Group #20274228 — subgroups

Subgroups — #20274228

Order 6784.

166 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16811028	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810828	C₂	1	2	yes	maximal
3	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810368	C₂	1	2	yes	maximal, complemented
5	C₅₃×#436	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719923	—	2	2	no	—
7	#13133217	C₂²	1	2	yes	—
8	#13133151	—	2	1	no	—
9	#13133151	—	2	1	no	—
10	C₄⋊#88322	C₂²	1	2	yes	—
11	#13133140	—	2	2	no	—
12	C₂×M₁₆×C₅₃	—	2	1	no	—
13	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	—	2	1	no	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	M₁₆×C₅₃	—	4	2	no	—
21	C₂×C₈×C₅₃	—	2	1	no	—
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²⋊C₄×C₅₃	—	2	1	no	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	#11719923	—	4	4	no	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂³×C₅₃	—	2	1	no	—
32	D₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
39	C₂×Dic₅₃	—	4	2	no	—
40	Dic₁₀₆	—	4	2	no	—
41	Dic₁₀₆	—	4	2	no	—
42	C₈×C₅₃	—	2	1	no	—
43	C₈×C₅₃	—	4	1	no	—
44	C₈×C₅₃	SD₁₆	1	2	yes	complemented
45	#88322	—	4	2	no	—
46	C₂²×C₅₃	—	4	1	no	—
47	C₂²×C₅₃	—	4	1	no	—
48	C₂²×C₅₃	—	4	1	no	—
49	C₂²×C₅₃	#121	1	1	yes	characteristic
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	4	1	no	—
52	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
53	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₅₃	#137	1	1	yes	characteristic
55	Dic₅₃	—	8	2	no	—
56	#1693	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	8	1	no	—
58	C₂×C₅₃	#461	1	1	yes	characteristic
59	C₂×C₅₃	#469	1	1	yes	characteristic
60	C₂×C₅₃	#473	1	1	yes	characteristic
61	#500	—	53	1	no	—
62	#461	—	53	2	no	—
63	#436	D₅₃	1	1	yes	characteristic
64	#334	—	53	1	no	—
65	#332	—	53	2	no	—
66	C₅₃	#1693	1	1	yes	characteristic, complemented, Sylow
67	C₂×SD₁₆	—	106	2	no	—
68	C₂×M₁₆	—	2	1	no	—
69	C₄⋊Q₈	—	53	2	no	—
70	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
71	#107	—	106	1	no	—
72	#107	—	106	1	no	—
73	C₄⋊C₈	—	53	2	no	—
74	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
75	#103	—	106	2	no	—
76	C₂²⋊C₈	—	2	1	no	—
77	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
78	C₂×Q₈	—	106	2	no	—
79	C₂×D₄	D₂₁₂	1	1	yes	characteristic
80	C₂²×C₄	—	2	1	no	—
81	SD₁₆	—	212	4	no	—
82	M₁₆	—	4	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	106	2	no	—
85	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
87	C₄⋊C₄	—	106	2	no	—
88	C₄⋊C₄	—	106	2	no	—
89	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
90	C₂²⋊C₄	—	2	1	no	—
91	C₄²	C₂²×D₅₃	1	1	yes	characteristic
92	C₂³	—	2	1	no	—
93	Q₈	—	212	2	no	—
94	Q₈	—	212	2	no	—
95	D₄	—	2	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	212	2	no	—
100	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
101	C₂×C₄	#11719945	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	4	1	no	—
105	C₈	—	212	2	no	—
106	C₈	#11719923	1	2	yes	complemented
107	C₂²	—	4	1	no	—
108	C₂²	—	4	1	no	—
109	C₂²	—	4	1	no	—
110	C₂²	#13133191	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	4	1	no	—
113	C₄	—	424	2	no	—
114	C₄	C₂×#11719923	1	1	yes	characteristic
115	C₄	#13133245	1	2	yes	—
116	C₄	#13133247	1	1	yes	characteristic
117	C₂	—	8	1	no	—
118	C₂	#16810828	1	1	yes	characteristic
119	C₂	#16810876	1	1	yes	characteristic
120	C₂	#16810898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811028

C₂

yes

maximal, characteristic, complemented

#16810828

C₂

yes

maximal

#16810374

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, complemented

C₅₃×#436

C₂

yes

maximal, characteristic

C₂×#11719923

—

#13133217

C₂²

yes

—

#13133151

—

#13133151

—

C₄⋊#88322

C₂²

yes

—

#13133140

—

C₂×M₁₆×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719923

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

complemented

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

characteristic

Dic₅₃

—

#1693

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#334

—

#332

—

C₅₃

#1693

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₅₃

yes

characteristic

#107

—

106

—

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

D₂₁₂

yes

characteristic

C₂²×C₄

—

SD₁₆

—

212

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

D₂₁₂

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

100

C₂×C₄

C₂×D₂₁₂

yes

characteristic

101

C₂×C₄

#11719945

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

212

—

106

C₈

#11719923

yes

complemented

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

#13133191

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

424

—

114

C₄

C₂×#11719923

yes

characteristic

115

C₄

#13133245

yes

—

116

C₄

#13133247

yes

characteristic

117

C₂

—

118

C₂

#16810828

yes

characteristic

119

C₂

#16810876

yes

characteristic

120

C₂

#16810898

yes

characteristic

	subgroup	G/H
center	C₂²	#13133191
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#436	C₂
socle	C₂²×C₅₃	#121
hypercenter	#436	D₅₃
hypocenter	C₅₃	#1693
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #436

subgroup

G/H

center

C₂²

#13133191

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#436

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #436

N	H ≅ G/N	type
1. #16811028	C₂	semidirect
3. #16810374	C₂	semidirect
4. #16810368	C₂	semidirect
44. C₈×C₅₃	SD₁₆	semidirect
66. C₅₃	#1693	semidirect
106. C₈	#11719923	semidirect

H ≅ G/N

type

1. #16811028

C₂

semidirect

3. #16810374

C₂

semidirect

4. #16810368

C₂

semidirect

44. C₈×C₅₃

SD₁₆

semidirect

66. C₅₃

#1693

semidirect

106. C₈

#11719923

semidirect

Subgroups — #20274228

Conjugacy classes of subgroups

Complements