Group #27270926 — subgroups

Subgroups — #27270926

Order 13568.

262 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13133157	C₂	1	2	yes	maximal
2	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133157	C₄	1	4	yes	—
4	#16810399	—	2	2	no	—
5	C₄×#11719908	—	2	2	no	—
6	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#322	—	2	2	no	—
9	#13133157	—	2	4	no	—
10	#13133157	C₈	1	8	yes	complemented
11	C₂×#11719908	—	2	4	no	—
12	C₂²×C₈×C₅₃	C₈	1	4	yes	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₅₃	—	2	2	no	—
16	C₄×C₈×C₅₃	—	2	2	no	—
17	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₄×C₈×C₅₃	D₄	1	2	yes	—
19	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₅₃	—	2	4	no	—
23	C₂×C₈×C₅₃	—	4	4	no	—
24	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
27	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
28	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄²×C₅₃	M₁₆	1	2	yes	—
31	#11719908	—	2	8	no	—
32	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
36	C₂×C₄×C₅₃	#97	1	2	yes	—
37	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
39	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
40	C₈×C₅₃	—	4	4	no	—
41	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
42	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
43	#7139	—	53	1	no	maximal, Sylow
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	C₈²	1	1	yes	characteristic
46	C₂²×C₅₃	#322	1	1	yes	characteristic
47	C₂²×C₅₃	#336	1	1	yes	characteristic
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	#322	1	2	yes	—
50	C₄×C₅₃	#336	1	2	yes	—
51	C₄×C₅₃	#349	1	4	yes	—
52	C₄×M₃₂	—	53	2	no	—
53	C₂×#322	D₅₃	1	1	yes	characteristic
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	#1273	1	1	yes	characteristic
57	C₂×C₅₃	#1377	1	2	yes	—
58	C₂×M₃₂	—	53	4	no	—
59	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
60	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
61	#346	—	106	2	no	—
62	C₄×C₁₆	—	106	2	no	—
63	#322	—	2	2	no	—
64	C₅₃	#7139	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	#88322	1	4	yes	—
66	C₂²×C₈	C₄×D₅₃	1	2	yes	—
67	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
68	M₃₂	—	53	8	no	—
69	M₃₂	—	106	4	no	—
70	C₂×C₁₆	—	106	4	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
74	C₄×C₈	D₂₁₂	1	1	yes	characteristic
75	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
76	C₂²×C₄	C₂×#88322	1	2	yes	—
77	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	C₈×D₅₃	1	4	yes	—
81	C₂×C₈	#11719912	1	4	yes	—
82	C₂×C₈	#11719919	1	2	yes	—
83	C₂×C₈	#11719921	1	2	yes	—
84	C₄²	#11719917	1	2	yes	—
85	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
86	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
87	C₁₆	—	106	8	no	—
88	C₂³	C₄×#88322	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	#13133124	1	1	yes	characteristic
91	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
92	C₂×C₄	#13133127	1	2	yes	—
93	C₂×C₄	#13133128	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊#88322	1	2	yes	—
95	C₂×C₄	C₄⋊#88322	1	2	yes	—
96	C₈	—	4	4	no	—
97	C₈	#13133129	1	4	yes	—
98	C₈	#13133146	1	4	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	C₈×#88322	1	1	yes	characteristic
101	C₂²	#16810344	1	1	yes	characteristic
102	C₂²	#16810378	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	#16810345	1	2	yes	—
105	C₄	#16810377	1	2	yes	—
106	C₄	#16810405	1	4	yes	—
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	#20272542	1	1	yes	characteristic
110	C₂	#20272799	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133157

C₂

yes

maximal

C₂×C₅₃×#322

C₂

yes

maximal, characteristic

C₂×#13133157

C₄

yes

—

#16810399

—

C₄×#11719908

—

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

—

#13133157

—

#13133157

C₈

yes

complemented

C₂×#11719908

—

C₂²×C₈×C₅₃

C₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

M₁₆

yes

—

#11719908

—

C₂³×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₈

yes

—

C₈×C₅₃

C₄⋊C₈

yes

—

#7139

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₈²

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

#322

yes

—

C₄×C₅₃

#336

yes

—

C₄×C₅₃

#349

yes

—

C₄×M₃₂

—

C₂×#322

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1273

yes

characteristic

C₂×C₅₃

#1377

yes

—

C₂×M₃₂

—

C₂×C₄×C₈

Dic₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#346

—

106

—

C₄×C₁₆

—

106

—

#322

—

C₅₃

#7139

yes

characteristic, complemented, Sylow

C₂²×C₈

#88322

yes

—

C₂²×C₈

C₄×D₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₀₆

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂×#88322

yes

—

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₅₃

yes

—

C₂×C₈

#11719912

yes

—

C₂×C₈

#11719919

yes

—

C₂×C₈

#11719921

yes

—

C₄²

#11719917

yes

—

C₄²

C₄⋊Dic₅₃

yes

characteristic

C₄²

C₂²⋊Dic₅₃

yes

characteristic

C₁₆

—

106

—

C₂³

C₄×#88322

yes

characteristic

C₂×C₄

—

C₂×C₄

#13133124

yes

characteristic

C₂×C₄

C₈×Dic₅₃

yes

—

C₂×C₄

#13133127

yes

—

C₂×C₄

#13133128

yes

characteristic

C₂×C₄

C₂²⋊#88322

yes

—

C₂×C₄

C₄⋊#88322

yes

—

C₈

—

C₈

#13133129

yes

—

C₈

#13133146

yes

—

C₂²

—

100

C₂²

C₈×#88322

yes

characteristic

101

C₂²

#16810344

yes

characteristic

102

C₂²

#16810378

yes

characteristic

103

C₄

—

104

C₄

#16810345

yes

—

105

C₄

#16810377

yes

—

106

C₄

#16810405

yes

—

107

C₂

—

108

C₂

—

109

C₂

#20272542

yes

characteristic

110

C₂

#20272799

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₈²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#7139
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

subgroup

G/H

center

C₄²

C₂²⋊Dic₅₃

commutator

C₂²×C₅₃

C₈²

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₂×C₅₃×#322

C₂

socle

C₂²×C₅₃

#336

prime radical

C₂³×C₅₃

C₄×C₈

cosocle

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

N	H ≅ G/N	type
10. #13133157	C₈	semidirect
64. C₅₃	#7139	semidirect

H ≅ G/N

type

10. #13133157

C₈

semidirect

64. C₅₃

#7139

semidirect

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

subgroup	index
C₄×#13133157	2
C₂×C₅₃×#322	2
#7139	53

subgroup

index

C₄×#13133157

C₂×C₅₃×#322

#7139

Subgroups — #27270926

Conjugacy classes of subgroups

Complements

Maximal subgroups