Group #27273406 — subgroups

Subgroups — #27273406

Order 13568.

322 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20276278	C₂	1	3	yes	maximal
2	#20272538	C₂	1	3	yes	maximal
3	C₅₃×#2810	C₂	1	1	yes	maximal, characteristic
4	#16810493	C₂²	1	1	yes	characteristic
5	#16810385	—	2	3	no	—
6	#16810385	—	2	3	no	—
7	#16810378	C₂²	1	3	yes	—
8	C₅₃×#400	—	2	3	no	—
9	C₅₃×#389	C₂²	1	3	yes	—
10	C₅₃×#387	—	2	3	no	—
11	C₂²×#88322	—	2	3	no	—
12	C₂×C₄⋊Dic₅₃	—	2	3	no	—
13	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
14	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
15	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
16	C₂×C₄²×C₅₃	—	2	1	no	—
17	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	#95×C₅₃	—	2	3	no	—
19	#95×C₅₃	—	2	3	no	—
20	#95×C₅₃	—	2	1	no	—
21	#95×C₅₃	—	2	3	no	—
22	C₂²×C₄×C₅₃	—	2	3	no	—
23	C₂²×C₄×C₅₃	—	2	3	no	—
24	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
25	C₂²×Dic₅₃	—	4	1	no	—
26	C₄⋊C₄×C₅₃	—	2	3	no	—
27	C₄⋊C₄×C₅₃	—	2	3	no	—
28	C₄⋊C₄×C₅₃	—	4	3	no	—
29	C₄⋊C₄×C₅₃	—	4	3	no	—
30	C₄⋊C₄×C₅₃	—	4	3	no	—
31	C₄²×C₅₃	—	4	1	no	—
32	C₄²×C₅₃	—	8	1	no	—
33	C₄⋊Dic₅₃	—	4	6	no	—
34	C₂×#88322	—	4	3	no	—
35	C₂×#88322	—	4	3	no	—
36	C₂×#88322	—	4	3	no	—
37	C₂³×C₅₃	#120	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	3	no	—
39	C₂×C₄×C₅₃	—	2	3	no	—
40	C₂×C₄×C₅₃	—	4	3	no	—
41	C₂×C₄×C₅₃	—	4	3	no	—
42	C₂×C₄×C₅₃	—	4	3	no	—
43	C₂×C₄×C₅₃	—	4	3	no	—
44	C₂×C₄×C₅₃	—	4	3	no	—
45	C₂×C₄×C₅₃	—	4	3	no	—
46	C₂×C₄×C₅₃	#135	1	3	yes	—
47	C₂×C₄×C₅₃	#137	1	3	yes	—
48	C₂×Dic₅₃	—	8	3	no	—
49	#88322	—	8	3	no	—
50	#7832	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	#452	1	3	yes	—
52	C₂²×C₅₃	#454	1	3	yes	—
53	C₂²×C₅₃	#458	1	1	yes	characteristic
54	C₄×C₅₃	—	2	3	no	—
55	C₄×C₅₃	—	2	3	no	—
56	C₄×C₅₃	—	8	3	no	—
57	C₄×C₅₃	—	8	3	no	—
58	Dic₅₃	—	16	1	no	—
59	#2810	D₅₃	1	1	yes	characteristic
60	#2100	—	53	3	no	—
61	#1271	—	53	3	no	—
62	C₂×C₅₃	#1615	1	3	yes	—
63	C₂×C₅₃	#1638	1	3	yes	—
64	C₂×C₅₃	#1658	1	1	yes	characteristic
65	#400	—	2	3	no	—
66	#389	C₂×D₅₃	1	3	yes	—
67	#387	—	2	3	no	—
68	#384	—	53	1	no	—
69	#340	—	106	3	no	—
70	#340	—	106	3	no	—
71	#336	—	53	3	no	—
72	C₅₃	#7832	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	106	3	no	—
74	C₂×C₄⋊C₄	—	2	3	no	—
75	C₂×C₄⋊C₄	—	106	3	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
77	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
80	#95	—	2	3	no	—
81	#95	—	2	3	no	—
82	#95	—	2	1	no	—
83	#95	—	2	3	no	—
84	C₂²×C₄	—	2	3	no	—
85	C₂²×C₄	—	2	3	no	—
86	C₂²×C₄	—	212	1	no	—
87	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
88	C₂×C₈	—	212	3	no	—
89	C₂×C₈	—	212	3	no	—
90	C₂×C₈	—	212	3	no	—
91	C₄⋊C₄	—	2	3	no	—
92	C₄⋊C₄	—	2	3	no	—
93	C₄⋊C₄	—	4	3	no	—
94	C₄⋊C₄	—	4	3	no	—
95	C₄⋊C₄	—	4	3	no	—
96	C₄⋊C₄	—	212	6	no	—
97	C₄²	—	4	1	no	—
98	C₄²	—	8	1	no	—
99	C₂³	#13133186	1	1	yes	characteristic
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	4	3	no	—
103	C₂×C₄	—	4	3	no	—
104	C₂×C₄	—	4	3	no	—
105	C₂×C₄	—	4	3	no	—
106	C₂×C₄	—	4	3	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	—	424	3	no	—
109	C₂×C₄	#13133237	1	3	yes	—
110	C₂×C₄	#13133247	1	3	yes	—
111	C₈	—	424	3	no	—
112	C₂²	#16810785	1	3	yes	—
113	C₂²	#16810796	1	3	yes	—
114	C₂²	#16810813	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	2	3	no	—
117	C₄	—	8	3	no	—
118	C₄	—	8	3	no	—
119	C₄	—	848	1	no	—
120	C₂	#20273746	1	3	yes	—
121	C₂	#20273887	1	3	yes	—
122	C₂	#20274017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276278

C₂

yes

maximal

#20272538

C₂

yes

maximal

C₅₃×#2810

C₂

yes

maximal, characteristic

#16810493

C₂²

yes

—

—

C₂²

yes

—

C₅₃×#400

—

C₅₃×#389

C₂²

yes

—

C₅₃×#387

—

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

—

—

C₂²×C₅₃

yes

—

C₂²×C₅₃

#454

yes

—

C₂²×C₅₃

#458

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#2810

D₅₃

yes

characteristic

#2100

—

#1271

—

C₂×C₅₃

#1615

yes

—

C₂×C₅₃

#1638

yes

—

C₂×C₅₃

yes

—

C₂×D₅₃

yes

—

#387

—

#384

—

#340

—

106

—

#340

—

106

—

#336

—

C₅₃

#7832

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄²

—

C₄²

—

C₂³

#13133186

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

424

—

109

C₂×C₄

#13133237

yes

—

110

C₂×C₄

#13133247

yes

—

111

C₈

—

424

—

112

C₂²

#16810785

yes

—

113

C₂²

#16810796

yes

—

114

C₂²

#16810813

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

848

—

120

C₂

#20273746

yes

—

121

C₂

#20273887

yes

—

122

C₂

#20274017

yes

characteristic

	subgroup	G/H
center	C₂³	#13133186
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2810	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#2810	D₅₃
hypocenter	C₅₃	#7832
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2810

subgroup

G/H

center

C₂³

#13133186

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2810

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂³×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2810

N	H ≅ G/N	type
72. C₅₃	#7832	semidirect

H ≅ G/N

type

72. C₅₃

#7832

semidirect

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

subgroup	index
#20276278	2
#20272538	2
C₅₃×#2810	2
#7832	53

subgroup

index

#20276278

#20272538

C₅₃×#2810

#7832

Subgroups — #27273406

Conjugacy classes of subgroups

Complements

Maximal subgroups