Group #27286858 — subgroups

Subgroups — #27286858

Order 13568.

478 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20286109	C₂	1	3	yes	maximal
2	#20275691	C₂	1	3	yes	maximal
3	C₅₃×#1309	C₂	1	1	yes	maximal, characteristic
4	#16811180	C₂²	1	3	yes	—
5	#16810909	—	2	6	no	—
6	#16810881	—	2	6	no	—
7	#16810834	—	2	6	no	—
8	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
9	#16810384	—	2	6	no	—
10	C₅₃×#423	C₂²	1	3	yes	—
11	C₂×Dic₂₁₂	—	2	6	no	—
12	#13133217	D₄	1	6	yes	—
13	#13133209	—	2	3	no	—
14	#13133196	—	2	6	no	—
15	C₄×Dic₁₀₆	—	2	6	no	—
16	#13133176	—	2	3	no	—
17	C₂×C₄×Dic₅₃	—	2	3	no	—
18	#13133149	—	2	6	no	—
19	#13133142	—	2	12	no	—
20	C₂×M₁₆×C₅₃	D₄	1	6	yes	—
21	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	—	2	6	no	—
23	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
24	C₂²×Dic₅₃	—	4	1	no	—
25	C₂×Dic₁₀₆	—	2	6	no	—
26	M₁₆×C₅₃	—	4	6	no	—
27	C₂×C₈×C₅₃	—	2	6	no	—
28	C₄²×C₅₃	C₂×D₄	1	3	yes	—
29	C₄²×C₅₃	#41	1	1	yes	characteristic
30	C₂²⋊Dic₅₃	—	4	3	no	—
31	C₄⋊Dic₅₃	—	2	6	no	—
32	C₄⋊Dic₅₃	—	2	6	no	—
33	#11719919	—	4	6	no	—
34	C₄×Dic₅₃	—	2	6	no	—
35	C₄×Dic₅₃	—	4	3	no	—
36	C₄×Dic₅₃	—	4	3	no	—
37	Dic₂₁₂	—	8	6	no	—
38	C₂³×C₅₃	#127	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	3	no	—
40	C₂×C₄×C₅₃	—	2	3	no	—
41	C₂×C₄×C₅₃	#120	1	3	yes	—
42	C₂×C₄×C₅₃	#121	1	3	yes	—
43	C₂×Dic₅₃	—	4	6	no	—
44	C₂×Dic₅₃	—	4	6	no	—
45	Dic₁₀₆	—	4	6	no	—
46	Dic₁₀₆	—	4	6	no	—
47	C₈×C₅₃	—	4	6	no	—
48	#10303	—	53	1	no	maximal, Sylow
49	C₂²×C₅₃	—	2	3	no	—
50	C₂²×C₅₃	#392	1	1	yes	characteristic
51	C₄×C₅₃	—	2	3	no	—
52	C₄×C₅₃	#455	1	3	yes	—
53	C₄×C₅₃	#456	1	3	yes	—
54	Dic₅₃	—	4	4	no	—
55	Dic₅₃	—	8	6	no	—
56	#3255	—	53	3	no	—
57	#1994	—	53	3	no	—
58	#1309	D₅₃	1	1	yes	characteristic
59	C₂×C₅₃	—	4	1	no	—
60	C₂×C₅₃	#2007	1	3	yes	—
61	#533	—	53	3	no	—
62	#475	—	106	6	no	—
63	#470	—	106	6	no	—
64	#462	—	106	6	no	—
65	#423	C₂×D₅₃	1	3	yes	—
66	C₄³	—	53	1	no	—
67	#339	—	106	6	no	—
68	C₅₃	#10303	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₁₆	—	106	6	no	—
70	C₂×M₁₆	D₂₁₂	1	6	yes	—
71	C₄⋊Q₈	—	53	6	no	—
72	#125	—	106	3	no	—
73	C₂²⋊Q₈	—	106	6	no	—
74	C₄×Q₈	—	106	6	no	—
75	#117	—	106	3	no	—
76	C₂×C₄²	—	106	3	no	—
77	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
78	#106	—	106	6	no	—
79	C₄⋊C₈	—	2	6	no	—
80	#103	—	106	12	no	—
81	C₂×Q₈	—	106	6	no	—
82	C₂²×C₄	—	212	1	no	—
83	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
84	Q₁₆	—	424	6	no	—
85	M₁₆	—	4	6	no	—
86	C₂×C₈	—	2	6	no	—
87	C₄⋊C₄	—	106	6	no	—
88	C₄⋊C₄	—	106	6	no	—
89	C₄⋊C₄	—	212	6	no	—
90	C₂²⋊C₄	—	212	3	no	—
91	C₄²	—	106	6	no	—
92	C₄²	—	212	3	no	—
93	C₄²	—	212	3	no	—
94	C₄²	D₄×D₅₃	1	3	yes	—
95	C₄²	#11719949	1	1	yes	characteristic
96	C₂³	#13133214	1	1	yes	characteristic
97	Q₈	—	212	6	no	—
98	Q₈	—	212	6	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	212	6	no	—
102	C₂×C₄	—	212	6	no	—
103	C₂×C₄	#13133187	1	3	yes	—
104	C₂×C₄	#13133192	1	3	yes	—
105	C₈	—	4	6	no	—
106	C₂²	—	2	3	no	—
107	C₂²	#16810533	1	1	yes	characteristic
108	C₄	—	2	3	no	—
109	C₄	—	424	6	no	—
110	C₄	—	212	4	no	—
111	C₄	#16810803	1	3	yes	—
112	C₄	#16810808	1	3	yes	—
113	C₂	—	4	1	no	—
114	C₂	#20275764	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20286109

