yes	—
47	C₄×C₅₃	#432	1	6	yes	—
48	Dic₅₃	—	4	1	no	—
49	Dic₅₃	—	4	3	no	—
50	C₄×#97	—	53	3	no	—
51	#1273	—	53	3	no	—
52	#1273	D₅₃	1	1	yes	characteristic
53	C₂×C₅₃	#1455	1	1	yes	characteristic
54	C₂×C₅₃	#1455	1	3	yes	—
55	C₂×C₅₃	#1745	1	3	yes	—
56	C₂×#97	—	53	6	no	—
57	C₂×#97	—	53	6	no	—
58	C₂×C₄×C₈	—	53	3	no	—
59	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
60	C₄³	—	53	1	no	—
61	C₅₃	#8136	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₈	—	53	6	no	—
63	C₂²×C₈	C₄×D₅₃	1	6	yes	—
64	C₂×C₄²	—	53	3	no	—
65	C₂×C₄²	—	53	3	no	—
66	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
67	#97	—	106	12	no	—
68	#97	—	106	12	no	—
69	C₄×C₈	—	2	6	no	—
70	C₄×C₈	—	106	3	no	—
71	C₄×C₈	—	106	3	no	—
72	C₂²×C₄	—	53	1	no	—
73	C₂²×C₄	—	53	3	no	—
74	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
75	C₂×C₈	—	2	6	no	—
76	C₂×C₈	—	2	6	no	—
77	C₂×C₈	—	53	12	no	—
78	C₂×C₈	—	106	6	no	—
79	C₂×C₈	—	106	6	no	—
80	C₂×C₈	#11719912	1	12	yes	—
81	C₄²	—	106	6	no	—
82	C₄²	—	106	6	no	—
83	C₄²	#11719947	1	3	yes	—
84	C₄²	#11719949	1	1	yes	characteristic
85	C₂³	C₄²×D₅₃	1	1	yes	characteristic
86	C₂×C₄	—	106	3	no	—
87	C₂×C₄	—	106	3	no	—
88	C₂×C₄	—	106	6	no	—
89	C₂×C₄	#13133173	1	3	yes	—
90	C₂×C₄	#13133175	1	3	yes	—
91	C₂×C₄	M₁₆×D₅₃	1	6	yes	—
92	C₂×C₄	C₂×#11719912	1	6	yes	—
93	C₈	—	2	12	no	—
94	C₈	—	106	12	no	—
95	C₂²	#16810468	1	1	yes	characteristic
96	C₂²	C₄×#11719912	1	3	yes	—
97	C₂²	#16810575	1	3	yes	—
98	C₄	—	212	1	no	—
99	C₄	—	212	3	no	—
100	C₄	#16810677	1	6	yes	—
101	C₄	#16810679	1	6	yes	—
102	C₂	#20273017	1	1	yes	characteristic
103	C₂	#20273019	1	3	yes	—
104	C₂	#20274462	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133127

C₂

yes

maximal

#20272542

C₂

yes

maximal

C₅₃×#1273

C₂

yes

maximal, characteristic

C₂×#13133127

C₄

yes

—

C₂×#13133127

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

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

M₁₆

yes

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

M₁₆

yes

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

C₄×M₁₆

yes

—

C₂²×C₅₃

C₄×M₁₆

yes

—

C₄×C₅₃

#432

yes

—

C₄×C₅₃

#432

yes

—

Dic₅₃

—

Dic₅₃

—

C₄×#97

—

#1273

—

#1273

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1745

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

—

C₅₃

#8136

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#97

—

106

—

#97

—

106

—

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

#11719912

yes

—

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

M₁₆×D₅₃

yes

—

C₂×C₄

C₂×#11719912

yes

—

C₈

—

C₈

—

106

—

C₂²

#16810468

yes

characteristic

C₂²

C₄×#11719912

yes

—

C₂²

#16810575

yes

—

C₄

—

212

—

C₄

—

212

—

100

C₄

#16810677

yes

—

101

C₄

#16810679

yes

—

102

C₂

#20273017

yes

characteristic

103

C₂

#20273019

yes

—

104

C₂

#20274462

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	#8136
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
61. C₅₃	#8136	semidirect

H ≅ G/N

type

61. C₅₃

#8136

semidirect

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

subgroup	index
C₄×#13133127	2
#20272542	2
C₅₃×#1273	2
#8136	53

subgroup

index

C₄×#13133127

#20272542

C₅₃×#1273

#8136

Subgroups — #27274528

Conjugacy classes of subgroups

Complements

Maximal subgroups