Group #27271168 — subgroups

Subgroups — #27271168

Order 13568.

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

normal subgroups only

This group has 270 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	#20274753	C₂	1	2	yes	maximal
2	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133153	—	2	2	no	—
4	#16810587	—	2	2	no	—
5	#16810388	C₄	1	4	yes	—
6	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
8	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
9	C₂×#11719917	—	2	4	no	—
10	#13133153	—	4	4	no	—
11	C₂²⋊#88322	—	4	2	no	—
12	C₂²⋊#88322	—	4	2	no	—
13	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	—	4	2	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	D₄	1	2	yes	—
18	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	—	2	2	no	—
20	C₂×C₁₆×C₅₃	—	4	2	no	—
21	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	—	4	1	no	—
23	C₂²×C₄×C₅₃	—	4	1	no	—
24	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	4	no	—
30	C₂×C₈×C₅₃	—	4	2	no	—
31	C₂×C₈×C₅₃	—	4	1	no	—
32	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	#11719917	—	4	4	no	—
36	#11719917	—	4	4	no	—
37	C₂×#88322	—	4	4	no	—
38	C₁₆×C₅₃	—	4	2	no	—
39	C₂³×C₅₃	—	2	2	no	—
40	C₂³×C₅₃	—	4	1	no	—
41	C₂³×C₅₃	—	4	1	no	—
42	C₂³×C₅₃	#103	1	2	yes	—
43	C₂³×C₅₃	#107	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#102	1	2	yes	—
52	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
53	C₈×C₅₃	—	2	1	no	—
54	C₈×C₅₃	—	2	2	no	—
55	C₈×C₅₃	#99	1	2	yes	—
56	#88322	—	8	4	no	—
57	#7231	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	—	2	2	no	—
61	C₂²×C₅₃	—	2	2	no	—
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	—	4	4	no	—
64	C₂²×C₅₃	#340	1	1	yes	characteristic
65	C₂²×C₅₃	#368	1	2	yes	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
69	C₄×C₅₃	#342	1	1	yes	characteristic
70	C₂×C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
71	#1810	—	53	2	no	—
72	C₂×C₅₃	—	2	2	no	—
73	C₂×C₅₃	—	4	2	no	—
74	C₂×C₅₃	#1289	1	1	yes	characteristic
75	C₂×C₅₃	#1381	1	1	yes	characteristic
76	C₂×C₅₃	#1385	1	1	yes	characteristic
77	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
78	C₂²×C₁₆	Dic₅₃	1	2	yes	—
79	C₂×#108	—	106	2	no	—
80	#407	—	106	2	no	—
81	C₂²⋊C₁₆	—	2	2	no	—
82	#341	—	53	4	no	—
83	C₅₃	#7231	1	1	yes	characteristic, complemented, Sylow
84	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
85	C₂×M₁₆	—	106	4	no	—
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
88	C₂²×C₈	C₄×D₅₃	1	2	yes	—
89	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
90	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	4	2	no	—
93	#108	—	212	4	no	—
94	C₂²⋊C₈	—	212	2	no	—
95	C₂²⋊C₈	—	212	2	no	—
96	C₂⁴	Dic₂₁₂	1	1	yes	characteristic
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	4	1	no	—
99	C₂²×C₄	#11719913	1	2	yes	—
100	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
101	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
102	C₂²×C₄	#11719921	1	2	yes	—
103	M₁₆	—	212	4	no	—
104	M₁₆	—	212	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	212	4	no	—
110	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
111	C₂×C₈	#11719919	1	2	yes	—
112	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
113	C₁₆	—	4	2	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	4	1	no	—
116	C₂³	—	4	1	no	—
117	C₂³	#13133142	1	2	yes	—
118	C₂³	#13133151	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	#13133124	1	1	yes	characteristic
126	C₂×C₄	#13133139	1	2	yes	—
127	C₂×C₄	#13133149	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	2	no	—
130	C₈	—	424	4	no	—
131	C₈	#13133132	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	4	no	—
138	C₂²	#16810386	1	1	yes	characteristic
139	C₂²	#16810453	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	#16810380	1	1	yes	characteristic
143	C₄	#16810390	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	—	4	2	no	—
146	C₂	#20272583	1	1	yes	characteristic
147	C₂	#20272809	1	1	yes	characteristic
148	C₂	#20272819	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274753

C₂

yes

maximal

C₂×C₅₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#13133153

—

#16810587

—

#16810388

C₄

yes

—

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₄

yes

—

C₅₃×C₂²⋊C₁₆

—

—

—

—

—

C₂³×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂⁴×C₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#103

yes

—

C₂³×C₅₃

#107

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#99

yes

—

#88322

—

#7231

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₂²×C₅₃

#368

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₂²⋊C₁₆

D₅₃

yes

characteristic

#1810

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂³×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₁₆

Dic₅₃

yes

—

C₂×#108

—

106

—

#407

—

106

—

C₂²⋊C₁₆

—

#341

—

C₅₃

#7231

yes

characteristic, complemented, Sylow

C₂³×C₄

D₂₁₂

yes

characteristic

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

C₄×D₅₃

yes

—

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

212

—

C₂²⋊C₈

—

212

—

C₂²⋊C₈

—

212

—

C₂⁴

Dic₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#11719913

yes

—

100

C₂²×C₄

D₄₂₄

yes

characteristic

101

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

102

C₂²×C₄

#11719921

yes

—

103

M₁₆

—

212

—

104

M₁₆

—

212

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

212

—

110

C₂×C₈

C₄×Dic₅₃

yes

characteristic

111

C₂×C₈

#11719919

yes

—

112

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

113

C₁₆

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

#13133142

yes

—

118

C₂³

#13133151

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#13133124

yes

characteristic

126

C₂×C₄

#13133139

yes

—

127

C₂×C₄

#13133149

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

424

—

131

C₈

#13133132

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#16810386

yes

characteristic

139

C₂²

#16810453

yes

—

140

C₄

—

141

C₄

—

142

C₄

#16810380

yes

characteristic

143

C₄

#16810390

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#20272583

yes

characteristic

147

C₂

#20272809

yes

characteristic

148

C₂

#20272819

yes

characteristic

	subgroup	G/H
center	C₂²	#16810386
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂³×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂³×C₈×C₅₃	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#7231
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#16810386

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂³×C₈

C₂×D₅₃

Fitting

C₂×C₅₃×C₂²⋊C₁₆

C₂

socle

C₂²×C₅₃

#340

prime radical

C₂⁴×C₅₃

Q₁₆

cosocle

C₂³×C₈×C₅₃

C₂²

hypercenter

C₂×C₂²⋊C₁₆

D₅₃

hypocenter

C₅₃

#7231

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

N	H ≅ G/N	type
83. C₅₃	#7231	semidirect

H ≅ G/N

type

83. C₅₃

#7231

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274753	2
C₂×C₅₃×C₂²⋊C₁₆	2
#7231	53

subgroup

index

#20274753

C₂×C₅₃×C₂²⋊C₁₆

#7231

Subgroups — #27271168

Conjugacy classes of subgroups

Complements

Maximal subgroups