Group #20276612 — subgroups

Subgroups — #20276612

Order 6784.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16811061	C₂	1	1	yes	maximal, characteristic
2	#16810744	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810734	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810435	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810432	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810400	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133239	C₂²	1	1	yes	characteristic
9	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
10	#13133225	—	2	1	no	—
11	#13133179	—	2	1	no	—
12	#13133162	C₄	1	2	yes	complemented
13	D₈₄₈	C₄	1	2	yes	complemented
14	#13133157	C₂²	1	1	yes	characteristic
15	#13133143	—	2	1	no	—
16	#13133137	—	2	1	no	—
17	#13133127	C₂²	1	1	yes	characteristic
18	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
19	#108×C₅₃	C₂²	1	1	yes	characteristic
20	#106×C₅₃	C₂²	1	1	yes	characteristic
21	#11719945	—	2	1	no	—
22	C₂×D₂₁₂	—	2	1	no	—
23	C₂×C₄×D₅₃	—	4	1	no	—
24	M₁₆×C₅₃	—	2	1	no	—
25	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₃	—	2	1	no	—
27	#11719921	—	4	1	no	—
28	C₄×Dic₅₃	D₄	1	1	yes	characteristic
29	C₂×#88322	D₄	1	1	yes	characteristic
30	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic, complemented
31	D₄₂₄	—	2	1	no	—
32	D₄₂₄	C₂×C₄	1	2	yes	—
33	D₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
34	#11719913	—	2	1	no	—
35	#11719912	—	4	1	no	—
36	C₈×D₅₃	—	4	1	no	—
37	C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
38	#11719908	—	2	1	no	—
39	C₂²×D₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²⋊D₅₃	—	4	1	no	—
43	C₂×Dic₅₃	—	2	1	no	—
44	D₂₁₂	—	2	2	no	—
45	D₂₁₂	—	2	1	no	—
46	D₂₁₂	—	4	1	no	—
47	C₄×D₅₃	—	4	1	no	—
48	C₄×D₅₃	—	8	2	no	—
49	Dic₁₀₆	—	2	1	no	—
50	C₈×C₅₃	—	4	1	no	—
51	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₅₃	#41	1	1	yes	characteristic
53	#88322	—	2	1	no	—
54	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₅₃	—	4	1	no	—
56	C₂×D₅₃	—	4	2	no	—
57	C₂×D₅₃	—	8	1	no	—
58	C₂×D₅₃	—	8	1	no	—
59	C₄×C₅₃	—	8	1	no	—
60	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₅₃	—	4	1	no	—
63	Dic₅₃	—	4	1	no	—
64	AGL(1,ℤ/16)	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	#442	1	1	yes	characteristic
67	D₅₃	—	8	1	no	—
68	D₅₃	—	8	2	no	—
69	#509	—	53	1	no	—
70	#444	—	53	1	no	—
71	#442	—	53	1	no	—
72	#365	D₅₃	1	1	yes	characteristic, complemented
73	#361	—	53	1	no	—
74	#360	—	53	1	no	—
75	#347	—	53	1	no	—
76	C₅₃	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
77	#135	—	53	1	no	—
78	C₂×D₈	—	53	1	no	—
79	#131	—	106	1	no	—
80	C₄×D₄	—	106	1	no	—
81	SD₃₂	—	53	2	no	—
82	D₁₆	—	53	2	no	—
83	M₃₂	—	53	1	no	—
84	M₃₂	C₂×D₅₃	1	1	yes	characteristic
85	#108	C₂×D₅₃	1	1	yes	characteristic
86	#106	C₂×D₅₃	1	1	yes	characteristic
87	#104	—	106	1	no	—
88	#102	—	106	1	no	—
89	#97	—	53	1	no	—
90	#41	—	106	1	no	—
91	C₂×D₄	—	106	1	no	—
92	C₂²×C₄	—	212	1	no	—
93	Q₁₆	—	53	1	no	—
94	SD₁₆	—	106	1	no	—
95	D₈	—	53	1	no	—
96	D₈	—	53	2	no	—
97	D₈	—	106	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	212	1	no	—
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	212	1	no	—
102	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	212	1	no	—
105	C₄²	—	53	1	no	—
106	C₁₆	—	106	1	no	—
107	C₁₆	C₄×D₅₃	1	2	yes	complemented
108	C₂³	—	212	1	no	—
109	Q₈	—	106	1	no	—
110	D₄	—	106	2	no	—
111	D₄	—	106	1	no	—
112	D₄	—	212	1	no	—
113	D₄	—	212	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	106	1	no	—
116	C₂×C₄	—	424	2	no	—
117	C₂×C₄	—	212	1	no	—
118	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	106	1	no	—
121	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
122	C₈	#11719949	1	1	yes	characteristic
123	C₂²	—	424	1	no	—
124	C₂²	—	424	1	no	—
125	C₂²	—	212	1	no	—
126	C₂²	—	212	2	no	—
127	C₂²	#13133244	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	212	1	no	—
130	C₄	—	212	1	no	—
131	C₄	#13133179	1	1	yes	characteristic
132	C₄	#13133244	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	424	2	no	—
135	C₂	—	424	1	no	—
136	C₂	#16810734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811061

