Group #20276604 — subgroups

Subgroups — #20276604

Order 6784.

142 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16811064	C₂	1	1	yes	maximal, characteristic
2	#16810746	C₂	1	1	yes	maximal, characteristic
3	#16810445	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810439	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810432	C₂	1	1	yes	maximal, characteristic
6	#16810401	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133239	C₂²	1	1	yes	characteristic
9	#13133223	—	2	1	no	—
10	#13133163	C₄	1	2	yes	complemented
11	#13133160	C₄	1	2	yes	complemented
12	#13133157	C₂²	1	1	yes	characteristic
13	#13133157	C₂²	1	1	yes	characteristic
14	#13133153	C₂²	1	1	yes	characteristic
15	#13133145	—	2	1	no	—
16	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₅₃	—	2	1	no	—
18	#106×C₅₃	C₂²	1	1	yes	characteristic
19	#103×C₅₃	—	2	1	no	—
20	#97×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	#11719945	—	2	1	no	—
23	Q₁₆×C₅₃	—	2	1	no	—
24	Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	4	1	no	—
29	C₄²×C₅₃	—	4	1	no	—
30	C₄²×C₅₃	D₄	1	1	yes	characteristic
31	#11719917	—	2	1	no	—
32	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
33	D₄₂₄	C₂×C₄	1	1	yes	characteristic
34	#11719913	—	2	1	no	—
35	#11719912	—	4	1	no	—
36	C₈×D₅₃	—	4	1	no	—
37	#11719908	—	2	1	no	—
38	#11719908	C₂×C₄	1	2	yes	—
39	Q₈×C₅₃	—	2	2	no	—
40	Q₈×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²⋊D₅₃	—	4	1	no	—
46	D₂₁₂	—	2	1	no	—
47	C₄×D₅₃	—	4	1	no	—
48	Dic₁₀₆	—	2	1	no	—
49	C₈×C₅₃	—	2	1	no	—
50	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₅₃	#41	1	1	yes	characteristic
52	#88322	—	4	1	no	—
53	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×D₅₃	—	4	1	no	—
55	C₄×C₅₃	—	4	1	no	—
56	C₄×C₅₃	—	4	2	no	—
57	C₄×C₅₃	—	8	1	no	—
58	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	Dic₅₃	—	4	1	no	—
61	#2180	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	#442	1	1	yes	characteristic
64	D₅₃	—	8	1	no	—
65	#510	—	53	1	no	—
66	#444	—	53	1	no	—
67	#441	D₅₃	1	1	yes	characteristic, complemented
68	#365	—	53	1	no	—
69	#362	—	53	1	no	—
70	#360	—	53	1	no	—
71	#347	—	53	1	no	—
72	C₅₃	#2180	1	1	yes	characteristic, complemented, Sylow
73	#135	—	53	1	no	—
74	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
75	#131	—	106	1	no	—
76	C₄×Q₈	—	2	1	no	—
77	Q₃₂	—	53	2	no	—
78	SD₃₂	—	53	2	no	—
79	M₃₂	—	53	1	no	—
80	M₃₂	—	53	1	no	—
81	#108	—	53	1	no	—
82	#106	C₂×D₅₃	1	1	yes	characteristic
83	#104	—	106	1	no	—
84	#103	—	2	1	no	—
85	#97	C₂×D₅₃	1	1	yes	characteristic
86	#41	—	106	1	no	—
87	C₂×Q₈	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	53	1	no	—
90	Q₁₆	C₄×D₅₃	1	2	yes	—
91	SD₁₆	—	106	1	no	—
92	D₈	—	53	1	no	—
93	M₁₆	—	106	1	no	—
94	M₁₆	—	212	1	no	—
95	C₂×C₈	—	212	1	no	—
96	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
97	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄²	—	4	1	no	—
101	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₁₆	—	53	2	no	—
103	C₁₆	—	106	1	no	—
104	Q₈	—	2	2	no	—
105	Q₈	—	4	1	no	—
106	Q₈	—	106	1	no	—
107	D₄	—	106	1	no	—
108	D₄	—	212	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	212	1	no	—
113	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	212	1	no	—
116	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
117	C₈	#11719945	1	1	yes	characteristic
118	C₂²	—	212	1	no	—
119	C₂²	#13133243	1	1	yes	characteristic
120	C₄	—	4	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	8	1	no	—
123	C₄	—	212	1	no	—
124	C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
125	C₄	#13133242	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	424	1	no	—
128	C₂	#16810733	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811064

