Group #27271193 — subgroups

Subgroups — #27271193

Order 13568.

190 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20273478	C₂	1	2	yes	maximal
2	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#16810680	—	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×C₄⋊C₁₆	—	2	2	no	—
9	C₄⋊#88322	—	2	4	no	—
10	C₂²⋊#88322	—	4	2	no	—
11	#13133128	—	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	—	4	2	no	—
17	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	—	4	1	no	—
25	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
29	C₄²×C₅₃	SD₁₆	1	2	yes	—
30	C₂×#88322	—	4	4	no	—
31	C₁₆×C₅₃	—	4	2	no	—
32	C₂³×C₅₃	#107	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	2	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#102	1	2	yes	—
39	C₂×C₄×C₅₃	#103	1	2	yes	—
40	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
41	C₈×C₅₃	—	2	2	no	—
42	C₈×C₅₃	—	2	1	no	—
43	C₈×C₅₃	#100	1	1	yes	characteristic
44	C₈×C₅₃	#101	1	1	yes	characteristic
45	#88322	—	8	4	no	—
46	#7242	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	#340	1	1	yes	characteristic
50	C₂²×C₅₃	#368	1	2	yes	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	#338	1	1	yes	characteristic
55	C₄×C₅₃	#343	1	1	yes	characteristic
56	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
57	#1568	—	53	2	no	—
58	C₂×C₅₃	—	2	2	no	—
59	C₂×C₅₃	#1292	1	1	yes	characteristic
60	C₂×C₅₃	#1381	1	1	yes	characteristic
61	C₂×C₅₃	#1384	1	1	yes	characteristic
62	C₂²×C₁₆	Dic₅₃	1	2	yes	—
63	#432	—	106	2	no	—
64	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	2	no	—
66	#335	—	106	2	no	—
67	#334	—	53	4	no	—
68	C₅₃	#7242	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
70	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
71	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₄⋊C₈	—	106	4	no	—
75	C₂²⋊C₈	—	212	2	no	—
76	#97	—	212	2	no	—
77	C₄×C₈	C₄×D₅₃	1	2	yes	—
78	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
79	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
80	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
81	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	212	4	no	—
86	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
87	C₂×C₈	#11719919	1	2	yes	—
88	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
89	C₄²	#11719913	1	2	yes	—
90	C₄²	#11719921	1	2	yes	—
91	C₁₆	—	4	2	no	—
92	C₂³	#13133151	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	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	#13133149	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	2	1	no	—
103	C₈	—	424	4	no	—
104	C₈	#13133134	1	1	yes	characteristic
105	C₈	#13133136	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#16810386	1	1	yes	characteristic
109	C₂²	#16810453	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	#16810381	1	1	yes	characteristic
114	C₄	#16810392	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#20272590	1	1	yes	characteristic
117	C₂	#20272809	1	1	yes	characteristic
118	C₂	#20272817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273478

C₂

yes

maximal

C₂×C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#16810680

—

#16810376

—

#16810374

C₄

yes

—

C₂²×C₁₆×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₄⋊#88322

—

C₂²⋊#88322

—

#13133128

—

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

—

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

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₂²×C₅₃

#368

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₄⋊C₁₆

D₅₃

yes

characteristic

#1568

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

Dic₅₃

yes

—

#432

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

#335

—

106

—

#334

—

C₅₃

#7242

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

212

—

#97

—

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

—

C₂×C₈

—

C₂×C₈

—

212

—

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

—

C₂×C₄

—

C₂×C₄

#13133124

yes

characteristic

C₂×C₄

#13133139

yes

—

C₂×C₄

#13133142

yes

—

100

C₂×C₄

#13133149

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

424

—

104

C₈

#13133134

yes

characteristic

105

C₈

#13133136

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

#16810386

yes

characteristic

109

C₂²

#16810453

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#16810381

yes

characteristic

114

C₄

#16810392

yes

characteristic

115

C₂

—

116

C₂

#20272590

yes

characteristic

117

C₂

#20272809

yes

characteristic

118

C₂

#20272817

yes

characteristic

	subgroup	G/H
center	C₂²	#16810386
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#107
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#7242
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#16810386

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₂×C₅₃×C₄⋊C₁₆

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×C₈×C₅₃

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₅₃

hypocenter

C₅₃

#7242

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
68. C₅₃	#7242	semidirect

H ≅ G/N

type

68. C₅₃

#7242

semidirect

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

subgroup	index
#20273478	2
C₂×C₅₃×C₄⋊C₁₆	2
#7242	53

subgroup

index

#20273478

C₂×C₅₃×C₄⋊C₁₆

#7242

Subgroups — #27271193

Conjugacy classes of subgroups

Complements

Maximal subgroups