Group #27270941 — subgroups

Subgroups — #27270941

Order 13568.

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

normal subgroups only

This group has 222 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	C₂×C₅₃×#322	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	—
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#322	—	2	2	no	—
10	C₂×#11719908	—	2	4	no	—
11	C₂²×C₈×C₅₃	—	2	1	no	—
12	C₂²×C₈×C₅₃	C₈	1	2	yes	—
13	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄×C₈×C₅₃	—	2	1	no	—
17	C₄×C₈×C₅₃	—	2	1	no	—
18	C₄×C₈×C₅₃	—	2	1	no	—
19	C₄×C₈×C₅₃	C₈	1	2	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	D₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	4	1	no	—
29	C₂×C₈×C₅₃	—	4	1	no	—
30	C₂×C₈×C₅₃	—	4	2	no	—
31	C₂×C₈×C₅₃	C₁₆	1	4	yes	complemented
32	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
33	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₄²×C₅₃	C₄²	1	1	yes	characteristic
35	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
38	#11719908	—	4	4	no	—
39	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
51	C₈×C₅₃	—	2	2	no	—
52	C₈×C₅₃	—	2	2	no	—
53	C₈×C₅₃	—	2	4	no	—
54	C₈×C₅₃	—	4	2	no	—
55	#7145	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	#336	1	1	yes	characteristic
59	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
63	C₄×C₅₃	#338	1	1	yes	characteristic
64	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
65	C₄×C₅₃	#346	1	1	yes	characteristic
66	C₄×C₅₃	#347	1	2	yes	—
67	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
68	#2162	—	53	2	no	—
69	C₂×#322	D₅₃	1	1	yes	characteristic
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	#1275	1	1	yes	characteristic
72	C₂×C₅₃	#1374	1	1	yes	characteristic
73	C₂×C₅₃	#1375	1	1	yes	characteristic
74	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
75	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
77	C₄⋊C₁₆	—	106	2	no	—
78	C₂²⋊C₁₆	—	106	2	no	—
79	C₄×C₁₆	—	53	4	no	—
80	#322	—	2	2	no	—
81	C₅₃	#7145	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	#88322	1	2	yes	—
84	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
85	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂×C₁₆	—	106	4	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	#88322	1	2	yes	—
92	C₄×C₈	C₄×D₅₃	1	2	yes	—
93	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
94	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
95	C₂²×C₄	#11719917	1	1	yes	characteristic
96	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	#11719908	1	4	yes	complemented
104	C₂×C₈	#11719919	1	2	yes	—
105	C₂×C₈	#11719921	1	2	yes	—
106	C₄²	C₂×#88322	1	1	yes	characteristic
107	C₄²	#11719917	1	1	yes	characteristic
108	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
109	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
110	C₁₆	—	212	4	no	—
111	C₂³	C₄⋊#88322	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	#13133124	1	1	yes	characteristic
116	C₂×C₄	C₄×#88322	1	1	yes	characteristic
117	C₂×C₄	#13133128	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
120	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
121	C₂×C₄	C₂×#11719908	1	2	yes	—
122	C₂×C₄	#13133157	1	2	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	2	4	no	—
126	C₈	—	4	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#16810378	1	1	yes	characteristic
130	C₂²	C₄⋊#11719908	1	2	yes	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	#16810380	1	1	yes	characteristic
134	C₄	#16810381	1	1	yes	characteristic
135	C₄	C₄×#11719908	1	1	yes	characteristic
136	C₄	#16810399	1	1	yes	characteristic
137	C₄	#16810401	1	2	yes	—
138	C₄	C₂²⋊#11719908	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#20272546	1	1	yes	characteristic
141	C₂	#20272793	1	1	yes	characteristic
142	C₂	#20272795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276514

C₂

yes

maximal

C₂×C₅₃×#322

C₂

yes

maximal, characteristic

C₄⋊#11719908

—

C₂²⋊#11719908

—

C₄×#11719908

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

—

C₂×#11719908

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₈

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₄⋊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

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₁₆

yes

complemented

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

#11719908

—

C₂³×C₅₃

C₄⋊C₈

yes

characteristic

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

C₂²⋊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

—

C₈×C₅₃

—

#7145

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#336

yes

characteristic

C₂²×C₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₄×C₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

C₂²⋊C₁₆

yes

—

#2162

—

C₂×#322

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×C₈

Dic₅₃

yes

characteristic

C₂×C₄×C₈

Dic₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

106

—

C₄×C₁₆

—

#322

—

C₅₃

#7145

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

#88322

yes

—

C₂²×C₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

D₂₁₂

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#88322

yes

—

C₄×C₈

C₄×D₅₃

yes

—

C₄×C₈

C₂²⋊D₅₃

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

#11719908

yes

complemented

104

C₂×C₈

#11719919

yes

—

105

C₂×C₈

#11719921

yes

—

106

C₄²

C₂×#88322

yes

characteristic

107

C₄²

#11719917

yes

characteristic

108

C₄²

C₄×Dic₅₃

yes

characteristic

109

C₄²

C₂²⋊Dic₅₃

yes

characteristic

110

C₁₆

—

212

—

111

C₂³

C₄⋊#88322

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

#13133124

yes

characteristic

116

C₂×C₄

C₄×#88322

yes

characteristic

117

C₂×C₄

#13133128

yes

characteristic

118

C₂×C₄

C₂²⋊#88322

yes

characteristic

119

C₂×C₄

C₂²⋊#88322

yes

characteristic

120

C₂×C₄

C₄⋊#88322

yes

characteristic

121

C₂×C₄

C₂×#11719908

yes

—

122

C₂×C₄

#13133157

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

#16810378

yes

characteristic

130

C₂²

C₄⋊#11719908

yes

—

131

C₄

—

132

C₄

—

133

C₄

#16810380

yes

characteristic

134

C₄

#16810381

yes

characteristic

135

C₄

C₄×#11719908

yes

characteristic

136

C₄

#16810399

yes

characteristic

137

C₄

#16810401

yes

—

138

C₄

C₂²⋊#11719908

yes

—

139

C₂

—

140

C₂

#20272546

yes

characteristic

141

C₂

#20272793

yes

characteristic

142

C₂

#20272795

yes

characteristic

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#7145
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×#322

subgroup

G/H

center

C₄²

C₄×Dic₅₃

commutator

C₄×C₅₃

C₄×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₅₃

Fitting

C₂×C₅₃×#322

C₂

socle

C₂²×C₅₃

#336

prime radical

C₂³×C₅₃

C₄⋊C₈

cosocle

C₂×C₄×C₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ C₂×#322

N	H ≅ G/N	type
31. C₂×C₈×C₅₃	C₁₆	semidirect
81. C₅₃	#7145	semidirect
103. C₂×C₈	#11719908	semidirect

H ≅ G/N

type

31. C₂×C₈×C₅₃

C₁₆

semidirect

81. C₅₃

#7145

semidirect

103. C₂×C₈

#11719908

semidirect

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

subgroup	index
#20276514	2
C₂×C₅₃×#322	2
#7145	53

subgroup

index

#20276514

C₂×C₅₃×#322

#7145

Subgroups — #27270941

Conjugacy classes of subgroups

Complements

Maximal subgroups