Group #27300480 — subgroups

Subgroups — #27300480

Order 13568.

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

normal subgroups only

This group has 282 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	C₂×C₁₆×Dic₅₃	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	C₁₆×Dic₅₃	C₄	1	2	yes	—
12	C₁₆×Dic₅₃	C₄	1	2	yes	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₃	C₂²	1	2	yes	—
15	C₂×#11719917	—	2	2	no	—
16	C₂²×#88322	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	C₂×C₄	1	2	yes	—
20	#13133153	—	2	2	no	—
21	#13133153	—	2	2	no	—
22	#13133152	—	4	4	no	—
23	#13133146	—	4	4	no	—
24	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
27	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	#108×C₅₃	—	2	2	no	—
33	#108×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	Q₁₆	1	2	yes	—
36	M₁₆×C₅₃	—	4	2	no	—
37	M₁₆×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	#41	1	2	yes	—
42	C₂×C₈×C₅₃	#41	1	2	yes	—
43	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₅₃	D₈	1	2	yes	—
46	#11719917	—	4	2	no	—
47	#11719917	—	4	2	no	—
48	C₂×#88322	—	4	2	no	—
49	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#88322	SD₁₆	1	4	yes	—
52	C₁₆×C₅₃	—	2	2	no	—
53	#11719908	—	2	2	no	—
54	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#124	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
59	C₂×Dic₅₃	—	2	4	no	—
60	C₂×Dic₅₃	#107	1	2	yes	—
61	C₂×Dic₅₃	#107	1	2	yes	—
62	C₈×C₅₃	—	8	2	no	—
63	C₈×C₅₃	#117	1	1	yes	characteristic
64	C₈×C₅₃	#117	1	1	yes	characteristic
65	C₈×C₅₃	#125	1	2	yes	—
66	#88322	—	8	2	no	—
67	#88322	#106	1	2	yes	—
68	#88322	#106	1	2	yes	—
69	#12891	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	—	2	2	no	—
71	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
72	C₂²×C₅₃	#487	1	2	yes	—
73	C₄×C₅₃	#382	1	1	yes	characteristic
74	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
75	C₄×C₅₃	#486	1	2	yes	—
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	#368	1	4	yes	—
78	C₂×C₄×C₁₆	—	53	1	no	—
79	#1848	—	53	2	no	—
80	#1381	—	53	1	no	—
81	#1381	—	53	2	no	—
82	#1381	D₅₃	1	1	yes	characteristic
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	#1834	1	1	yes	characteristic
85	C₂×C₅₃	C₂×#368	1	2	yes	—
86	C₂²×C₁₆	—	53	1	no	—
87	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
88	C₂×#108	—	53	2	no	—
89	C₂×#108	—	106	2	no	—
90	C₂×#108	C₂×D₅₃	1	2	yes	—
91	#423	—	106	2	no	—
92	C₂×C₄×C₈	—	53	1	no	—
93	C₄×C₁₆	—	53	2	no	—
94	C₄×C₁₆	—	53	2	no	—
95	C₅₃	#12891	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	106	2	no	—
98	C₂²×C₈	—	53	1	no	—
99	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
100	C₂×C₄²	—	53	1	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	53	2	no	—
103	C₂×C₁₆	—	106	2	no	—
104	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
105	#108	—	2	2	no	—
106	#108	—	2	2	no	—
107	#108	—	106	2	no	—
108	#108	—	106	2	no	—
109	#108	—	212	4	no	—
110	C₄⋊C₈	—	212	4	no	—
111	C₄×C₈	—	53	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	—	53	1	no	—
115	C₂²×C₄	—	53	2	no	—
116	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	212	2	no	—
120	M₁₆	—	212	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	53	4	no	—
125	C₂×C₈	—	212	2	no	—
126	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₈	#11719945	1	2	yes	—
128	C₂×C₈	#11719947	1	2	yes	—
129	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
130	C₄²	—	53	1	no	—
131	C₄²	—	53	1	no	—
132	C₄²	—	53	2	no	—
133	C₁₆	—	2	2	no	—
134	C₁₆	—	106	2	no	—
135	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
136	C₂×C₄	—	53	2	no	—
137	C₂×C₄	—	53	2	no	—
138	C₂×C₄	—	106	4	no	—
139	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
140	C₂×C₄	#13133203	1	2	yes	—
141	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
142	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	53	2	no	—
145	C₈	—	53	2	no	—
146	C₈	—	424	2	no	—
147	C₈	#13133173	1	1	yes	characteristic
148	C₈	#13133174	1	1	yes	characteristic
149	C₈	#13133207	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	#107×D₅₃	1	1	yes	characteristic
152	C₂²	#16810972	1	2	yes	—
153	C₄	—	53	4	no	—
154	C₄	—	106	2	no	—
155	C₄	#16810487	1	1	yes	characteristic
156	C₄	#106×D₅₃	1	1	yes	characteristic
157	C₄	#16810968	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#20274841	1	1	yes	characteristic
160	C₂	#368×D₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×Dic₅₃

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

C₁₆×Dic₅₃

C₄

yes

—

C₁₆×Dic₅₃

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

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

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

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

D₈

yes

—

—

—

—

C₄⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

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

—

C₂×Dic₅₃

#107

yes

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

#88322

—

#88322

#106

yes

—

#88322

#106

yes

—

#12891

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#107

yes

characteristic

C₂²×C₅₃

#487

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

#368

yes

—

C₂×C₄×C₁₆

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1834

yes

characteristic

C₂×C₅₃

C₂×#368

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂×#108

—

C₂×#108

—

106

—

C₂×#108

C₂×D₅₃

yes

—

#423

—

106

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₅₃

#12891

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

100

C₂×C₄²

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

106

—

104

C₂×C₁₆

C₄×D₅₃

yes

—

105

#108

—

106

#108

—

107

#108

—

106

—

108

#108

—

106

—

109

#108

—

212

—

110

C₄⋊C₈

—

212

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₂²×C₄

—

116

C₂²×C₄

D₄×D₅₃

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

212

—

120

M₁₆

—

212

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

212

—

126

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

127

C₂×C₈

#11719945

yes

—

128

C₂×C₈

#11719947

yes

—

129

C₂×C₈

Q₈×D₅₃

yes

characteristic

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₁₆

—

134

C₁₆

—

106

—

135

C₂³

Q₁₆×D₅₃

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

106

—

139

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

140

C₂×C₄

#13133203

yes

—

141

C₂×C₄

D₈×D₅₃

yes

characteristic

142

C₂×C₄

SD₁₆×D₅₃

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

424

—

147

C₈

#13133173

yes

characteristic

148

C₈

#13133174

yes

characteristic

149

C₈

#13133207

yes

—

150

C₂²

—

151

C₂²

#107×D₅₃

yes

characteristic

152

C₂²

#16810972

yes

—

153

C₄

—

154

C₄

—

106

—

155

C₄

#16810487

yes

characteristic

156

C₄

#106×D₅₃

yes

characteristic

157

C₄

#16810968

yes

—

158

C₂

—

159

C₂

#20274841

yes

characteristic

160

C₂

#368×D₅₃

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	#12891
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
95. C₅₃	#12891	semidirect

H ≅ G/N

type

95. C₅₃

#12891

semidirect

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

subgroup	index
C₂×C₁₆×Dic₅₃	2
#20274885	2
#20272809	2
#20272808	2
C₅₃×#1381	2
#12891	53

subgroup

index

C₂×C₁₆×Dic₅₃

C₅₃×#1381

Subgroups — #27300480

Conjugacy classes of subgroups

Complements

Maximal subgroups