Group #27300483 — subgroups

Subgroups — #27300483

Order 13568.

274 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16810398	C₂	1	1	yes	maximal, characteristic
2	#20274885	C₂	1	2	yes	maximal
3	#20272809	C₂	1	1	yes	maximal, characteristic
4	#20272808	C₂	1	2	yes	maximal
5	C₅₃×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133153	C₂²	1	2	yes	—
8	C₂×#13133152	—	2	2	no	—
9	#16810646	—	2	2	no	—
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	#16810398	C₄	1	4	yes	—
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#108×C₅₃	C₂²	1	2	yes	—
14	C₂×#11719917	—	2	2	no	—
15	C₂²×#88322	D₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	C₂×C₄	1	2	yes	—
19	#13133153	—	2	2	no	—
20	#13133153	—	2	2	no	—
21	#13133152	—	4	4	no	—
22	#13133146	—	4	4	no	—
23	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
24	C₈×Dic₅₃	Q₈	1	2	yes	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	—	2	2	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	#108×C₅₃	—	2	2	no	—
30	#108×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₅₃	Q₁₆	1	2	yes	—
33	M₁₆×C₅₃	—	4	2	no	—
34	M₁₆×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	#41	1	2	yes	—
39	C₂×C₈×C₅₃	#41	1	2	yes	—
40	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
41	C₄×Dic₅₃	D₈	1	2	yes	—
42	#11719917	—	4	2	no	—
43	#11719917	—	4	2	no	—
44	C₂×#88322	—	4	2	no	—
45	C₂×#88322	C₄⋊C₄	1	2	yes	—
46	C₂×#88322	SD₁₆	1	4	yes	—
47	C₁₆×C₅₃	—	2	2	no	—
48	#11719908	—	2	2	no	—
49	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#124	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
54	C₂×Dic₅₃	—	4	2	no	—
55	C₂×Dic₅₃	#107	1	4	yes	—
56	C₈×C₅₃	—	8	2	no	—
57	C₈×C₅₃	#117	1	1	yes	characteristic
58	C₈×C₅₃	#117	1	1	yes	characteristic
59	C₈×C₅₃	#125	1	2	yes	—
60	#88322	—	8	2	no	—
61	#88322	#106	1	4	yes	—
62	#12892	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
65	C₂²×C₅₃	#487	1	2	yes	—
66	C₄×C₅₃	#382	1	1	yes	characteristic
67	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
68	C₄×C₅₃	#486	1	2	yes	—
69	Dic₅₃	—	2	4	no	—
70	C₂×#346	—	53	1	no	—
71	#1848	—	53	2	no	—
72	#1381	—	53	1	no	—
73	#1381	—	53	2	no	—
74	#1381	D₅₃	1	1	yes	characteristic
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	#1834	1	1	yes	characteristic
77	C₂×C₅₃	#2161	1	2	yes	—
78	C₂²×C₁₆	—	53	1	no	—
79	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
80	C₂×#108	—	53	2	no	—
81	C₂×#108	—	106	2	no	—
82	C₂×#108	C₂×D₅₃	1	2	yes	—
83	#423	—	106	2	no	—
84	C₂×C₄×C₈	—	53	1	no	—
85	#346	—	53	4	no	—
86	C₅₃	#12892	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	106	2	no	—
89	C₂²×C₈	—	53	1	no	—
90	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
91	C₂×C₄²	—	53	1	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	53	2	no	—
94	C₂×C₁₆	—	106	2	no	—
95	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	106	2	no	—
99	#108	—	106	2	no	—
100	#108	—	212	4	no	—
101	C₄⋊C₈	—	212	4	no	—
102	C₄×C₈	—	53	2	no	—
103	C₄×C₈	—	53	2	no	—
104	C₂²×C₄	—	53	2	no	—
105	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	212	2	no	—
109	M₁₆	—	212	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	53	2	no	—
112	C₂×C₈	—	53	4	no	—
113	C₂×C₈	—	212	2	no	—
114	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₈	#11719945	1	2	yes	—
116	C₂×C₈	#11719947	1	2	yes	—
117	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
118	C₄²	—	53	2	no	—
119	C₄²	—	53	2	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	106	2	no	—
122	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
123	C₂×C₄	—	53	4	no	—
124	C₂×C₄	—	212	2	no	—
125	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₄	#13133203	1	2	yes	—
127	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
128	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
129	C₈	—	8	2	no	—
130	C₈	—	53	4	no	—
131	C₈	—	424	2	no	—
132	C₈	#13133173	1	1	yes	characteristic
133	C₈	#13133174	1	1	yes	characteristic
134	C₈	#13133207	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	#107×D₅₃	1	1	yes	characteristic
137	C₂²	#16810972	1	2	yes	—
138	C₄	—	106	4	no	—
139	C₄	#16810487	1	1	yes	characteristic
140	C₄	#106×D₅₃	1	1	yes	characteristic
141	C₄	#16810968	1	2	yes	—
142	C₂	—	2	2	no	—
143	C₂	#20274841	1	1	yes	characteristic
144	C₂	#20276508	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810398

