Group #20287854 — subgroups

Subgroups — #20287854

Order 6784.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 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	#16811428	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811059	C₂	1	3	yes	maximal, complemented
3	#16811058	C₂	1	3	yes	maximal, complemented
4	#16811053	C₂	1	3	yes	maximal, complemented
5	C₂×#13133158	C₂	1	3	yes	maximal, complemented
6	#16811040	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133263	—	2	1	no	—
9	#13133245	—	2	3	no	—
10	#13133239	C₂²	1	3	yes	complemented
11	C₂×D₄₂₄	C₂²	1	3	yes	complemented
12	#13133163	C₂²	1	1	yes	characteristic, complemented
13	#13133161	C₂²	1	3	yes	complemented
14	#13133160	C₂²	1	3	yes	complemented
15	#13133158	C₂²	1	3	yes	complemented
16	#13133158	C₂²	1	6	yes	complemented
17	#13133157	C₂²	1	3	yes	complemented
18	C₂×#11719908	C₂²	1	3	yes	complemented
19	H(5,𝔽₂)×C₅₃	—	2	1	no	—
20	AGL(1,ℤ/8)×C₅₃	—	2	3	no	—
21	#135×C₅₃	C₂²	1	3	yes	complemented
22	C₂×D₈×C₅₃	C₂²	1	3	yes	complemented
23	#131×C₅₃	C₂²	1	1	yes	characteristic
24	#41×C₅₃	—	2	3	no	—
25	#41×C₅₃	—	4	1	no	—
26	#41×C₅₃	D₄	1	1	yes	characteristic
27	C₂×D₄×C₅₃	—	2	3	no	—
28	C₂×D₄×C₅₃	—	4	3	no	—
29	#11719949	—	4	1	no	—
30	D₄×D₅₃	—	4	3	no	—
31	#11719945	—	2	3	no	—
32	C₂×D₂₁₂	—	2	3	no	—
33	Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
34	SD₁₆×C₅₃	—	2	3	no	—
35	D₈×C₅₃	—	2	3	no	—
36	D₈×C₅₃	C₂³	1	3	yes	—
37	M₁₆×C₅₃	D₄	1	3	yes	—
38	C₂×C₈×C₅₃	C₂³	1	3	yes	—
39	Dic₂₁₂	C₂³	1	1	yes	characteristic
40	D₄₂₄	—	2	3	no	—
41	D₄₂₄	C₂³	1	3	yes	—
42	#11719913	—	2	3	no	—
43	#11719908	C₂³	1	1	yes	characteristic
44	#11719908	C₂³	1	3	yes	—
45	C₂³×C₅₃	—	4	3	no	—
46	C₂²×D₅₃	—	4	3	no	—
47	Q₈×C₅₃	—	2	1	no	—
48	Q₈×C₅₃	D₈	1	1	yes	characteristic
49	D₄×C₅₃	—	2	3	no	—
50	D₄×C₅₃	—	4	3	no	—
51	D₄×C₅₃	—	4	3	no	—
52	D₄×C₅₃	D₈	1	3	yes	—
53	C₂×C₄×C₅₃	—	4	3	no	—
54	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
55	C₂²⋊D₅₃	—	4	3	no	—
56	D₂₁₂	—	2	3	no	—
57	D₂₁₂	—	4	3	no	—
58	C₄×D₅₃	—	4	3	no	—
59	Dic₁₀₆	—	2	1	no	—
60	C₈×C₅₃	C₂×D₄	1	3	yes	—
61	C₈×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂²×C₅₃	—	4	3	no	—
63	C₂²×C₅₃	—	16	3	no	—
64	C₂²×C₅₃	C₂×D₈	1	3	yes	—
65	C₂×D₅₃	—	4	3	no	—
66	C₂×D₅₃	—	16	3	no	—
67	C₄×C₅₃	—	4	1	no	—
68	C₄×C₅₃	C₂×D₈	1	3	yes	—
69	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	Dic₅₃	—	4	1	no	—
71	#3415	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	—	2	3	no	—
73	C₂×C₅₃	—	8	3	no	—
74	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
75	D₅₃	—	8	3	no	—
76	#576	—	53	1	no	—
77	#576	D₅₃	1	1	yes	characteristic, complemented
78	#509	—	53	3	no	—
79	#509	—	53	3	no	—
80	#508	—	53	3	no	—
81	C₂×D₁₆	—	53	3	no	—
82	#504	—	53	1	no	—
83	C₅₃	#3415	1	1	yes	characteristic, complemented, Sylow
84	H(5,𝔽₂)	—	2	1	no	—
85	H(5,𝔽₂)	—	106	1	no	—
86	AGL(1,ℤ/8)	—	2	3	no	—
87	AGL(1,ℤ/8)	—	106	3	no	—
88	#135	—	53	3	no	—
89	#135	C₂×D₅₃	1	3	yes	complemented
90	C₂×D₈	—	53	3	no	—
91	C₂×D₈	C₂×D₅₃	1	3	yes	complemented
92	#131	C₂×D₅₃	1	1	yes	characteristic
93	Q₃₂	—	53	1	no	—
94	SD₃₂	—	53	3	no	—
95	SD₃₂	—	53	3	no	—
96	D₁₆	—	53	3	no	—
97	D₁₆	—	53	6	no	—
98	M₃₂	—	53	3	no	—
99	C₂×C₁₆	—	53	3	no	—
100	#41	—	2	3	no	—
101	#41	—	4	1	no	—
102	#41	—	106	3	no	—
103	#41	—	212	1	no	—
104	#41	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	106	3	no	—
108	C₂×D₄	—	212	3	no	—
109	Q₁₆	—	53	1	no	—
110	Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	106	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	53	3	no	—
115	D₈	—	106	3	no	—
116	D₈	C₂²×D₅₃	1	3	yes	—
117	M₁₆	C₂²⋊D₅₃	1	3	yes	—
118	C₂×C₈	C₂²×D₅₃	1	3	yes	—
119	C₁₆	—	53	1	no	—
120	C₁₆	—	53	3	no	—
121	C₂³	—	4	3	no	—
122	C₂³	—	212	3	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	106	1	no	—
125	Q₈	#11719922	1	1	yes	characteristic
126	D₄	—	2	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	3	no	—
129	D₄	—	106	3	no	—
130	D₄	—	212	3	no	—
131	D₄	—	212	3	no	—
132	D₄	#11719922	1	3	yes	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	212	3	no	—
135	C₂×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
136	C₈	C₂×C₂²⋊D₅₃	1	3	yes	—
137	C₈	C₂³×D₅₃	1	1	yes	characteristic
138	C₂²	—	4	3	no	—
139	C₂²	—	16	3	no	—
140	C₂²	—	848	3	no	—
141	C₂²	—	212	3	no	—
142	C₂²	C₂×#11719922	1	3	yes	—
143	C₄	—	4	1	no	—
144	C₄	—	212	1	no	—
145	C₄	C₂×#11719922	1	3	yes	—
146	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂	—	2	3	no	—
148	C₂	—	8	3	no	—
149	C₂	—	424	3	no	—
150	C₂	C₂²×#11719922	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811428