C₂

yes

maximal, characteristic

#16810744

C₂

yes

maximal, characteristic, complemented

#16810734

C₂

yes

maximal, characteristic, complemented

#16810435

C₂

yes

maximal, characteristic, complemented

#16810432

C₂

yes

maximal, characteristic, complemented

#16810400

C₂

yes

maximal, characteristic, complemented

C₅₃×#365

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

characteristic

C₂×D₄₂₄

C₂²

yes

—

—

C₄

yes

complemented

D₈₄₈

C₄

yes

complemented

#13133157

C₂²

yes

—

—

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

characteristic

#108×C₅₃

C₂²

yes

characteristic

#106×C₅₃

C₂²

yes

characteristic

#11719945

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719921

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

C₂×C₄

yes

characteristic, complemented

D₄₂₄

—

D₄₂₄

C₂×C₄

yes

—

D₄₂₄

C₂×C₄

yes

characteristic, complemented

#11719913

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

C₂×C₄

yes

complemented

#11719908

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂²×C₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

AGL(1,ℤ/8)

yes

characteristic

Dic₅₃

—

Dic₅₃

—

AGL(1,ℤ/16)

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#442

yes

characteristic

D₅₃

—

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

#361

—

#360

—

#347

—

C₅₃

AGL(1,ℤ/16)

yes

characteristic, complemented, Sylow

#135

—

C₂×D₈

—

#131

—

106

—

C₄×D₄

—

106

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

characteristic

#108

C₂×D₅₃

yes

characteristic

#106

C₂×D₅₃

yes

characteristic

#104

—

106

—

#102

—

106

—

#97

—

#41

—

106

—

C₂×D₄

—

106

—

C₂²×C₄

—

212

—

Q₁₆

—

SD₁₆

—

106

—

D₈

—

D₈

—

D₈

—

106

—

M₁₆

—

M₁₆

—

212

—

100

C₂×C₈

—

101

C₂×C₈

—

212

—

102

C₂×C₈

C₂²×D₅₃

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

212

—

105

C₄²

—

106

C₁₆

—

106

—

107

C₁₆

C₄×D₅₃

yes

complemented

108

C₂³

—

212

—

109

Q₈

—

106

—

110

D₄

—

106

—

111

D₄

—

106

—

112

D₄

—

212

—

113

D₄

—

212

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

424

—

117

C₂×C₄

—

212

—

118

C₂×C₄

D₄×D₅₃

yes

characteristic

119

C₈

—

120

C₈

—

106

—

121

C₈

C₂×C₄×D₅₃

yes

characteristic

122

C₈

#11719949

yes

characteristic

123

C₂²

—

424

—

124

C₂²

—

424

—

125

C₂²

—

212

—

126

C₂²

—

212

—

127

C₂²

#13133244

yes

characteristic

128

C₄

—

129

C₄

—

212

—

130

C₄

—

212

—

131

C₄

#13133179

yes

characteristic

132

C₄

#13133244

yes

characteristic

133

C₂

—

134

C₂

—

424

—

135

C₂

—

424

—

136

C₂

#16810734

yes

characteristic

	subgroup	G/H
center	C₂	#16810734
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#365	C₂
socle	C₂×C₅₃	#442
hypercenter	#365	D₅₃
hypocenter	C₅₃	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#16810734

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#365

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
2. #16810744	C₂	semidirect
3. #16810734	C₂	semidirect
4. #16810435	C₂	semidirect
5. #16810432	C₂	semidirect
6. #16810400	C₂	semidirect
7. C₅₃×#365	C₂	semidirect
12. #13133162	C₄	semidirect
13. D₈₄₈	C₄	semidirect
30. Dic₂₁₂	C₂×C₄	semidirect
33. D₄₂₄	C₂×C₄	semidirect
37. C₁₆×C₅₃	C₂×C₄	semidirect
72. #365	D₅₃	semidirect
76. C₅₃	AGL(1,ℤ/16)	semidirect
107. C₁₆	C₄×D₅₃	semidirect

H ≅ G/N

type

2. #16810744

C₂

semidirect

3. #16810734

C₂

semidirect

4. #16810435

C₂

semidirect

5. #16810432

C₂

semidirect

6. #16810400

C₂

semidirect

7. C₅₃×#365

C₂

semidirect

12. #13133162

C₄

semidirect

13. D₈₄₈

C₄

semidirect

30. Dic₂₁₂

C₂×C₄

semidirect

33. D₄₂₄

C₂×C₄

semidirect

37. C₁₆×C₅₃

C₂×C₄

semidirect

72. #365

D₅₃

semidirect

76. C₅₃

AGL(1,ℤ/16)

semidirect

107. C₁₆

C₄×D₅₃

semidirect

Subgroups — #20276612

Conjugacy classes of subgroups

Complements