Group #27270922 — subgroups

Subgroups — #27270922

Order 13568.

266 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₂×#16810399	C₂	1	2	yes	maximal
2	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719908	C₄	1	4	yes	—
4	#16810399	—	2	2	no	—
5	#16810399	—	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	C₂×#11719908	—	2	4	no	—
10	C₂×#11719908	—	2	4	no	—
11	C₂×#11719908	C₈	1	8	yes	complemented
12	C₂²×C₈×C₅₃	—	2	2	no	—
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	C₈	1	4	yes	—
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	M₁₆	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	C₂×C₈	1	2	yes	—
31	#11719908	—	2	8	no	—
32	C₂³×C₅₃	#97	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	C₄×C₈	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#97	1	2	yes	—
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	#7137	—	53	1	no	maximal, Sylow
44	C₂²×C₅₃	#322	1	2	yes	—
45	C₂²×C₅₃	#336	1	1	yes	characteristic
46	C₂²×C₅₃	#347	1	4	yes	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	C₈²	1	1	yes	characteristic
50	C₄×C₅₃	#322	1	1	yes	characteristic
51	C₄×C₅₃	#336	1	2	yes	—
52	C₂×#346	—	53	2	no	—
53	C₂×#322	D₅₃	1	1	yes	characteristic
54	C₂×C₅₃	#1273	1	1	yes	characteristic
55	C₂×C₅₃	#1313	1	4	yes	—
56	C₂×C₅₃	#1375	1	2	yes	—
57	C₂²×C₁₆	—	53	4	no	—
58	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
59	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
60	#346	—	106	2	no	—
61	#346	—	106	2	no	—
62	#322	—	2	2	no	—
63	C₅₃	#7137	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	C₄×D₅₃	1	2	yes	—
66	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
67	C₂×C₁₆	—	53	8	no	—
68	C₂×C₁₆	—	106	4	no	—
69	C₂×C₁₆	—	106	4	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	#88322	1	4	yes	—
72	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
73	C₄×C₈	D₂₁₂	1	1	yes	characteristic
74	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
75	C₂²×C₄	#11719917	1	2	yes	—
76	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	C₈×D₅₃	1	4	yes	—
80	C₂×C₈	#11719912	1	4	yes	—
81	C₂×C₈	#11719919	1	2	yes	—
82	C₂×C₈	#11719921	1	2	yes	—
83	C₄²	C₂×#88322	1	2	yes	—
84	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
85	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
86	C₁₆	—	106	8	no	—
87	C₂³	#13133128	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	#13133124	1	1	yes	characteristic
90	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
91	C₂×C₄	C₄×#88322	1	1	yes	characteristic
92	C₂×C₄	#13133127	1	2	yes	—
93	C₂×C₄	C₂²⋊#88322	1	2	yes	—
94	C₂×C₄	C₄⋊#88322	1	2	yes	—
95	C₈	—	4	4	no	—
96	C₈	#13133129	1	4	yes	—
97	C₈	#13133146	1	4	yes	—
98	C₂²	#16810345	1	2	yes	—
99	C₂²	#16810378	1	1	yes	characteristic
100	C₂²	#16810401	1	4	yes	—
101	C₄	—	2	2	no	—
102	C₄	—	2	2	no	—
103	C₄	C₈×#88322	1	1	yes	characteristic
104	C₄	#16810344	1	1	yes	characteristic
105	C₄	#16810377	1	2	yes	—
106	C₂	#20272542	1	1	yes	characteristic
107	C₂	#20272636	1	4	yes	—
108	C₂	#20272795	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810399

C₂

yes

maximal

C₂×C₅₃×#322

C₂

yes

maximal, characteristic

C₂²×#11719908

C₄

yes

—

#16810399

—

#16810399

—

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

—

C₂×#11719908

—

C₂×#11719908

—

C₂×#11719908

C₈

yes

complemented

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

—

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

M₁₆

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

C₂×C₈

yes

—

#11719908

—

C₂³×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₈

yes

—

C₈×C₅₃

C₄⋊C₈

yes

—

#7137

—

maximal, Sylow

C₂²×C₅₃

#322

yes

—

C₂²×C₅₃

#336

yes

characteristic

C₂²×C₅₃

#347

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₈²

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#346

—

C₂×#322

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1375

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#346

—

106

—

#346

—

106

—

#322

—

C₅₃

#7137

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

#88322

yes

—

C₄×C₈

Dic₁₀₆

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

C₂²⋊D₅₃

yes

—

C₂²×C₄

#11719917

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

C₂×#88322

yes

—

C₄²

C₄⋊Dic₅₃

yes

characteristic

C₄²

C₂²⋊Dic₅₃

yes

characteristic

C₁₆

—

106

—

C₂³

#13133128

yes

characteristic

C₂×C₄

—

C₂×C₄

#13133124

yes

characteristic

C₂×C₄

C₈×Dic₅₃

yes

—

C₂×C₄

C₄×#88322

yes

characteristic

C₂×C₄

#13133127

yes

—

C₂×C₄

C₂²⋊#88322

yes

—

C₂×C₄

C₄⋊#88322

yes

—

C₈

—

C₈

#13133129

yes

—

C₈

#13133146

yes

—

C₂²

#16810345

yes

—

C₂²

#16810378

yes

characteristic

100

C₂²

#16810401

yes

—

101

C₄

—

102

C₄

—

103

C₄

C₈×#88322

yes

characteristic

104

C₄

#16810344

yes

characteristic

105

C₄

#16810377

yes

—

106

C₂

#20272542

yes

characteristic

107

C₂

#20272636

yes

—

108

C₂

#20272795

yes

—

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₅₃
commutator	C₄×C₅₃	C₈²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	#97
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#7137
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#322

subgroup

G/H

center

C₂²×C₄

C₄×Dic₅₃

commutator

C₄×C₅₃

C₈²

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₂×C₅₃×#322

C₂

socle, prime radical

C₂³×C₅₃

#97

cosocle

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×#322

N	H ≅ G/N	type
11. C₂×#11719908	C₈	semidirect
63. C₅₃	#7137	semidirect

H ≅ G/N

type

11. C₂×#11719908

C₈

semidirect

63. C₅₃

#7137

semidirect

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

subgroup	index
C₂×#16810399	2
C₂×C₅₃×#322	2
#7137	53

subgroup

index

C₂×#16810399

C₂×C₅₃×#322

#7137

Subgroups — #27270922

Conjugacy classes of subgroups

Complements

Maximal subgroups