Group #27305066 — subgroups

Subgroups — #27305066

Order 13568.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20276968	C₂	1	1	yes	maximal, characteristic
2	#20272895	C₂	1	4	yes	maximal
3	C₄×#13133123	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₂₄	—	2	1	no	—
6	C₂×Dic₄₂₄	C₂²	1	2	yes	—
7	#16811003	—	2	1	no	—
8	C₂×#13133123	C₂²	1	2	yes	—
9	C₅₃×#500	—	2	1	no	—
10	C₅₃×#366	—	2	1	no	—
11	C₅₃×#366	C₂²	1	2	yes	—
12	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₁₂	—	2	2	no	—
14	C₂×Dic₂₁₂	—	4	1	no	—
15	#13133217	—	4	1	no	—
16	Dic₄₂₄	—	2	4	no	—
17	Dic₄₂₄	—	4	2	no	—
18	#13133123	—	2	2	no	—
19	C₄⋊Q₈×C₅₃	—	4	1	no	—
20	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
22	#107×C₅₃	—	2	2	no	—
23	#107×C₅₃	—	4	1	no	—
24	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	—	8	1	no	—
26	C₂×Dic₁₀₆	—	4	2	no	—
27	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₅₃	—	4	2	no	—
30	C₄⋊C₄×C₅₃	—	8	1	no	—
31	C₄²×C₅₃	D₈	1	1	yes	characteristic
32	C₄⋊Dic₅₃	—	8	1	no	—
33	C₄⋊Dic₅₃	—	8	1	no	—
34	Dic₂₁₂	—	4	2	no	—
35	Dic₂₁₂	—	4	2	no	—
36	Dic₂₁₂	—	8	2	no	—
37	C₁₆×C₅₃	SD₁₆	1	2	yes	—
38	C₁₆×C₅₃	#41	1	2	yes	—
39	Q₈×C₅₃	—	16	2	no	—
40	C₂×C₄×C₅₃	—	8	2	no	—
41	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₅₃	—	8	2	no	—
45	Dic₁₀₆	—	8	2	no	—
46	Dic₁₀₆	—	8	2	no	—
47	C₈×C₅₃	SD₃₂	1	2	yes	—
48	C₈×C₅₃	#124	1	1	yes	characteristic
49	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
50	#13720	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
52	C₄×C₅₃	—	16	2	no	—
53	C₄×C₅₃	SD₆₄	1	2	yes	—
54	C₄×C₅₃	Q₆₄	1	2	yes	—
55	C₄×C₅₃	#486	1	1	yes	characteristic
56	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
57	Dic₅₃	—	16	2	no	—
58	#2252	D₅₃	1	1	yes	characteristic
59	#2247	—	53	1	no	—
60	#1416	—	53	4	no	—
61	C₄×C₃₂	—	53	1	no	—
62	C₂×C₅₃	#2240	1	1	yes	characteristic
63	C₂×C₅₃	C₂×SD₆₄	1	1	yes	characteristic
64	C₂×C₅₃	C₂×Q₆₄	1	1	yes	characteristic
65	C₂×Q₃₂	—	53	2	no	—
66	C₂×Q₃₂	—	106	1	no	—
67	#500	—	2	1	no	—
68	#494	—	106	1	no	—
69	C₂×C₃₂	—	53	2	no	—
70	#366	—	2	1	no	—
71	#366	C₂×D₅₃	1	2	yes	—
72	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
73	C₅₃	#13720	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₁₆	—	106	2	no	—
75	C₂×Q₁₆	—	212	1	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	212	1	no	—
78	Q₃₂	—	106	4	no	—
79	Q₃₂	—	212	2	no	—
80	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
82	#107	—	2	2	no	—
83	#107	—	4	1	no	—
84	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
85	C₃₂	—	106	2	no	—
86	C₂×Q₈	—	8	1	no	—
87	C₂×Q₈	—	212	2	no	—
88	Q₁₆	—	424	2	no	—
89	Q₁₆	—	212	2	no	—
90	Q₁₆	—	212	2	no	—
91	C₂×C₈	#11719922	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	424	1	no	—
96	C₄⋊C₄	—	424	1	no	—
97	C₄²	#11719922	1	1	yes	characteristic
98	C₁₆	#11719924	1	2	yes	—
99	C₁₆	#11719949	1	2	yes	—
100	Q₈	—	16	2	no	—
101	Q₈	—	424	2	no	—
102	Q₈	—	424	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	424	2	no	—
105	C₂×C₄	#13133158	1	1	yes	characteristic
106	C₂×C₄	#13133158	1	1	yes	characteristic
107	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
108	C₈	#13133160	1	2	yes	—
109	C₈	#13133206	1	1	yes	characteristic
110	C₈	C₂×#11719924	1	1	yes	characteristic
111	C₂²	C₂×#13133158	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	848	2	no	—
114	C₄	#16810461	1	2	yes	—
115	C₄	#16810463	1	2	yes	—
116	C₄	#16810971	1	1	yes	characteristic
117	C₄	C₂×#13133160	1	1	yes	characteristic
118	C₂	#20276940	1	1	yes	characteristic
119	C₂	C₂×#16810461	1	1	yes	characteristic
120	C₂	C₂×#16810463	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276968

