Group #27270703 — subgroups

Subgroups — #27270703

Order 13568.

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

normal subgroups only

This group has 304 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	#20274568	C₂	1	2	yes	maximal
2	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88322	—	2	2	no	—
4	C₂×#13133128	—	2	2	no	—
5	#16810378	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
7	C₄³×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#336	—	2	2	no	—
9	C₂²×#88322	—	2	4	no	—
10	C₄⋊#88322	—	2	8	no	—
11	#13133128	—	4	2	no	—
12	#13133128	—	4	2	no	—
13	C₂²×C₈×C₅₃	—	2	2	no	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	2	yes	—
18	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	—	4	2	no	—
20	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	4	2	no	—
26	C₂×C₈×C₅₃	—	4	4	no	—
27	C₄²×C₅₃	—	2	1	no	—
28	C₄²×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	—	2	4	no	—
30	C₄²×C₅₃	—	4	2	no	—
31	C₄²×C₅₃	—	4	1	no	—
32	C₄²×C₅₃	D₈	1	1	yes	characteristic
33	C₄²×C₅₃	SD₁₆	1	2	yes	—
34	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
35	C₂×#88322	—	2	8	no	—
36	C₂×#88322	—	4	4	no	—
37	C₂×#88322	—	4	4	no	—
38	C₂³×C₅₃	#95	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#102	1	2	yes	—
51	C₂×C₄×C₅₃	#103	1	2	yes	—
52	C₂×C₄×C₅₃	#104	1	4	yes	—
53	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
55	C₈×C₅₃	—	8	2	no	—
56	#88322	—	4	8	no	—
57	#7049	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	#338	1	1	yes	characteristic
59	C₂²×C₅₃	#339	1	2	yes	—
60	C₂²×C₅₃	#340	1	1	yes	characteristic
61	C₂²×C₅₃	#343	1	1	yes	characteristic
62	C₂²×C₅₃	#344	1	2	yes	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	4	2	no	—
68	C₄×C₅₃	—	4	1	no	—
69	C₄×C₅₃	#329	1	2	yes	—
70	C₄×C₅₃	#330	1	2	yes	—
71	#1831	D₅₃	1	1	yes	characteristic
72	#1767	—	53	2	no	—
73	C₂×C₅₃	#1291	1	2	yes	—
74	C₂×C₅₃	#1292	1	1	yes	characteristic
75	C₂×C₅₃	#1295	1	1	yes	characteristic
76	C₂×C₅₃	#1303	1	2	yes	—
77	C₂×C₅₃	#1307	1	1	yes	characteristic
78	C₂×C₄⋊C₈	—	106	2	no	—
79	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
80	C₂×#97	—	106	2	no	—
81	C₄³	C₂×D₅₃	1	1	yes	characteristic
82	#336	—	2	2	no	—
83	#336	—	53	4	no	—
84	C₅₃	#7049	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	106	4	no	—
87	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
88	C₂×C₄²	C₄×D₅₃	1	2	yes	—
89	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
90	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	—	106	8	no	—
94	#97	—	212	2	no	—
95	#97	—	212	2	no	—
96	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
97	C₂²×C₄	#11719919	1	2	yes	—
98	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
99	C₂²×C₄	#11719921	1	2	yes	—
100	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	106	8	no	—
104	C₂×C₈	—	212	4	no	—
105	C₂×C₈	—	212	4	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	4	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	4	2	no	—
111	C₄²	#11719913	1	2	yes	—
112	C₄²	D₄₂₄	1	1	yes	characteristic
113	C₄²	Dic₂₁₂	1	1	yes	characteristic
114	C₂³	#13133124	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	#13133134	1	1	yes	characteristic
125	C₂×C₄	#13133136	1	1	yes	characteristic
126	C₂×C₄	#13133139	1	2	yes	—
127	C₂×C₄	#13133142	1	2	yes	—
128	C₂×C₄	#13133144	1	4	yes	—
129	C₂×C₄	#13133149	1	1	yes	characteristic
130	C₂×C₄	#13133151	1	1	yes	characteristic
131	C₈	—	8	2	no	—
132	C₈	—	212	8	no	—
133	C₂²	#16810381	1	1	yes	characteristic
134	C₂²	#16810382	1	2	yes	—
135	C₂²	#16810386	1	1	yes	characteristic
136	C₂²	#16810392	1	1	yes	characteristic
137	C₂²	#16810395	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	#16810363	1	2	yes	—
145	C₄	#16810365	1	2	yes	—
146	C₂	#20272586	1	2	yes	—
147	C₂	#20272590	1	1	yes	characteristic
148	C₂	#20272596	1	1	yes	characteristic
149	C₂	#20272617	1	2	yes	—
150	C₂	#20272626	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274568

C₂

yes

maximal

C₅₃×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×#13133128

—

#16810378

C₄

yes

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂²×#88322

—

C₄⋊#88322

—

#13133128

—

#13133128

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

Q₁₆

yes

—

—

—

C₂³×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#329

yes

—

C₄×C₅₃

#330

yes

—

#1831

D₅₃

yes

characteristic

#1767

—

C₂×C₅₃

#1291

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1307

yes

characteristic

C₂×C₄⋊C₈

—

106

—

C₂×C₄⋊C₈

Dic₅₃

yes

—

C₂×#97

—

106

—

C₄³

C₂×D₅₃

yes

characteristic

#336

—

#336

—

C₅₃

#7049

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

Dic₁₀₆

yes

characteristic

C₂×C₄²

C₄×D₅₃

yes

—

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#97

—

212

—

#97

—

212

—

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

C₂²×C₄

#11719919

yes

—

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

C₂²×C₄

#11719921

yes

—

100

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

212

—

105

C₂×C₈

—

212

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

#11719913

yes

—

112

C₄²

D₄₂₄

yes

characteristic

113

C₄²

Dic₂₁₂

yes

characteristic

114

C₂³

#13133124

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#13133134

yes

characteristic

125

C₂×C₄

#13133136

yes

characteristic

126

C₂×C₄

#13133139

yes

—

127

C₂×C₄

#13133142

yes

—

128

C₂×C₄

#13133144

yes

—

129

C₂×C₄

#13133149

yes

characteristic

130

C₂×C₄

#13133151

yes

characteristic

131

C₈

—

132

C₈

—

212

—

133

C₂²

#16810381

yes

characteristic

134

C₂²

#16810382

yes

—

135

C₂²

#16810386

yes

characteristic

136

C₂²

#16810392

yes

characteristic

137

C₂²

#16810395

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#16810363

yes

—

145

C₄

#16810365

yes

—

146

C₂

#20272586

yes

—

147

C₂

#20272590

yes

characteristic

148

C₂

#20272596

yes

characteristic

149

C₂

#20272617

yes

—

150

C₂

#20272626

yes

characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#7049
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

#13133124

commutator

C₂²×C₄×C₅₃

C₄²

Frattini

C₄³

C₂×D₅₃

Fitting

C₅₃×#1831

C₂

socle, prime radical

C₂³×C₅₃

#95

cosocle

C₄³×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
84. C₅₃	#7049	semidirect

H ≅ G/N

type

84. C₅₃

#7049

semidirect

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

subgroup	index
#20274568	2
C₅₃×#1831	2
#7049	53

subgroup

index

#20274568

C₅₃×#1831

#7049

Subgroups — #27270703

Conjugacy classes of subgroups

Complements

Maximal subgroups