Group #27270934 — subgroups

Subgroups — #27270934

Order 13568.

246 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20276514	C₂	1	2	yes	maximal
2	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	—	2	2	no	—
4	C₂²⋊#11719908	—	2	2	no	—
5	C₄×#11719908	C₄	1	4	yes	complemented
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₄³×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×#336	—	2	2	no	—
10	C₂×#11719908	—	2	4	no	—
11	C₂²×C₈×C₅₃	—	2	1	no	—
12	C₂²×C₈×C₅₃	D₄	1	2	yes	—
13	C₂×C₄²×C₅₃	—	2	1	no	—
14	C₂×C₄²×C₅₃	—	2	1	no	—
15	C₂×C₄²×C₅₃	C₈	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄×C₈×C₅₃	—	2	1	no	—
18	C₄×C₈×C₅₃	C₈	1	2	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	4	2	no	—
28	C₂×C₈×C₅₃	—	4	2	no	—
29	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₄²×C₅₃	—	2	2	no	—
32	C₄²×C₅₃	—	4	2	no	—
33	C₄²×C₅₃	—	4	2	no	—
34	C₄²×C₅₃	C₁₆	1	4	yes	complemented
35	C₄²×C₅₃	C₄²	1	1	yes	characteristic
36	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
39	#11719908	—	4	4	no	—
40	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
54	C₈×C₅₃	—	2	2	no	—
55	C₈×C₅₃	—	2	2	no	—
56	C₈×C₅₃	—	4	2	no	—
57	#7142	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	2	1	no	—
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	#336	1	1	yes	characteristic
61	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	4	4	no	—
65	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
66	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
67	C₄×C₅₃	#346	1	1	yes	characteristic
68	C₄×C₅₃	#347	1	2	yes	—
69	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
70	#2162	—	53	2	no	—
71	#1744	D₅₃	1	1	yes	characteristic
72	C₂×C₅₃	—	2	2	no	—
73	C₂×C₅₃	#1274	1	1	yes	characteristic
74	C₂×C₅₃	#1374	1	1	yes	characteristic
75	C₂×C₅₃	#1375	1	1	yes	characteristic
76	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
78	C₄³	Dic₅₃	1	1	yes	characteristic
79	C₄⋊C₁₆	—	106	2	no	—
80	C₂²⋊C₁₆	—	106	2	no	—
81	C₄×C₁₆	—	53	4	no	—
82	#336	—	2	2	no	—
83	C₅₃	#7142	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
86	C₂×C₄²	—	2	1	no	—
87	C₂×C₄²	—	2	1	no	—
88	C₂×C₄²	#88322	1	2	yes	—
89	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
90	C₂×C₁₆	—	106	4	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	#88322	1	2	yes	—
93	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
94	C₄×C₈	C₄×D₅₃	1	2	yes	—
95	C₄×C₈	D₂₁₂	1	1	yes	characteristic
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
98	C₂²×C₄	#11719917	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	#11719919	1	2	yes	—
104	C₂×C₈	#11719921	1	2	yes	—
105	C₄²	—	2	2	no	—
106	C₄²	—	4	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	#11719908	1	4	yes	complemented
109	C₄²	C₂×#88322	1	1	yes	characteristic
110	C₄²	#11719917	1	1	yes	characteristic
111	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
112	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
113	C₁₆	—	212	4	no	—
114	C₂³	C₂²⋊#88322	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	#13133124	1	1	yes	characteristic
121	C₂×C₄	C₄×#88322	1	1	yes	characteristic
122	C₂×C₄	#13133128	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
124	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
125	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
126	C₂×C₄	C₂×#11719908	1	2	yes	—
127	C₂×C₄	#13133157	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	4	2	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#16810378	1	1	yes	characteristic
134	C₂²	C₂²⋊#11719908	1	2	yes	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	4	no	—
138	C₄	#16810379	1	2	yes	—
139	C₄	C₄×#11719908	1	1	yes	characteristic
140	C₄	#16810399	1	1	yes	characteristic
141	C₄	#16810401	1	2	yes	—
142	C₄	C₄⋊#11719908	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	#20272544	1	1	yes	characteristic
145	C₂	#20272793	1	1	yes	characteristic
146	C₂	#20272795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276514

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₄⋊#11719908

—

C₂²⋊#11719908

—

C₄×#11719908

C₄

yes

complemented

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄³×C₅₃

C₄

yes

characteristic

C₅₃×#336

—

C₂×#11719908

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₈

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₁₆

yes

complemented

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

#11719908

—

C₂³×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

#97

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

M₃₂

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#7142

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#336

yes

characteristic

C₂²×C₅₃

C₂²⋊C₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

H(3,ℤ/4)

yes

—

C₄×C₅₃

C₄×C₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

C₄⋊C₁₆

yes

—

#2162

—

#1744

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×C₈

Dic₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄³

Dic₅₃

yes

characteristic

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

106

—

C₄×C₁₆

—

#336

—

C₅₃

#7142

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#88322

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

#88322

yes

—

C₄×C₈

Dic₁₀₆

yes

characteristic

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

D₂₁₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

#11719919

yes

—

104

C₂×C₈

#11719921

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

#11719908

yes

complemented

109

C₄²

C₂×#88322

yes

characteristic

110

C₄²

#11719917

yes

characteristic

111

C₄²

C₄×Dic₅₃

yes

characteristic

112

C₄²

C₄⋊Dic₅₃

yes

characteristic

113

C₁₆

—

212

—

114

C₂³

C₂²⋊#88322

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#13133124

yes

characteristic

121

C₂×C₄

C₄×#88322

yes

characteristic

122

C₂×C₄

#13133128

yes

characteristic

123

C₂×C₄

C₂²⋊#88322

yes

characteristic

124

C₂×C₄

C₄⋊#88322

yes

characteristic

125

C₂×C₄

C₄⋊#88322

yes

characteristic

126

C₂×C₄

C₂×#11719908

yes

—

127

C₂×C₄

#13133157

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

#16810378

yes

characteristic

134

C₂²

C₂²⋊#11719908

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#16810379

yes

—

139

C₄

C₄×#11719908

yes

characteristic

140

C₄

#16810399

yes

characteristic

141

C₄

#16810401

yes

—

142

C₄

C₄⋊#11719908

yes

—

143

C₂

—

144

C₂

#20272544

yes

characteristic

145

C₂

#20272793

yes

characteristic

146

C₂

#20272795

yes

characteristic

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#1744	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#7142
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1744

subgroup

G/H

center

C₄²

C₄×Dic₅₃

commutator

C₄×C₅₃

C₄×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₅₃×#1744

C₂

socle

C₂²×C₅₃

#336

prime radical

C₂³×C₅₃

C₂²⋊C₈

cosocle

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ #1744

N	H ≅ G/N	type
5. C₄×#11719908	C₄	semidirect
34. C₄²×C₅₃	C₁₆	semidirect
83. C₅₃	#7142	semidirect
108. C₄²	#11719908	semidirect

H ≅ G/N

type

5. C₄×#11719908

C₄

semidirect

34. C₄²×C₅₃

C₁₆

semidirect

83. C₅₃

#7142

semidirect

108. C₄²

#11719908

semidirect

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

subgroup	index
#20276514	2
C₅₃×#1744	2
#7142	53

subgroup

index

#20276514

C₅₃×#1744

#7142

Subgroups — #27270934

Conjugacy classes of subgroups

Complements

Maximal subgroups