Group #20273003 — subgroups

Subgroups — #20273003

Order 6784.

208 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 101 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	#16810674	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719912	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719912	C₄	1	2	yes	—
9	C₂×C₈×D₅₃	C₄	1	2	yes	—
10	C₄²×D₅₃	C₂²	1	1	yes	characteristic
11	#13133146	C₄	1	2	yes	—
12	#13133129	C₄	1	2	yes	—
13	#13133127	C₄	1	2	yes	—
14	C₄×#88322	C₂²	1	1	yes	characteristic
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₈×Dic₅₃	C₄	1	2	yes	—
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂×#88322	C₂×C₄	1	2	yes	—
34	#11719912	—	2	4	no	—
35	C₈×D₅₃	—	2	4	no	—
36	C₂²×D₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
41	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
42	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
43	C₄×D₅₃	—	2	2	no	—
44	C₄×D₅₃	—	2	2	no	—
45	C₄×D₅₃	M₁₆	1	4	yes	—
46	C₈×C₅₃	—	2	2	no	—
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	—	2	2	no	—
49	#88322	—	2	2	no	—
50	#88322	—	2	2	no	—
51	#88322	M₁₆	1	4	yes	complemented
52	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×D₅₃	—	2	1	no	—
54	C₂×D₅₃	—	2	1	no	—
55	C₂×D₅₃	#97	1	2	yes	—
56	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
57	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
58	C₄×C₅₃	#131	1	2	yes	—
59	Dic₅₃	—	2	1	no	—
60	Dic₅₃	—	2	1	no	—
61	Dic₅₃	#97	1	2	yes	—
62	#1451	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
64	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
65	C₂×C₅₃	#405	1	1	yes	characteristic
66	D₅₃	—	2	2	no	—
67	#431	—	53	1	no	—
68	C₄×M₁₆	—	53	1	no	—
69	C₂×C₄×C₈	—	53	1	no	—
70	#322	—	53	1	no	—
71	#322	—	53	1	no	—
72	#322	—	53	1	no	—
73	#322	D₅₃	1	1	yes	characteristic, complemented
74	C₅₃	#1451	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	53	2	no	—
76	C₂²×C₈	—	53	2	no	—
77	C₂×C₄²	—	53	1	no	—
78	C₄⋊C₈	—	53	2	no	—
79	C₂²⋊C₈	—	53	2	no	—
80	#97	—	53	2	no	—
81	C₄×C₈	—	53	2	no	—
82	C₄×C₈	—	53	1	no	—
83	C₄×C₈	—	53	1	no	—
84	C₄×C₈	—	53	1	no	—
85	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	C₂²×C₄	—	53	1	no	—
89	C₂²×C₄	—	53	1	no	—
90	C₂²×C₄	—	53	1	no	—
91	M₁₆	—	106	4	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	—	106	4	no	—
96	C₂×C₈	C₄×D₅₃	1	2	yes	—
97	C₂×C₈	C₄×D₅₃	1	2	yes	—
98	C₂×C₈	C₄×D₅₃	1	2	yes	—
99	C₄²	—	53	1	no	—
100	C₄²	—	53	1	no	—
101	C₄²	—	53	1	no	—
102	C₄²	C₂²×D₅₃	1	1	yes	characteristic
103	C₂³	—	53	1	no	—
104	C₂×C₄	—	53	1	no	—
105	C₂×C₄	—	53	4	no	—
106	C₂×C₄	—	53	1	no	—
107	C₂×C₄	—	53	1	no	—
108	C₂×C₄	—	106	2	no	—
109	C₂×C₄	—	106	2	no	—
110	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	53	4	no	—
117	C₈	—	106	2	no	—
118	C₈	—	106	2	no	—
119	C₂²	—	53	2	no	—
120	C₂²	—	106	1	no	—
121	C₂²	—	106	1	no	—
122	C₂²	C₄²×D₅₃	1	1	yes	characteristic
123	C₄	—	53	2	no	—
124	C₄	—	106	1	no	—
125	C₄	—	106	1	no	—
126	C₄	M₁₆×D₅₃	1	2	yes	—
127	C₄	M₁₆×D₅₃	1	2	yes	—
128	C₄	#13133223	1	2	yes	—
129	C₂	—	106	2	no	—
130	C₂	#97×D₅₃	1	1	yes	characteristic
131	C₂	#16810575	1	1	yes	characteristic
132	C₂	#16810577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810674

C₂

yes

maximal, characteristic

C₄×#11719912

C₂

yes

maximal, characteristic

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, characteristic, complemented

#16810345

C₂

yes

maximal, characteristic, complemented

#16810345

C₂

yes

maximal, characteristic, complemented

C₅₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

C₄

yes

—

C₂×C₈×D₅₃

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133146

C₄

yes

—

#13133129

C₄

yes

—

#13133127

C₄

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

#11719912

—

C₈×D₅₃

—

C₂²×D₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₂×Dic₅₃

C₄²

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

M₁₆

yes

complemented

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#97

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

#131

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#97

yes

—

#1451

—

maximal, Sylow

C₂×C₅₃

C₂×#97

yes

characteristic

C₂×C₅₃

C₄×M₁₆

yes

characteristic

C₂×C₅₃

#405

yes

characteristic

D₅₃

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1451

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×D₅₃

yes

characteristic

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

111

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

112

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

106

—

118

C₈

—

106

—

119

C₂²

—

120

C₂²

—

106

—

121

C₂²

—

106

—

122

C₂²

C₄²×D₅₃

yes

characteristic

123

C₄

—

124

C₄

—

106

—

125

C₄

—

106

—

126

C₄

M₁₆×D₅₃

yes

—

127

C₄

M₁₆×D₅₃

yes

—

128

C₄

#13133223

yes

—

129

C₂

—

106

—

130

C₂

#97×D₅₃

yes

characteristic

131

C₂

#16810575

yes

characteristic

132

C₂

#16810577

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₅₃×#322	C₂
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1451
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂×C₄²

Fitting

C₅₃×#322

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #16810345	C₂	semidirect
5. #16810345	C₂	semidirect
6. #16810345	C₂	semidirect
7. C₅₃×#322	C₂	semidirect
51. #88322	M₁₆	semidirect
73. #322	D₅₃	semidirect
74. C₅₃	#1451	semidirect

H ≅ G/N

type

4. #16810345

C₂

semidirect

5. #16810345

C₂

semidirect

6. #16810345

C₂

semidirect

7. C₅₃×#322

C₂

semidirect

51. #88322

M₁₆

semidirect

73. #322

D₅₃

semidirect

74. C₅₃

#1451

semidirect

Subgroups — #20273003

Conjugacy classes of subgroups

Complements