C₂

yes

maximal, characteristic

#20272895

C₂

yes

maximal

C₄×#13133123

C₂

yes

maximal, characteristic

C₅₃×#2252

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

—

C₂×Dic₄₂₄

C₂²

yes

—

#16811003

—

C₂×#13133123

C₂²

yes

—

C₅₃×#500

—

C₅₃×#366

—

C₅₃×#366

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

—

Dic₄₂₄

—

Dic₄₂₄

—

#13133123

—

C₄⋊Q₈×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#107×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₈

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

SD₁₆

yes

—

C₁₆×C₅₃

#41

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₁₆

yes

characteristic

C₂×C₄×C₅₃

D₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

SD₃₂

yes

—

C₈×C₅₃

#124

yes

characteristic

C₈×C₅₃

C₂×SD₁₆

yes

characteristic

#13720

—

maximal, Sylow

C₂²×C₅₃

C₂×D₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

SD₆₄

yes

—

C₄×C₅₃

Q₆₄

yes

—

C₄×C₅₃

#486

yes

characteristic

C₄×C₅₃

C₂×SD₃₂

yes

characteristic

Dic₅₃

—

#2252

D₅₃

yes

characteristic

#2247

—

#1416

—

C₄×C₃₂

—

C₂×C₅₃

#2240

yes

characteristic

C₂×C₅₃

C₂×SD₆₄

yes

characteristic

C₂×C₅₃

C₂×Q₆₄

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

106

—

#500

—

#494

—

106

—

C₂×C₃₂

—

#366

—

#366

C₂×D₅₃

yes

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₅₃

#13720

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

212

—

C₄⋊Q₈

—

C₄⋊Q₈

—

212

—

Q₃₂

—

106

—

Q₃₂

—

212

—

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

#107

—

#107

—

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

C₃₂

—

106

—

C₂×Q₈

—

C₂×Q₈

—

212

—

Q₁₆

—

424

—

Q₁₆

—

212

—

Q₁₆

—

212

—

C₂×C₈

#11719922

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

424

—

C₄⋊C₄

—

424

—

C₄²

#11719922

yes

characteristic

C₁₆

#11719924

yes

—

C₁₆

#11719949

yes

—

100

Q₈

—

101

Q₈

—

424

—

102

Q₈

—

424

—

103

C₂×C₄

—

104

C₂×C₄

—

424

—

105

C₂×C₄

#13133158

yes

characteristic

106

C₂×C₄

#13133158

yes

characteristic

107

C₂×C₄

C₂×#11719922

yes

characteristic

108

C₈

#13133160

yes

—

109

C₈

#13133206

yes

characteristic

110

C₈

C₂×#11719924

yes

characteristic

111

C₂²

C₂×#13133158

yes

characteristic

112

C₄

—

113

C₄

—

848

—

114

C₄

#16810461

yes

—

115

C₄

#16810463

yes

—

116

C₄

#16810971

yes

characteristic

117

C₄

C₂×#13133160

yes

characteristic

118

C₂

#20276940

yes

characteristic

119

C₂

C₂×#16810461

yes

characteristic

120

C₂

C₂×#16810463

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13133158
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#2252	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	#2252	D₅₃
hypocenter	C₅₃	#13720
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

subgroup

G/H

center

C₂²

C₂×#13133158

commutator, cosocle

C₂×C₁₆×C₅₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₅₃×#2252

C₂

socle, prime radical

C₂²×C₅₃

C₂×D₁₆

D₅₃

C₅₃

G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

N	H ≅ G/N	type
73. C₅₃	#13720	semidirect

H ≅ G/N

type

73. C₅₃

#13720

semidirect

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

subgroup	index
#20276968	2
#20272895	2
C₄×#13133123	2
C₅₃×#2252	2
#13720	53

subgroup

index

C₅₃×#2252

Subgroups — #27305066

Conjugacy classes of subgroups

Complements

Maximal subgroups