C₂

yes

maximal, characteristic

#16810746

C₂

yes

maximal, characteristic

#16810445

C₂

yes

maximal, characteristic, complemented

#16810439

C₂

yes

maximal, characteristic, complemented

#16810432

C₂

yes

maximal, characteristic

#16810401

C₂

yes

maximal, characteristic, complemented

C₅₃×#441

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

characteristic

#13133223

—

#13133163

C₄

yes

complemented

#13133160

C₄

yes

complemented

#13133157

C₂²

yes

characteristic

#13133157

C₂²

yes

characteristic

#13133153

C₂²

yes

characteristic

#13133145

—

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

#106×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

#97×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

#11719945

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

#11719917

—

Dic₂₁₂

C₂×C₄

yes

characteristic

D₄₂₄

C₂×C₄

yes

characteristic

#11719913

—

#11719912

—

C₈×D₅₃

—

#11719908

—

#11719908

C₂×C₄

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂²×C₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

Dic₅₃

—

#2180

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#442

yes

characteristic

D₅₃

—

#510

—

#444

—

#441

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#2180

yes

characteristic, complemented, Sylow

#135

—

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

#131

—

106

—

C₄×Q₈

—

Q₃₂

—

SD₃₂

—

M₃₂

—

M₃₂

—

#108

—

#106

C₂×D₅₃

yes

characteristic

#104

—

106

—

#103

—

#97

C₂×D₅₃

yes

characteristic

#41

—

106

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₄×D₅₃

yes

—

SD₁₆

—

106

—

D₈

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

C₂²⋊D₅₃

yes

characteristic

102

C₁₆

—

103

C₁₆

—

106

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

106

—

107

D₄

—

106

—

108

D₄

—

212

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

212

—

113

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

114

C₈

—

115

C₈

—

212

—

116

C₈

C₂×C₄×D₅₃

yes

characteristic

117

C₈

#11719945

yes

characteristic

118

C₂²

—

212

—

119

C₂²

#13133243

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

212

—

124

C₄

C₄×C₂²⋊D₅₃

yes

characteristic

125

C₄

#13133242

yes

characteristic

126

C₂

—

127

C₂

—

424

—

128

C₂

#16810733

yes

characteristic

	subgroup	G/H
center	C₂	#16810733
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#441	C₂
socle	C₂×C₅₃	#442
hypercenter	#441	D₅₃
hypocenter	C₅₃	#2180
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #441

subgroup

G/H

center

C₂

#16810733

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#441

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #441

N	H ≅ G/N	type
3. #16810445	C₂	semidirect
4. #16810439	C₂	semidirect
6. #16810401	C₂	semidirect
7. C₅₃×#441	C₂	semidirect
10. #13133163	C₄	semidirect
11. #13133160	C₄	semidirect
67. #441	D₅₃	semidirect
72. C₅₃	#2180	semidirect

H ≅ G/N

type

3. #16810445

C₂

semidirect

4. #16810439

C₂

semidirect

6. #16810401

C₂

semidirect

7. C₅₃×#441

C₂

semidirect

10. #13133163

C₄

semidirect

11. #13133160

C₄

semidirect

67. #441

D₅₃

semidirect

72. C₅₃

#2180

semidirect

Subgroups — #20276604

Conjugacy classes of subgroups

Complements