Group #20276274 — subgroups

Subgroups — #20276274

Order 6784.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#16810521	C₂	1	1	yes	maximal, characteristic
2	#16810493	C₂	1	1	yes	maximal, characteristic
3	#16810386	C₂	1	2	yes	maximal
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	#13133124	—	2	1	no	—
13	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	1	no	—
19	C₂²×Dic₅₃	—	2	1	no	—
20	C₂²×Dic₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₄²×C₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	—	2	1	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	2	4	no	—
28	C₄⋊Dic₅₃	—	4	2	no	—
29	C₄⋊Dic₅₃	—	4	2	no	—
30	C₄⋊Dic₅₃	Q₈	1	2	yes	—
31	#11719919	—	4	2	no	—
32	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	D₈	1	2	yes	—
38	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	2	yes	—
41	C₂×C₄×C₅₃	#41	1	2	yes	—
42	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	—	4	1	no	—
45	C₂×Dic₅₃	—	4	1	no	—
46	C₂×Dic₅₃	—	4	1	no	—
47	C₂×Dic₅₃	—	4	2	no	—
48	C₂×Dic₅₃	—	4	1	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	4	1	no	—
52	C₈×C₅₃	—	4	2	no	—
53	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₅₃	#123	1	1	yes	characteristic
55	C₂²×C₅₃	#124	1	1	yes	characteristic
56	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₅₃	#137	1	1	yes	characteristic
59	C₂²×C₅₃	#137	1	1	yes	characteristic
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	#125	1	2	yes	—
63	C₄×C₅₃	#126	1	2	yes	—
64	Dic₅₃	—	4	2	no	—
65	Dic₅₃	—	8	1	no	—
66	Dic₅₃	—	8	2	no	—
67	#2099	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	#400	1	1	yes	characteristic
69	C₂×C₅₃	#474	1	1	yes	characteristic
70	C₂×C₅₃	#475	1	1	yes	characteristic
71	C₂×C₅₃	#480	1	1	yes	characteristic
72	C₂×C₅₃	#483	1	1	yes	characteristic
73	C₂×C₅₃	#486	1	1	yes	characteristic
74	C₂×C₅₃	#491	1	1	yes	characteristic
75	#389	—	53	1	no	—
76	#384	—	53	1	no	—
77	#340	—	53	2	no	—
78	#340	—	53	2	no	—
79	#336	D₅₃	1	1	yes	characteristic
80	C₅₃	#2099	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	C₂×D₅₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	53	1	no	—
83	C₂×C₄⋊C₄	—	53	2	no	—
84	C₂×C₄⋊C₄	—	53	1	no	—
85	C₂×C₄⋊C₄	—	106	1	no	—
86	C₂×C₄⋊C₄	—	106	1	no	—
87	C₂×C₄⋊C₄	—	106	1	no	—
88	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
89	#95	—	106	1	no	—
90	C₂²×C₄	—	106	1	no	—
91	C₂²×C₄	—	106	1	no	—
92	C₂²×C₄	—	106	2	no	—
93	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
94	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	53	2	no	—
100	C₄⋊C₄	—	106	1	no	—
101	C₄⋊C₄	—	106	4	no	—
102	C₄⋊C₄	—	106	2	no	—
103	C₄⋊C₄	—	212	2	no	—
104	C₄⋊C₄	—	212	2	no	—
105	C₄⋊C₄	—	212	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	106	2	no	—
113	C₂×C₄	—	212	1	no	—
114	C₂×C₄	—	212	2	no	—
115	C₂×C₄	—	212	1	no	—
116	C₂×C₄	—	212	2	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	—	212	1	no	—
119	C₂×C₄	—	212	1	no	—
120	C₂×C₄	—	212	1	no	—
121	C₂×C₄	#11719913	1	2	yes	—
122	C₂×C₄	D₄₂₄	1	2	yes	—
123	C₂×C₄	#11719945	1	2	yes	—
124	C₂×C₄	#11719947	1	2	yes	—
125	C₂×C₄	#11719947	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	#13133197	1	1	yes	characteristic
129	C₂²	#13133201	1	1	yes	characteristic
130	C₂²	#13133205	1	1	yes	characteristic
131	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
132	C₂²	C₂×#11719913	1	1	yes	characteristic
133	C₂²	#13133250	1	1	yes	characteristic
134	C₂²	#13133250	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	424	1	no	—
138	C₄	—	424	2	no	—
139	C₄	—	212	2	no	—
140	C₄	#13133207	1	2	yes	—
141	C₄	#13133210	1	2	yes	—
142	C₂	#16810563	1	1	yes	characteristic
143	C₂	#16810904	1	1	yes	characteristic
144	C₂	#16810909	1	1	yes	characteristic
145	C₂	#16810937	1	1	yes	characteristic
146	C₂	#16810954	1	1	yes	characteristic
147	C₂	#16810970	1	1	yes	characteristic
148	C₂	#16810992	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810521

C₂

yes

maximal, characteristic

#16810493

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal

#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

characteristic

C₂×#11719919

—

#13133124

—

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

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

Q₈

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

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

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

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#125

yes

—

C₄×C₅₃

#126

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

#2099

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

#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

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

212

—

104

C₄⋊C₄

—

212

—

105

C₄⋊C₄

—

212

—

106

C₄²

—

107

C₂³

C₂×D₂₁₂

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

106

—

113

C₂×C₄

—

212

—

114

C₂×C₄

—

212

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

—

212

—

118

C₂×C₄

—

212

—

119

C₂×C₄

—

212

—

120

C₂×C₄

—

212

—

121

C₂×C₄

#11719913

yes

—

122

C₂×C₄

D₄₂₄

yes

—

123

C₂×C₄

#11719945

yes

—

124

C₂×C₄

#11719947

yes

—

125

C₂×C₄

#11719947

yes

characteristic

126

C₂×C₄

Q₈×D₅₃

yes

characteristic

127

C₈

—

128

C₂²

#13133197

yes

characteristic

129

C₂²

#13133201

yes

characteristic

130

C₂²

#13133205

yes

characteristic

131

C₂²

C₂×D₄₂₄

yes

characteristic

132

C₂²

C₂×#11719913

yes

characteristic

133

C₂²

#13133250

yes

characteristic

134

C₂²

#13133250

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

424

—

138

C₄

—

424

—

139

C₄

—

212

—

140

C₄

#13133207

yes

—

141

C₄

#13133210

yes

—

142

C₂

#16810563

yes

characteristic

143

C₂

#16810904

yes

characteristic

144

C₂

#16810909

yes

characteristic

145

C₂

#16810937

yes

characteristic

146

C₂

#16810954

yes

characteristic

147

C₂

#16810970

yes

characteristic

148

C₂

#16810992

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	#2099
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
80. C₅₃	#2099	semidirect

H ≅ G/N

type

80. C₅₃

#2099

semidirect

Subgroups — #20276274

Conjugacy classes of subgroups

Complements