Group #27270943 — subgroups

Subgroups — #27270943

Order 13568.

210 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20276518	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	#16810399	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	4	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	C₂²⋊C₄	1	2	yes	—
32	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₄²×C₅₃	C₄²	1	1	yes	characteristic
34	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
37	#11719908	—	4	4	no	—
38	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
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	C₈×C₅₃	—	2	4	no	—
50	C₈×C₅₃	—	2	2	no	—
51	C₈×C₅₃	—	2	2	no	—
52	C₈×C₅₃	—	4	2	no	—
53	#7146	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	#336	1	1	yes	characteristic
57	C₂²×C₅₃	#364	1	2	yes	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
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	#349	1	2	yes	—
65	#2163	—	53	2	no	—
66	C₂×#322	D₅₃	1	1	yes	characteristic
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	#1275	1	1	yes	characteristic
69	C₂×C₅₃	#1376	1	1	yes	characteristic
70	C₂×C₅₃	#1377	1	1	yes	characteristic
71	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
72	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
74	C₄⋊C₁₆	—	106	2	no	—
75	C₂²⋊C₁₆	—	106	2	no	—
76	#346	—	53	4	no	—
77	#322	—	2	2	no	—
78	C₅₃	#7146	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	#88322	1	2	yes	—
81	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
82	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
83	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
84	C₂×C₁₆	—	106	4	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	#88322	1	2	yes	—
89	C₄×C₈	C₄×D₅₃	1	2	yes	—
90	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
91	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
92	C₂²×C₄	#11719917	1	1	yes	characteristic
93	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	#11719919	1	2	yes	—
101	C₂×C₈	#11719921	1	2	yes	—
102	C₄²	C₂×#88322	1	1	yes	characteristic
103	C₄²	#11719917	1	1	yes	characteristic
104	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
105	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
106	C₁₆	—	212	4	no	—
107	C₂³	C₄⋊#88322	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	#13133124	1	1	yes	characteristic
113	C₂×C₄	C₄×#88322	1	1	yes	characteristic
114	C₂×C₄	#13133128	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
116	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
117	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
118	C₈	—	2	4	no	—
119	C₈	—	2	2	no	—
120	C₈	—	2	2	no	—
121	C₈	—	4	2	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#16810378	1	1	yes	characteristic
125	C₂²	#16810444	1	2	yes	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	#16810380	1	1	yes	characteristic
131	C₄	#16810381	1	1	yes	characteristic
132	C₄	#16810405	1	2	yes	—
133	C₂	—	2	2	no	—
134	C₂	#20272546	1	1	yes	characteristic
135	C₂	#20272797	1	1	yes	characteristic
136	C₂	#20272799	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276518

C₂

yes

maximal

C₂×C₅₃×#322

C₂

yes

maximal, characteristic

C₄⋊#11719908

—

C₂²⋊#11719908

—

#16810399

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

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

—

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

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#7146

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#336

yes

characteristic

C₂²×C₅₃

#364

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2163

—

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

—

#346

—

#322

—

C₅₃

#7146

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

#11719919

yes

—

101

C₂×C₈

#11719921

yes

—

102

C₄²

C₂×#88322

yes

characteristic

103

C₄²

#11719917

yes

characteristic

104

C₄²

C₄×Dic₅₃

yes

characteristic

105

C₄²

C₂²⋊Dic₅₃

yes

characteristic

106

C₁₆

—

212

—

107

C₂³

C₄⋊#88322

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#13133124

yes

characteristic

113

C₂×C₄

C₄×#88322

yes

characteristic

114

C₂×C₄

#13133128

yes

characteristic

115

C₂×C₄

C₂²⋊#88322

yes

characteristic

116

C₂×C₄

C₂²⋊#88322

yes

characteristic

117

C₂×C₄

C₄⋊#88322

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#16810378

yes

characteristic

125

C₂²

#16810444

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#16810380

yes

characteristic

131

C₄

#16810381

yes

characteristic

132

C₄

#16810405

yes

—

133

C₂

—

134

C₂

#20272546

yes

characteristic

135

C₂

#20272797

yes

characteristic

136

C₂

#20272799

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₄×C₈
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	#7146
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13133124

commutator

C₂×C₄×C₅₃

C₄×C₈

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 ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

N	H ≅ G/N	type
78. C₅₃	#7146	semidirect

H ≅ G/N

type

78. C₅₃

#7146

semidirect

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

subgroup	index
#20276518	2
C₂×C₅₃×#322	2
#7146	53

subgroup

index

#20276518

C₂×C₅₃×#322

#7146

Subgroups — #27270943

Conjugacy classes of subgroups

Complements

Maximal subgroups