Group #20272569 — subgroups

Subgroups — #20272569

Order 6784.

226 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810486	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#407	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
5	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
6	C₂×C₄×Dic₅₃	C₄	1	2	yes	complemented
7	#13133124	—	2	1	no	—
8	#13133124	—	2	1	no	—
9	#13133124	C₄	1	2	yes	complemented
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₅₃	—	2	1	no	—
12	C₂²⋊C₈×C₅₃	—	2	1	no	—
13	C₂²⋊C₈×C₅₃	C₄	1	2	yes	complemented
14	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	—	4	2	no	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₂²×Dic₅₃	—	2	2	no	—
24	M₁₆×C₅₃	—	4	2	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂²⋊Dic₅₃	—	2	4	no	—
27	C₂²⋊Dic₅₃	—	4	2	no	—
28	C₄×Dic₅₃	—	4	2	no	—
29	C₂³×C₅₃	—	2	2	no	—
30	C₂³×C₅₃	—	2	2	no	—
31	C₂³×C₅₃	—	4	1	no	—
32	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₅₃	—	2	4	no	—
46	C₂×Dic₅₃	—	4	2	no	—
47	C₂×Dic₅₃	—	4	2	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₈×C₅₃	—	4	2	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	—	4	4	no	—
60	C₂²×C₅₃	#95	1	1	yes	characteristic
61	C₂²×C₅₃	#99	1	2	yes	—
62	C₂²×C₅₃	#104	1	2	yes	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	Dic₅₃	—	4	4	no	—
68	Dic₅₃	—	8	2	no	—
69	#1284	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	—	2	2	no	—
72	C₂×C₅₃	—	4	1	no	—
73	C₂×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
74	C₂×C₅₃	#339	1	1	yes	characteristic
75	C₂×C₅₃	#342	1	1	yes	characteristic
76	#407	D₅₃	1	1	yes	characteristic
77	#381	—	53	1	no	—
78	C₄×C₂²⋊C₄	—	53	1	no	—
79	C₅₃	#1284	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
81	C₂×M₁₆	—	2	1	no	—
82	C₂×C₂²⋊C₄	—	106	1	no	—
83	C₂×C₂²⋊C₄	—	106	1	no	—
84	C₂×C₄²	—	53	2	no	—
85	C₂²⋊C₈	—	2	1	no	—
86	C₂²⋊C₈	Dic₅₃	1	2	yes	complemented
87	#95	—	53	2	no	—
88	#95	—	106	1	no	—
89	#95	—	106	1	no	—
90	C₂⁴	D₂₁₂	1	1	yes	characteristic
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	—	106	2	no	—
93	C₂²×C₄	—	106	2	no	—
94	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
96	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂²⋊C₄	—	106	4	no	—
103	C₂²⋊C₄	—	212	2	no	—
104	C₄²	—	212	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
109	C₂³	#11719921	1	1	yes	characteristic
110	C₂³	#11719921	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	106	4	no	—
118	C₂×C₄	—	212	2	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	—	212	2	no	—
121	C₂×C₄	—	212	2	no	—
122	C₂×C₄	—	212	2	no	—
123	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
124	C₂×C₄	#11719919	1	1	yes	characteristic
125	C₂×C₄	#11719919	1	1	yes	characteristic
126	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	4	4	no	—
136	C₂²	#13133124	1	1	yes	characteristic
137	C₂²	#13133131	1	2	yes	—
138	C₂²	#13133143	1	2	yes	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	424	2	no	—
144	C₄	—	212	4	no	—
145	C₂	—	2	2	no	—
146	C₂	—	2	2	no	—
147	C₂	—	4	1	no	—
148	C₂	#16810380	1	1	yes	characteristic
149	C₂	#16810384	1	1	yes	characteristic
150	C₂	#16810389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810486

C₂

yes

maximal, characteristic

C₄×C₂²⋊Dic₅₃

C₂

yes

maximal, characteristic

C₅₃×#407

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₅₃

—

C₂×C₂²⋊Dic₅₃

—

C₂×C₄×Dic₅₃

C₄

yes

—

—

C₄

yes

complemented

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

C₄

yes

complemented

C₂⁴×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂³×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#95

yes

characteristic

C₂²×C₅₃

#99

yes

—

C₂²×C₅₃

#104

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

#1284

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

#381

—

C₄×C₂²⋊C₄

—

C₅₃

#1284

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×C₂²⋊C₄

—

106

—

C₂×C₂²⋊C₄

—

106

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₅₃

yes

complemented

#95

—

#95

—

106

—

#95

—

106

—

C₂⁴

D₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂²⋊C₄

—

106

—

103

C₂²⋊C₄

—

212

—

104

C₄²

—

212

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₄⋊Dic₅₃

yes

characteristic

109

C₂³

#11719921

yes

characteristic

110

C₂³

#11719921

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

212

—

119

C₂×C₄

—

212

—

120

C₂×C₄

—

212

—

121

C₂×C₄

—

212

—

122

C₂×C₄

—

212

—

123

C₂×C₄

C₄×Dic₅₃

yes

characteristic

124

C₂×C₄

#11719919

yes

characteristic

125

C₂×C₄

#11719919

yes

characteristic

126

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#13133124

yes

characteristic

137

C₂²

#13133131

yes

—

138

C₂²

#13133143

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

424

—

144

C₄

—

212

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#16810380

yes

characteristic

149

C₂

#16810384

yes

characteristic

150

C₂

#16810389

yes

characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂³×C₄	C₂×D₅₃
Fitting	C₅₃×#407	C₂
socle	C₂²×C₅₃	#95
hypercenter	#407	D₅₃
hypocenter	C₅₃	#1284
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

#13133124

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂³×C₄

C₂×D₅₃

Fitting

C₅₃×#407

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
6. C₂×C₄×Dic₅₃	C₄	semidirect
9. #13133124	C₄	semidirect
13. C₂²⋊C₈×C₅₃	C₄	semidirect
79. C₅₃	#1284	semidirect
86. C₂²⋊C₈	Dic₅₃	semidirect

H ≅ G/N

type

6. C₂×C₄×Dic₅₃

C₄

semidirect

9. #13133124

C₄

semidirect

13. C₂²⋊C₈×C₅₃

C₄

semidirect

79. C₅₃

#1284

semidirect

86. C₂²⋊C₈

Dic₅₃

semidirect

Subgroups — #20272569

Conjugacy classes of subgroups

Complements