C₂

yes

maximal

#20275691

C₂

yes

maximal

C₅₃×#1309

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

—

#16810909

—

#16810881

—

#16810834

—

C₄²×Dic₅₃

C₂²

yes

characteristic

#16810384

—

C₅₃×#423

C₂²

yes

—

C₂×Dic₂₁₂

—

#13133217

D₄

yes

—

#13133209

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

—

C₂×C₄×Dic₅₃

—

#13133149

—

#13133142

—

C₂×M₁₆×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

Dic₂₁₂

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#10303

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#392

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#455

yes

—

C₄×C₅₃

#456

yes

—

Dic₅₃

—

Dic₅₃

—

#3255

—

#1994

—

#1309

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#2007

yes

—

#533

—

#475

—

106

—

#470

—

106

—

#462

—

106

—

#423

C₂×D₅₃

yes

—

C₄³

—

#339

—

106

—

C₅₃

#10303

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×M₁₆

D₂₁₂

yes

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#106

—

106

—

C₄⋊C₈

—

#103

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×D₂₁₂

yes

—

Q₁₆

—

424

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₄²

—

106

—

C₄²

—

212

—

C₄²

—

212

—

C₄²

D₄×D₅₃

yes

—

C₄²

yes

C₂³

yes

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

212

—

102

C₂×C₄

—

212

—

103

C₂×C₄

#13133187

yes

—

104

C₂×C₄

#13133192

yes

—

105

C₈

—

106

C₂²

—

107

C₂²

#16810533

yes

characteristic

108

C₄

—

109

C₄

—

424

—

110

C₄

—

212

—

111

C₄

#16810803

yes

—

112

C₄

#16810808

yes

—

113

C₂

—

114

C₂

#20275764

yes

—

	subgroup	G/H
center	C₂²	#16810533
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1309	C₂
socle	C₂²×C₅₃	#392
prime radical	C₂³×C₅₃	#127
hypercenter	#1309	D₅₃
hypocenter	C₅₃	#10303
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#16810533

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1309

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
68. C₅₃	#10303	semidirect

H ≅ G/N

type

68. C₅₃

#10303

semidirect

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

subgroup	index
#20286109	2
#20275691	2
C₅₃×#1309	2
#10303	53

subgroup

index

#20286109

#20275691

C₅₃×#1309

#10303

Subgroups — #27286858

Conjugacy classes of subgroups

Complements

Maximal subgroups