Group #27270968 — subgroups

Subgroups — #27270968

Order 13568.

214 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20273463	C₂	1	2	yes	maximal
2	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
3	#16810676	—	2	2	no	—
4	#16810376	—	2	2	no	—
5	#16810374	C₄	1	4	yes	—
6	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#346	—	2	2	no	—
9	C₄⋊#88322	—	2	4	no	—
10	C₂²⋊#88322	—	4	2	no	—
11	C₄×#88322	—	4	2	no	—
12	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	—	2	2	no	—
16	C₂×C₁₆×C₅₃	—	2	2	no	—
17	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
29	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₅₃	SD₁₆	1	2	yes	—
31	C₂×#88322	—	4	4	no	—
32	C₁₆×C₅₃	—	2	4	no	—
33	C₂³×C₅₃	#107	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#102	1	2	yes	—
38	C₂×C₄×C₅₃	#103	1	2	yes	—
39	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
40	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#108	1	2	yes	—
42	C₈×C₅₃	—	2	2	no	—
43	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
44	C₈×C₅₃	#97	1	1	yes	characteristic
45	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
46	#88322	—	8	4	no	—
47	#7157	—	53	1	no	maximal, Sylow
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	—	2	1	no	—
50	C₂²×C₅₃	#335	1	2	yes	—
51	C₂²×C₅₃	#340	1	1	yes	characteristic
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	#334	1	2	yes	—
55	C₄×C₅₃	#336	1	1	yes	characteristic
56	C₄×C₅₃	#341	1	1	yes	characteristic
57	C₂×#346	D₅₃	1	1	yes	characteristic
58	#1564	—	53	2	no	—
59	C₂×C₅₃	—	2	2	no	—
60	C₂×C₅₃	#1277	1	1	yes	characteristic
61	C₂×C₅₃	#1384	1	1	yes	characteristic
62	C₂×C₅₃	#1385	1	1	yes	characteristic
63	C₂²×C₁₆	Dic₅₃	1	2	yes	—
64	#431	—	106	2	no	—
65	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
66	#346	—	2	2	no	—
67	#335	—	106	2	no	—
68	#334	—	53	4	no	—
69	C₅₃	#7157	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
71	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
72	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	#88322	1	4	yes	complemented
76	C₄⋊C₈	—	106	4	no	—
77	C₂²⋊C₈	—	212	2	no	—
78	C₄×C₈	—	212	2	no	—
79	C₄×C₈	C₄×D₅₃	1	2	yes	—
80	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
81	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
82	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
83	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	212	4	no	—
86	C₂×C₈	C₂×#88322	1	2	yes	—
87	C₂×C₈	#11719917	1	2	yes	—
88	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
89	C₂×C₈	#11719919	1	2	yes	—
90	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
91	C₄²	#11719913	1	2	yes	—
92	C₄²	#11719921	1	2	yes	—
93	C₁₆	—	2	4	no	—
94	C₂³	#13133151	1	1	yes	characteristic
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	#13133124	1	1	yes	characteristic
98	C₂×C₄	#13133139	1	2	yes	—
99	C₂×C₄	#13133142	1	2	yes	—
100	C₂×C₄	C₄⋊#88322	1	2	yes	—
101	C₂×C₄	#13133149	1	1	yes	characteristic
102	C₂×C₄	#13133153	1	2	yes	—
103	C₈	—	2	2	no	—
104	C₈	—	424	4	no	—
105	C₈	C₄×#88322	1	1	yes	characteristic
106	C₈	#13133128	1	1	yes	characteristic
107	C₈	C₂²⋊#88322	1	2	yes	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	#16810376	1	2	yes	—
111	C₂²	#16810386	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	2	no	—
114	C₄	#16810374	1	2	yes	—
115	C₄	#16810378	1	1	yes	characteristic
116	C₄	#16810388	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#20272550	1	1	yes	characteristic
119	C₂	#20272817	1	1	yes	characteristic
120	C₂	#20272819	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273463

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

#16810676

—

#16810376

—

#16810374

C₄

yes

—

C₂²×C₁₆×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

—

C₄⋊#88322

—

C₂²⋊#88322

—

C₄×#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₈

yes

complemented

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

SD₁₆

yes

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

#107

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

#108

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

characteristic

C₈×C₅₃

#97

yes

characteristic

C₈×C₅₃

C₂²⋊C₈

yes

—

#88322

—

#7157

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#335

yes

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#334

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1564

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

Dic₅₃

yes

—

#431

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#346

—

#335

—

106

—

#334

—

C₅₃

#7157

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#88322

yes

complemented

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

212

—

C₄×C₈

—

212

—

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

C₂²⋊D₅₃

yes

—

C₂²×C₄

D₄₂₄

yes

characteristic

C₂²×C₄

Dic₂₁₂

yes

characteristic

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#88322

yes

—

C₂×C₈

#11719917

yes

—

C₂×C₈

C₄×Dic₅₃

yes

characteristic

C₂×C₈

#11719919

yes

—

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

C₄²

#11719913

yes

—

C₄²

#11719921

yes

—

C₁₆

—

C₂³

#13133151

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13133124

yes

characteristic

C₂×C₄

#13133139

yes

—

C₂×C₄

#13133142

yes

—

100

C₂×C₄

C₄⋊#88322

yes

—

101

C₂×C₄

#13133149

yes

characteristic

102

C₂×C₄

#13133153

yes

—

103

C₈

—

104

C₈

—

424

—

105

C₈

C₄×#88322

yes

characteristic

106

C₈

#13133128

yes

characteristic

107

C₈

C₂²⋊#88322

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

#16810376

yes

—

111

C₂²

#16810386

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#16810374

yes

—

115

C₄

#16810378

yes

characteristic

116

C₄

#16810388

yes

characteristic

117

C₂

—

118

C₂

#20272550

yes

characteristic

119

C₂

#20272817

yes

characteristic

120

C₂

#20272819

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133149
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#107
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#7157
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

#13133149

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₂×C₅₃×#346

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
17. C₂×C₁₆×C₅₃	C₈	semidirect
69. C₅₃	#7157	semidirect
75. C₂×C₁₆	#88322	semidirect

H ≅ G/N

type

17. C₂×C₁₆×C₅₃

C₈

semidirect

69. C₅₃

#7157

semidirect

75. C₂×C₁₆

#88322

semidirect

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

subgroup	index
#20273463	2
C₂×C₅₃×#346	2
#7157	53

subgroup

index

#20273463

C₂×C₅₃×#346

#7157

Subgroups — #27270968

Conjugacy classes of subgroups

Complements

Maximal subgroups