C₂

yes

maximal, characteristic, complemented

#16811059

C₂

yes

maximal, complemented

#16811058

C₂

yes

maximal, complemented

#16811053

C₂

yes

maximal, complemented

C₂×#13133158

C₂

yes

maximal, complemented

#16811040

C₂

yes

maximal, characteristic, complemented

C₅₃×#576

C₂

yes

maximal, characteristic, complemented

#13133263

—

#13133245

—

#13133239

C₂²

yes

complemented

C₂×D₄₂₄

C₂²

yes

complemented

#13133163

C₂²

yes

characteristic, complemented

#13133161

C₂²

yes

complemented

#13133160

C₂²

yes

complemented

#13133158

C₂²

yes

complemented

#13133158

C₂²

yes

complemented

#13133157

C₂²

yes

complemented

C₂×#11719908

C₂²

yes

complemented

H(5,𝔽₂)×C₅₃

—

AGL(1,ℤ/8)×C₅₃

—

#135×C₅₃

C₂²

yes

complemented

C₂×D₈×C₅₃

C₂²

yes

complemented

#131×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

#41×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

#11719949

—

D₄×D₅₃

—

#11719945

—

C₂×D₂₁₂

—

Q₁₆×C₅₃

C₂³

yes

characteristic

SD₁₆×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

C₂³

yes

—

M₁₆×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

Dic₂₁₂

C₂³

yes

characteristic

D₄₂₄

—

D₄₂₄

C₂³

yes

—

#11719913

—

#11719908

C₂³

yes

characteristic

#11719908

C₂³

yes

—

C₂³×C₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

D₈

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂²⋊D₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

