Group #20272616 — subgroups

Subgroups — #20272616

Order 6784.

176 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₃	—	2	1	no	—
5	C₂×C₄×Dic₅₃	—	2	1	no	—
6	C₄⋊#88322	C₄	1	2	yes	complemented
7	C₂²⋊#88322	—	2	1	no	—
8	C₄×#88322	—	2	1	no	—
9	#13133124	C₄	1	2	yes	—
10	C₂×M₁₆×C₅₃	—	2	1	no	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₅₃	—	2	1	no	—
13	C₄⋊C₈×C₅₃	C₄	1	2	yes	complemented
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	2	no	—
18	M₁₆×C₅₃	—	4	1	no	—
19	M₁₆×C₅₃	—	4	1	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₄²×C₅₃	Q₈	1	1	yes	characteristic
25	C₄⋊Dic₅₃	—	2	4	no	—
26	C₄×Dic₅₃	—	4	1	no	—
27	C₄×Dic₅₃	—	4	1	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	4	1	no	—
32	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
41	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
42	C₂×Dic₅₃	—	2	4	no	—
43	C₂×Dic₅₃	—	4	2	no	—
44	C₂×Dic₅₃	—	4	2	no	—
45	C₈×C₅₃	—	4	2	no	—
46	#88322	—	4	2	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	#95	1	1	yes	characteristic
50	C₂²×C₅₃	#102	1	1	yes	characteristic
51	C₂²×C₅₃	#103	1	1	yes	characteristic
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	#102	1	1	yes	characteristic
56	C₄×C₅₃	#103	1	1	yes	characteristic
57	C₄×C₅₃	#104	1	2	yes	—
58	C₄×C₅₃	#106	1	1	yes	characteristic
59	C₄×C₅₃	#107	1	1	yes	characteristic
60	Dic₅₃	—	4	4	no	—
61	#1303	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	—	2	1	no	—
64	C₂×C₅₃	#339	1	1	yes	characteristic
65	C₂×C₅₃	#340	1	1	yes	characteristic
66	C₂×C₅₃	#344	1	1	yes	characteristic
67	#431	—	53	1	no	—
68	#423	D₅₃	1	1	yes	characteristic
69	C₄×C₄⋊C₄	—	53	1	no	—
70	C₅₃	#1303	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	2	1	no	—
72	C₂×C₄⋊C₄	—	106	1	no	—
73	C₂×C₄²	—	106	1	no	—
74	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	53	2	no	—
77	C₄⋊C₈	Dic₅₃	1	2	yes	complemented
78	C₂²⋊C₈	—	106	1	no	—
79	C₄×C₈	—	106	1	no	—
80	#95	—	53	2	no	—
81	C₂²×C₄	—	106	2	no	—
82	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
83	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
85	M₁₆	—	4	1	no	—
86	M₁₆	—	4	1	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	106	2	no	—
89	C₄⋊C₄	—	106	4	no	—
90	C₄²	—	212	1	no	—
91	C₄²	—	212	1	no	—
92	C₄²	Dic₁₀₆	1	1	yes	characteristic
93	C₄²	C₄×D₅₃	1	1	yes	characteristic
94	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
95	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
96	C₂³	#11719921	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	106	4	no	—
100	C₂×C₄	—	212	2	no	—
101	C₂×C₄	—	212	2	no	—
102	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
103	C₂×C₄	#11719919	1	1	yes	characteristic
104	C₂×C₄	#11719919	1	1	yes	characteristic
105	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
106	C₂×C₄	#11719921	1	1	yes	characteristic
107	C₂×C₄	#11719922	1	1	yes	characteristic
108	C₂×C₄	#11719923	1	1	yes	characteristic
109	C₂×C₄	#11719924	1	1	yes	characteristic
110	C₂×C₄	#11719925	1	1	yes	characteristic
111	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	212	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#13133124	1	1	yes	characteristic
117	C₂²	#13133137	1	1	yes	characteristic
118	C₂²	#13133140	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	212	4	no	—
123	C₄	#13133138	1	1	yes	characteristic
124	C₄	#13133141	1	1	yes	characteristic
125	C₄	#13133143	1	2	yes	—
126	C₄	#13133148	1	1	yes	characteristic
127	C₄	#13133150	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#16810384	1	1	yes	characteristic
131	C₂	#16810385	1	1	yes	characteristic
132	C₂	#16810394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

C₅₃×#423

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

—

C₂×C₄×Dic₅₃

—

C₄⋊#88322

C₄

yes

—

—

C₄

yes

—

C₂×M₁₆×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×C₄×C₅₃

Q₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1303

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

C₅₃

#1303

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₅₃

yes

complemented

C₂²⋊C₈

—

106

—

C₄×C₈

—

106

—

#95

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

212

—

C₄²

—

212

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

#11719921

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

100

C₂×C₄

—

212

—

101

C₂×C₄

—

212

—

102

C₂×C₄

C₄×Dic₅₃

yes

characteristic

103

C₂×C₄

#11719919

yes

characteristic

104

C₂×C₄

#11719919

yes

characteristic

105

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

106

C₂×C₄

#11719921

yes

characteristic

107

C₂×C₄

#11719922

yes

characteristic

108

C₂×C₄

#11719923

yes

characteristic

109

C₂×C₄

#11719924

yes

characteristic

110

C₂×C₄

#11719925

yes

characteristic

111

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

112

C₈

—

113

C₈

—

212

—

114

C₂²

—

115

C₂²

—

116

C₂²

#13133124

yes

characteristic

117

C₂²

#13133137

yes

characteristic

118

C₂²

#13133140

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

212

—

123

C₄

#13133138

yes

characteristic

124

C₄

#13133141

yes

characteristic

125

C₄

#13133143

yes

—

126

C₄

#13133148

yes

characteristic

127

C₄

#13133150

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#16810384

yes

characteristic

131

C₂

#16810385

yes

characteristic

132

C₂

#16810394

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	#95
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1303
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₅₃

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₅₃×#423

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
6. C₄⋊#88322	C₄	semidirect
13. C₄⋊C₈×C₅₃	C₄	semidirect
70. C₅₃	#1303	semidirect
77. C₄⋊C₈	Dic₅₃	semidirect

H ≅ G/N

type

6. C₄⋊#88322

C₄

semidirect

13. C₄⋊C₈×C₅₃

C₄

semidirect

70. C₅₃

#1303

semidirect

77. C₄⋊C₈

Dic₅₃

semidirect

Subgroups — #20272616

Conjugacy classes of subgroups

Complements