Group #20276188 — subgroups

Subgroups — #20276188

Order 6784.

230 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₂×#13133151	C₂	1	2	yes	maximal
2	#16810521	C₂	1	1	yes	maximal, characteristic
3	#16810493	C₂	1	1	yes	maximal, characteristic
4	#16810386	C₂	1	2	yes	maximal
5	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₅₃	—	2	1	no	—
7	C₂×C₄⋊Dic₅₃	—	2	1	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×#11719919	—	2	1	no	—
12	#13133151	—	2	4	no	—
13	#13133124	—	2	1	no	—
14	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	—	2	1	no	—
20	C₂²×Dic₅₃	—	2	1	no	—
21	C₂²×Dic₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	Q₈	1	4	yes	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	1	no	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	—	4	2	no	—
31	C₄⋊Dic₅₃	—	4	2	no	—
32	C₄⋊Dic₅₃	D₄	1	2	yes	—
33	#11719919	—	4	2	no	—
34	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	—	2	1	no	—
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	D₈	1	2	yes	—
40	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₅₃	#41	1	2	yes	—
44	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×Dic₅₃	—	2	2	no	—
46	C₂×Dic₅₃	—	4	1	no	—
47	C₂×Dic₅₃	—	4	1	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	1	no	—
50	C₂×Dic₅₃	—	4	1	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	C₂×Dic₅₃	—	4	1	no	—
54	C₈×C₅₃	—	2	4	no	—
55	C₂²×C₅₃	#121	1	1	yes	characteristic
56	C₂²×C₅₃	#123	1	1	yes	characteristic
57	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₅₃	#137	1	1	yes	characteristic
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	—	2	2	no	—
64	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₅₃	#125	1	2	yes	—
66	Dic₅₃	—	4	2	no	—
67	Dic₅₃	—	8	2	no	—
68	Dic₅₃	—	8	1	no	—
69	#2083	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	#398	1	1	yes	characteristic
71	C₂×C₅₃	#459	1	1	yes	characteristic
72	C₂×C₅₃	#462	1	1	yes	characteristic
73	C₂×C₅₃	#480	1	1	yes	characteristic
74	C₂×C₅₃	#484	1	1	yes	characteristic
75	C₂×C₅₃	#500	1	1	yes	characteristic
76	C₂×C₅₃	#501	1	1	yes	characteristic
77	C₂×#107	—	53	2	no	—
78	#389	—	53	1	no	—
79	#384	—	53	1	no	—
80	#340	—	53	2	no	—
81	#336	D₅₃	1	1	yes	characteristic
82	C₅₃	#2083	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	C₂×D₅₃	1	2	yes	—
84	C₂×C₄⋊C₄	—	53	1	no	—
85	C₂×C₄⋊C₄	—	53	1	no	—
86	C₂×C₄⋊C₄	—	53	2	no	—
87	C₂×C₄⋊C₄	—	106	1	no	—
88	C₂×C₄⋊C₄	—	106	1	no	—
89	C₂×C₄⋊C₄	—	106	1	no	—
90	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
91	#107	—	106	4	no	—
92	#95	—	106	1	no	—
93	C₂²×C₄	—	106	1	no	—
94	C₂²×C₄	—	106	1	no	—
95	C₂²×C₄	—	106	2	no	—
96	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
97	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	Dic₁₀₆	1	4	yes	—
102	C₄⋊C₄	—	53	2	no	—
103	C₄⋊C₄	—	106	1	no	—
104	C₄⋊C₄	—	106	2	no	—
105	C₄⋊C₄	—	106	2	no	—
106	C₄⋊C₄	—	106	2	no	—
107	C₄⋊C₄	—	212	2	no	—
108	C₄⋊C₄	—	212	2	no	—
109	C₄⋊C₄	—	212	2	no	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	106	2	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	—	212	1	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	—	212	1	no	—
121	C₂×C₄	—	212	2	no	—
122	C₂×C₄	—	212	1	no	—
123	C₂×C₄	—	212	1	no	—
124	C₂×C₄	—	212	1	no	—
125	C₂×C₄	D₄₂₄	1	2	yes	—
126	C₂×C₄	Dic₂₁₂	1	2	yes	—
127	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
128	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
129	C₂×C₄	#11719947	1	2	yes	—
130	C₂×C₄	#11719949	1	1	yes	characteristic
131	C₈	—	2	4	no	—
132	C₂²	#13133192	1	1	yes	characteristic
133	C₂²	#13133201	1	1	yes	characteristic
134	C₂²	#13133217	1	1	yes	characteristic
135	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
136	C₂²	C₂×Dic₂₁₂	1	1	yes	characteristic
137	C₂²	#13133245	1	1	yes	characteristic
138	C₂²	#13133250	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	424	2	no	—
142	C₄	—	424	1	no	—
143	C₄	—	212	2	no	—
144	C₄	#13133194	1	2	yes	—
145	C₄	#13133208	1	2	yes	—
146	C₂	#16810555	1	1	yes	characteristic
147	C₂	#16810819	1	1	yes	characteristic
148	C₂	#16810834	1	1	yes	characteristic
149	C₂	#16810937	1	1	yes	characteristic
150	C₂	#16810959	1	1	yes	characteristic
151	C₂	#16811028	1	1	yes	characteristic
152	C₂	#16811032	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal

#16810521

C₂

yes

maximal, characteristic

#16810493

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal

C₅₃×#336

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

#11719919

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#125

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#2083

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

106

—

#95

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Dic₁₀₆

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

—

106

—

107

C₄⋊C₄

—

212

—

108

C₄⋊C₄

—

212

—

109

C₄⋊C₄

—

212

—

110

C₄²

—

111

C₂³

C₂×D₂₁₂

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

212

—

118

C₂×C₄

—

212

—

119

C₂×C₄

—

212

—

120

C₂×C₄

—

212

—

121

C₂×C₄

—

212

—

122

C₂×C₄

—

212

—

123

C₂×C₄

—

212

—

124

C₂×C₄

—

212

—

125

C₂×C₄

D₄₂₄

yes

—

126

C₂×C₄

Dic₂₁₂

yes

—

127

C₂×C₄

C₂×Dic₁₀₆

yes

—

128

C₂×C₄

D₄×D₅₃

yes

characteristic

129

C₂×C₄

#11719947

yes

—

130

C₂×C₄

#11719949

yes

characteristic

131

C₈

—

132

C₂²

#13133192

yes

characteristic

133

C₂²

#13133201

yes

characteristic

134

C₂²

#13133217

yes

characteristic

135

C₂²

C₂×D₄₂₄

yes

characteristic

136

C₂²

C₂×Dic₂₁₂

yes

characteristic

137

C₂²

#13133245

yes

characteristic

138

C₂²

#13133250

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

424

—

142

C₄

—

424

—

143

C₄

—

212

—

144

C₄

#13133194

yes

—

145

C₄

#13133208

yes

—

146

C₂

#16810555

yes

characteristic

147

C₂

#16810819

yes

characteristic

148

C₂

#16810834

yes

characteristic

149

C₂

#16810937

yes

characteristic

150

C₂

#16810959

yes

characteristic

151

C₂

#16811028

yes

characteristic

152

C₂

#16811032

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#2083
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
82. C₅₃	#2083	semidirect

H ≅ G/N

type

82. C₅₃

#2083

semidirect

Subgroups — #20276188

Conjugacy classes of subgroups

Complements