Group #20276486 — subgroups

Subgroups — #20276486

Order 6784.

144 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#108×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810658	C₂	1	1	yes	maximal, characteristic
4	#16810447	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810446	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810445	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×D₅₃	—	2	1	no	—
9	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#13133175	—	2	1	no	—
11	#13133157	C₂²	1	1	yes	characteristic
12	#13133156	C₄	1	2	yes	complemented
13	C₁₆×D₅₃	C₄	1	2	yes	complemented
14	#13133153	C₂²	1	1	yes	characteristic
15	#13133152	—	2	1	no	—
16	#13133150	—	2	1	no	—
17	#13133149	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	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
22	M₁₆×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	2	1	no	—
25	#11719921	—	4	1	no	—
26	C₄⋊Dic₅₃	—	2	1	no	—
27	C₄×Dic₅₃	—	4	1	no	—
28	#11719917	—	2	1	no	—
29	C₂×#88322	D₄	1	1	yes	characteristic
30	#11719912	—	4	1	no	—
31	C₈×D₅₃	—	4	1	no	—
32	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
33	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₈×D₅₃	Q₈	1	1	yes	characteristic
35	C₈×D₅₃	Q₈	1	1	yes	characteristic
36	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
37	#11719908	C₂×C₄	1	2	yes	—
38	C₂²×D₅₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₅₃	—	4	1	no	—
42	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
43	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
46	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
47	C₈×C₅₃	—	4	1	no	—
48	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	#88322	—	4	1	no	—
51	#88322	C₄⋊C₄	1	1	yes	characteristic
52	#88322	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×D₅₃	—	4	1	no	—
55	C₂×D₅₃	#106	1	1	yes	characteristic
56	C₂×D₅₃	#106	1	1	yes	characteristic
57	C₄×C₅₃	—	8	1	no	—
58	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
60	Dic₅₃	—	8	1	no	—
61	Dic₅₃	#106	1	1	yes	characteristic
62	Dic₅₃	#106	1	1	yes	characteristic
63	#2155	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	C₂×#106	1	1	yes	characteristic
66	D₅₃	—	2	1	no	—
67	D₅₃	—	2	1	no	—
68	C₂×M₃₂	—	53	1	no	—
69	C₂×#108	—	53	1	no	—
70	#427	—	53	1	no	—
71	#365	—	53	1	no	—
72	#365	—	53	1	no	—
73	#365	—	53	1	no	—
74	#365	D₅₃	1	1	yes	characteristic, complemented
75	C₅₃	#2155	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	106	1	no	—
77	C₂²×C₈	—	53	1	no	—
78	#117	—	106	1	no	—
79	M₃₂	—	53	1	no	—
80	M₃₂	—	53	2	no	—
81	M₃₂	C₂×D₅₃	1	1	yes	characteristic
82	C₂×C₁₆	—	53	2	no	—
83	#108	—	53	1	no	—
84	#108	—	106	1	no	—
85	#108	C₂×D₅₃	1	1	yes	characteristic
86	#107	—	106	1	no	—
87	#106	—	53	1	no	—
88	#106	C₂×D₅₃	1	1	yes	characteristic
89	C₂²×C₄	—	53	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	106	1	no	—
92	M₁₆	—	212	1	no	—
93	C₂×C₈	—	53	1	no	—
94	C₂×C₈	—	53	1	no	—
95	C₂×C₈	—	53	1	no	—
96	C₂×C₈	—	53	1	no	—
97	C₂×C₈	—	53	1	no	—
98	C₂×C₈	—	212	1	no	—
99	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	106	1	no	—
102	C₂²⋊C₄	—	212	1	no	—
103	C₄²	—	212	1	no	—
104	C₁₆	—	53	2	no	—
105	C₁₆	C₄×D₅₃	1	2	yes	—
106	C₂³	—	53	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	53	1	no	—
109	C₂×C₄	—	53	1	no	—
110	C₂×C₄	—	53	1	no	—
111	C₂×C₄	—	53	1	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	212	1	no	—
114	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
115	C₈	—	4	1	no	—
116	C₈	—	53	1	no	—
117	C₈	—	53	1	no	—
118	C₈	—	212	1	no	—
119	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
120	C₈	Q₈×D₅₃	1	1	yes	characteristic
121	C₂²	—	53	1	no	—
122	C₂²	—	53	1	no	—
123	C₂²	—	212	1	no	—
124	C₂²	SD₁₆×D₅₃	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	53	1	no	—
127	C₄	—	53	1	no	—
128	C₄	—	424	1	no	—
129	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
130	C₄	SD₁₆×D₅₃	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	106	1	no	—
133	C₂	—	106	1	no	—
134	C₂	#106×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₅₃