C₂

yes

maximal, characteristic

#20274885

C₂

yes

maximal

#20272809

C₂

yes

maximal, characteristic

#20272808

C₂

yes

maximal

C₅₃×#1381

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

characteristic

C₂×#13133153

C₂²

yes

—

C₂×#13133152

—

#16810646

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810398

C₄

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×#108×C₅₃

C₂²

yes

—

C₂×#11719917

—

C₂²×#88322

D₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

#108×C₅₃

—

#108×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

Q₁₆

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

C₄×Dic₅₃

D₈

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#88322

SD₁₆

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#107

yes

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

#88322

—

#88322

#106

yes

—

#12892

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

#487

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1834

yes

characteristic

C₂×C₅₃

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂×#108

—

C₂×#108

—

106

—

C₂×#108

C₂×D₅₃

yes

—

#423

—

106

—

C₂×C₄×C₈

—

#346

—

C₅₃

#12892

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₄×D₅₃

yes

—

#108

—

#108

—

#108

—

106

—

#108

—

106

—

100

#108

—

212

—

101

C₄⋊C₈

—

212

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₂²×C₄

—

105

C₂²×C₄

D₄×D₅₃

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

212

—

109

M₁₆

—

212

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

212

—

114

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

115

C₂×C₈

#11719945

yes

—

116

C₂×C₈

#11719947

yes

—

117

C₂×C₈

Q₈×D₅₃

yes

characteristic

118

C₄²

—

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

106

—

122

C₂³

Q₁₆×D₅₃

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

212

—

125

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

126

C₂×C₄

#13133203

yes

—

127

C₂×C₄

D₈×D₅₃

yes

characteristic

128

C₂×C₄

SD₁₆×D₅₃

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

424

—

132

C₈

#13133173

yes

characteristic

133

C₈

#13133174

yes

characteristic

134

C₈

#13133207

yes

—

135

C₂²

—

136

C₂²

#107×D₅₃

yes

characteristic

137

C₂²

#16810972

yes

—

138

C₄

—

106

—

139

C₄

#16810487

yes

characteristic

140

C₄

#106×D₅₃

yes

characteristic

141

C₄

#16810968

yes

—

142

C₂

—

143

C₂

#20274841

yes

characteristic

144

C₂

#20276508

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1381	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1381	D₅₃
hypocenter	C₅₃	#12892
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1381

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂³×C₅₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
86. C₅₃	#12892	semidirect

H ≅ G/N

type

86. C₅₃

#12892

semidirect

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

subgroup	index
C₂×#16810398	2
#20274885	2
#20272809	2
#20272808	2
C₅₃×#1381	2
#12892	53

subgroup

index

C₅₃×#1381

Subgroups — #27300483

Conjugacy classes of subgroups

Complements

Maximal subgroups