Group #20276258 — subgroups

Subgroups — #20276258

Order 6784.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13133148	C₂	1	2	yes	maximal
2	C₂×#13133141	C₂	1	2	yes	maximal
3	#16810493	C₂	1	1	yes	maximal, characteristic
4	#16810378	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	—	2	2	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	#13133148	—	2	4	no	—
10	#13133141	—	2	4	no	—
11	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	#95×C₅₃	—	2	2	no	—
15	C₂×Q₈×C₅₃	—	2	1	no	—
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	Q₈	1	2	yes	—
19	C₂²×C₄×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	D₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	4	4	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	D₄	1	4	yes	—
32	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₅₃	—	2	4	no	—
34	Q₈×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	2	2	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	—	4	2	no	—
42	C₂×C₄×C₅₃	SD₁₆	1	4	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂×Dic₅₃	—	4	2	no	—
48	C₂×Dic₅₃	—	4	4	no	—
49	#88322	—	2	4	no	—
50	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₅₃	#123	1	1	yes	characteristic
52	C₂²×C₅₃	#127	1	1	yes	characteristic
53	C₂²×C₅₃	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₅₃	#137	1	2	yes	—
55	C₄×C₅₃	—	2	4	no	—
56	C₄×C₅₃	—	4	2	no	—
57	C₄×C₅₃	—	8	1	no	—
58	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₅₃	#123	1	2	yes	—
60	Dic₅₃	—	8	2	no	—
61	#2096	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	#399	1	1	yes	characteristic
63	C₂×C₅₃	#475	1	2	yes	—
64	C₂×C₅₃	#483	1	2	yes	—
65	C₂×C₅₃	#492	1	1	yes	characteristic
66	C₂×C₅₃	#497	1	1	yes	characteristic
67	C₂×#106	—	53	2	no	—
68	C₂×#103	—	53	2	no	—
69	#391	D₅₃	1	1	yes	characteristic
70	#384	—	53	1	no	—
71	#336	—	53	1	no	—
72	C₅₃	#2096	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
74	C₂²×C₈	—	53	2	no	—
75	C₂×C₄⋊C₄	—	53	2	no	—
76	C₂×C₄⋊C₄	—	106	2	no	—
77	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
79	#106	—	106	4	no	—
80	#103	—	106	4	no	—
81	#95	—	2	2	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	Dic₁₀₆	1	2	yes	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	106	2	no	—
89	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
91	C₂×C₈	—	53	4	no	—
92	C₂×C₈	—	106	2	no	—
93	C₂×C₈	—	106	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	106	2	no	—
96	C₄⋊C₄	—	106	2	no	—
97	C₄⋊C₄	—	212	4	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	212	2	no	—
110	C₂×C₄	—	212	4	no	—
111	C₂×C₄	#11719924	1	4	yes	—
112	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
113	C₂×C₄	#11719945	1	1	yes	characteristic
114	C₂×C₄	D₄×D₅₃	1	2	yes	—
115	C₂×C₄	#11719947	1	2	yes	—
116	C₈	—	106	4	no	—
117	C₂²	#13133196	1	1	yes	characteristic
118	C₂²	#13133198	1	1	yes	characteristic
119	C₂²	#13133213	1	1	yes	characteristic
120	C₂²	C₂×#11719924	1	2	yes	—
121	C₂²	#13133247	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	1	no	—
125	C₄	—	424	2	no	—
126	C₄	#13133194	1	2	yes	—
127	C₄	#13133199	1	2	yes	—
128	C₂	#16810559	1	1	yes	characteristic
129	C₂	#16810908	1	2	yes	—
130	C₂	#16810951	1	2	yes	—
131	C₂	#16810997	1	1	yes	characteristic
132	C₂	#16811018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133148

C₂

yes

maximal

C₂×#13133141

C₂

yes

maximal

#16810493

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal, characteristic

C₅₃×#391

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133148

—

#13133141

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

C₂²⋊Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×SD₁₆

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#123

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#483

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#384

—

#336

—

C₅₃

#2096

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

#106

—

106

—

#103

—

106

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₀₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

212

—

110

C₂×C₄

—

212

—

111

C₂×C₄

#11719924

yes

—

112

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

113

C₂×C₄

#11719945

yes

characteristic

114

C₂×C₄

D₄×D₅₃

yes

—

115

C₂×C₄

#11719947

yes

—

116

C₈

—

106

—

117

C₂²

#13133196

yes

characteristic

118

C₂²

#13133198

yes

characteristic

119

C₂²

#13133213

yes

characteristic

120

C₂²

C₂×#11719924

yes

—

121

C₂²

#13133247

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

424

—

126

C₄

#13133194

yes

—

127

C₄

#13133199

yes

—

128

C₂

#16810559

yes

characteristic

129

C₂

#16810908

yes

—

130

C₂

#16810951

yes

—

131

C₂

#16810997

yes

characteristic

132

C₂

#16811018

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#391	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#391	D₅₃
hypocenter	C₅₃	#2096
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#391

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
72. C₅₃	#2096	semidirect

H ≅ G/N

type

72. C₅₃

#2096

semidirect

Subgroups — #20276258

Conjugacy classes of subgroups

Complements