C₂

yes

maximal, characteristic

#108×D₅₃

C₂

yes

maximal, characteristic

#16810658

C₂

yes

maximal, characteristic

#16810447

C₂

yes

maximal, characteristic, complemented

#16810446

C₂

yes

maximal, characteristic, complemented

#16810445

C₂

yes

maximal, characteristic, complemented

C₅₃×#365

C₂

yes

maximal, characteristic, complemented

M₁₆×D₅₃

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133175

—

#13133157

C₂²

yes

characteristic

#13133156

C₄

yes

complemented

C₁₆×D₅₃

C₄

yes

complemented

#13133153

C₂²

yes

—

—

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

characteristic

#108×C₅₃

C₂²

yes

characteristic

#106×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

D₄

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

C₈×D₅₃

Q₈

yes

characteristic

C₁₆×C₅₃

C₂×C₄

yes

—

#11719908

C₂×C₄

yes

—

C₂²×D₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

—

#88322

C₄⋊C₄

yes

characteristic

#88322

C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#106

yes

characteristic

D₅₃

—

D₅₃

—

C₂×M₃₂

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2155

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

#117

—

106

—

M₃₂

—

M₃₂

—

M₃₂

C₂×D₅₃

yes

characteristic

C₂×C₁₆

—

#108

—

#108

—

106

—

#108

C₂×D₅₃

yes

characteristic

#107

—

106

—

#106

—

#106

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

—

106

—

102

C₂²⋊C₄

—

212

—

103

C₄²

—

212

—

104

C₁₆

—

105

C₁₆

C₄×D₅₃

yes

—

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

212

—

114

C₂×C₄

D₄×D₅₃

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

212

—

119

C₈

C₂×C₄×D₅₃

yes

characteristic

120

C₈

Q₈×D₅₃

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

212

—

124

C₂²

SD₁₆×D₅₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

424

—

129

C₄

C₄⋊C₄×D₅₃

yes

characteristic

130

C₄

SD₁₆×D₅₃

yes

characteristic

131

C₂

—

132

C₂

—

106

—

133

C₂

—

106

—

134

C₂

#106×D₅₃

yes

characteristic

	subgroup	G/H
center	C₂	#106×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#365	C₂
socle	C₂×C₅₃	C₂×#106
hypercenter	#365	D₅₃
hypocenter	C₅₃	#2155
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#106×D₅₃

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
4. #16810447	C₂	semidirect
5. #16810446	C₂	semidirect
6. #16810445	C₂	semidirect
7. C₅₃×#365	C₂	semidirect
12. #13133156	C₄	semidirect
13. C₁₆×D₅₃	C₄	semidirect
74. #365	D₅₃	semidirect
75. C₅₃	#2155	semidirect

H ≅ G/N

type

4. #16810447

C₂

semidirect

5. #16810446

C₂

semidirect

6. #16810445

C₂

semidirect

7. C₅₃×#365

C₂

semidirect

12. #13133156

C₄

semidirect

13. C₁₆×D₅₃

C₄

semidirect

74. #365

D₅₃

semidirect

75. C₅₃

#2155

semidirect

Subgroups — #20276486

Conjugacy classes of subgroups

Complements