Group #27277804 — subgroups

Subgroups — #27277804

Order 13568.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20273478	C₂	1	1	yes	maximal, characteristic
2	#20273007	C₂	1	1	yes	maximal, characteristic
3	#20272578	C₂	1	2	yes	maximal
4	#20272576	C₂	1	2	yes	maximal
5	C₅₃×#1568	C₂	1	1	yes	maximal, characteristic
6	#16810680	C₂²	1	2	yes	—
7	#16810676	C₂²	1	1	yes	characteristic
8	#16810676	C₂²	1	1	yes	characteristic
9	#16810376	—	2	1	no	—
10	#16810374	—	2	1	no	—
11	#16810345	C₄	1	4	yes	—
12	C₅₃×#432	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#335	—	2	1	no	—
15	C₅₃×#334	—	2	1	no	—
16	C₄⋊#88322	—	2	1	no	—
17	C₄⋊#88322	—	2	1	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₂²⋊#88322	—	2	1	no	—
20	C₂²⋊#88322	—	2	1	no	—
21	C₂²⋊#88322	—	2	2	no	—
22	#13133128	—	2	2	no	—
23	C₄×#88322	C₂×C₄	1	2	yes	—
24	C₄×#88322	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₂²⋊C₈×C₅₃	—	2	2	no	—
29	#97×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₅₃	#41	1	1	yes	characteristic
43	C₄²×C₅₃	#41	1	1	yes	characteristic
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	—	4	2	no	—
47	C₂³×C₅₃	#125	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	—	2	2	no	—
59	C₈×C₅₃	—	8	2	no	—
60	#88322	—	4	2	no	—
61	#88322	—	4	2	no	—
62	#88322	—	8	2	no	—
63	#8798	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	#382	1	1	yes	characteristic
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	#430	1	2	yes	—
74	#1568	—	53	1	no	—
75	#1568	D₅₃	1	1	yes	characteristic
76	#1452	—	53	1	no	—
77	#1287	—	53	2	no	—
78	#1287	—	53	2	no	—
79	C₂×C₅₃	—	4	1	no	—
80	C₂×C₅₃	#1833	1	1	yes	characteristic
81	C₂×C₅₃	#1833	1	1	yes	characteristic
82	C₂×C₅₃	#1838	1	1	yes	characteristic
83	#432	—	53	2	no	—
84	#432	C₂×D₅₃	1	2	yes	—
85	#431	—	53	1	no	—
86	#431	—	53	1	no	—
87	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	#335	—	2	1	no	—
89	#335	—	106	1	no	—
90	#334	—	2	1	no	—
91	#334	—	106	1	no	—
92	#322	—	53	4	no	—
93	C₅₃	#8798	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	Dic₁₀₆	1	2	yes	—
95	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	106	1	no	—
98	C₄⋊C₈	—	106	1	no	—
99	C₄⋊C₈	—	106	2	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	106	1	no	—
102	C₂²⋊C₈	—	106	1	no	—
103	C₂²⋊C₈	—	106	2	no	—
104	#97	—	2	2	no	—
105	#97	—	106	2	no	—
106	C₄×C₈	—	53	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	C₄×D₅₃	1	2	yes	—
109	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
112	C₂²×C₄	#11719947	1	1	yes	characteristic
113	C₂²×C₄	#11719949	1	1	yes	characteristic
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	—	212	2	no	—
119	C₂×C₈	#11719919	1	2	yes	—
120	C₂×C₈	#11719919	1	2	yes	—
121	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
122	C₄²	#11719947	1	1	yes	characteristic
123	C₄²	#11719949	1	1	yes	characteristic
124	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂³	#13133208	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
130	C₂×C₄	#13133175	1	1	yes	characteristic
131	C₂×C₄	#13133175	1	1	yes	characteristic
132	C₂×C₄	#13133204	1	1	yes	characteristic
133	C₂×C₄	#13133206	1	1	yes	characteristic
134	C₂×C₄	#13133208	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	2	no	—
138	C₈	—	424	2	no	—
139	C₈	—	212	2	no	—
140	C₈	—	212	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#16810488	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#16810671	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	#20274838	1	1	yes	characteristic
153	C₂	#20274840	1	1	yes	characteristic
154	C₂	#20274855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273478

C₂

yes

maximal, characteristic

#20273007

C₂

yes

maximal, characteristic

#20272578

C₂

yes

maximal

#20272576

C₂

yes

maximal

C₅₃×#1568

C₂

yes

maximal, characteristic

#16810680

C₂²

yes

—

#16810676

C₂²

yes

characteristic

#16810676

C₂²

yes

—

—

C₄

yes

—

C₅₃×#432

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₅₃×#334

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

C₂²×C₄×C₅₃

yes

C₂²×C₄×C₅₃

yes

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

—

—

C₂³×C₅₃

#125

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#430

yes

—

#1568

—

#1568

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×D₅₃

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#335

—

#335

—

106

—

#334

—

#334

—

106

—

#322

—

C₅₃

#8798

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₁₀₆

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

106

—

102

C₂²⋊C₈

—

106

—

103

C₂²⋊C₈

—

106

—

104

#97

—

105

#97

—

106

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₄×D₅₃

yes

—

109

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

110

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

111

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

112

C₂²×C₄

#11719947

yes

characteristic

113

C₂²×C₄

#11719949

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

212

—

119

C₂×C₈

#11719919

yes

—

120

C₂×C₈

#11719919

yes

—

121

C₄²

C₂×C₄×D₅₃

yes

characteristic

122

C₄²

#11719947

yes

characteristic

123

C₄²

#11719949

yes

characteristic

124

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

125

C₂³

#13133208

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×#11719919

yes

characteristic

130

C₂×C₄

#13133175

yes

characteristic

131

C₂×C₄

#13133175

yes

characteristic

132

C₂×C₄

#13133204

yes

characteristic

133

C₂×C₄

#13133206

yes

characteristic

134

C₂×C₄

#13133208

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

424

—

139

C₈

—

212

—

140

C₈

—

212

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#16810488

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#16810671

yes

—

151

C₂

—

152

C₂

#20274838

yes

characteristic

153

C₂

#20274840

yes

characteristic

154

C₂

#20274855

yes

characteristic

	subgroup	G/H
center	C₂²	#16810488
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1568	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#125
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1568	D₅₃
hypocenter	C₅₃	#8798
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

subgroup

G/H

center

C₂²

#16810488

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1568

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

N	H ≅ G/N	type
93. C₅₃	#8798	semidirect

H ≅ G/N

type

93. C₅₃

#8798

semidirect

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

subgroup	index
#20273478	2
#20273007	2
#20272578	2
#20272576	2
C₅₃×#1568	2
#8798	53

subgroup

index

C₅₃×#1568

Subgroups — #27277804

Conjugacy classes of subgroups

Complements

Maximal subgroups