—

C₈×C₅₃

C₂⁴

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

Dic₅₃

—

#3415

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×D₈

yes

characteristic

D₅₃

—

#576

—

#576

D₅₃

yes

characteristic, complemented

#509

—

#509

—

#508

—

C₂×D₁₆

—

#504

—

C₅₃

#3415

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

H(5,𝔽₂)

—

106

—

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

—

106

—

#135

—

#135

C₂×D₅₃

yes

complemented

C₂×D₈

—

C₂×D₈

C₂×D₅₃

yes

complemented

#131

C₂×D₅₃

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

100

#41

—

101

#41

—

102

#41

—

106

—

103

#41

—

212

—

104

#41

C₂²⋊D₅₃

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

106

—

108

C₂×D₄

—

212

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₅₃

yes

characteristic

111

SD₁₆

—

112

SD₁₆

—

106

—

113

D₈

—

114

D₈

—

115

D₈

—

106

—

116

D₈

C₂²×D₅₃

yes

—

117

M₁₆

C₂²⋊D₅₃

yes

—

118

C₂×C₈

C₂²×D₅₃

yes

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

—

122

C₂³

—

212

—

123

Q₈

—

124

Q₈

—

106

—

125

Q₈

#11719922

yes

characteristic

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

106

—

130

D₄

—

212

—

131

D₄

—

212

—

132

D₄

#11719922

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

212

—

135

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

136

C₈

C₂×C₂²⋊D₅₃

yes

—

137

C₈

C₂³×D₅₃

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

848

—

141

C₂²

—

212

—

142

C₂²

C₂×#11719922

yes

—

143

C₄

—

144

C₄

—

212

—

145

C₄

C₂×#11719922

yes

—

146

C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

424

—

150

C₂

C₂²×#11719922

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#11719922
commutator	C₈×C₅₃	C₂⁴
Frattini	C₈	C₂³×D₅₃
Fitting	C₅₃×#576	C₂
socle	C₂×C₅₃	C₂²×D₈
hypercenter	#576	D₅₃
hypocenter	C₅₃	#3415
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

subgroup

G/H

center

C₂

C₂²×#11719922

commutator

C₈×C₅₃

C₂⁴

Frattini

C₈

C₂³×D₅₃

Fitting

C₅₃×#576

C₂

socle

C₂×C₅₃

C₂²×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

N	H ≅ G/N	type
1. #16811428	C₂	semidirect
2. #16811059	C₂	semidirect
3. #16811058	C₂	semidirect
4. #16811053	C₂	semidirect
5. C₂×#13133158	C₂	semidirect
6. #16811040	C₂	semidirect
7. C₅₃×#576	C₂	semidirect
10. #13133239	C₂²	semidirect
11. C₂×D₄₂₄	C₂²	semidirect
12. #13133163	C₂²	semidirect
13. #13133161	C₂²	semidirect
14. #13133160	C₂²	semidirect
15. #13133158	C₂²	semidirect
16. #13133158	C₂²	semidirect
17. #13133157	C₂²	semidirect
18. C₂×#11719908	C₂²	semidirect
21. #135×C₅₃	C₂²	semidirect
22. C₂×D₈×C₅₃	C₂²	semidirect
77. #576	D₅₃	semidirect
83. C₅₃	#3415	semidirect
89. #135	C₂×D₅₃	semidirect
91. C₂×D₈	C₂×D₅₃	semidirect

H ≅ G/N

type

1. #16811428

C₂

semidirect

2. #16811059

C₂

semidirect

3. #16811058

C₂

semidirect

4. #16811053

C₂

semidirect

5. C₂×#13133158

C₂

semidirect

6. #16811040

C₂

semidirect

7. C₅₃×#576

C₂

semidirect

10. #13133239

C₂²

semidirect

11. C₂×D₄₂₄

C₂²

semidirect

12. #13133163

C₂²

semidirect

13. #13133161

C₂²

semidirect

14. #13133160

C₂²

semidirect

15. #13133158

C₂²

semidirect

16. #13133158

C₂²

semidirect

17. #13133157

C₂²

semidirect

18. C₂×#11719908

C₂²

semidirect

21. #135×C₅₃

C₂²

semidirect

22. C₂×D₈×C₅₃

C₂²

semidirect

77. #576

D₅₃

semidirect

83. C₅₃

#3415

semidirect

89. #135

C₂×D₅₃

semidirect

91. C₂×D₈

C₂×D₅₃

semidirect

Subgroups — #20287854

Conjugacy classes of subgroups

Complements