Group #27271208 — subgroups

Subgroups — #27271208

Order 13568.

244 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20273059	C₂	1	2	yes	maximal
2	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic
3	#16810680	—	2	2	no	—
4	#16810349	—	2	2	no	—
5	#16810349	C₄	1	4	yes	—
6	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×#336	—	2	2	no	—
10	C₄⋊#88322	—	4	2	no	—
11	C₂²⋊#88322	—	2	4	no	—
12	#13133128	—	4	2	no	—
13	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	—	2	1	no	—
15	C₂²×C₈×C₅₃	—	2	1	no	—
16	C₄×Q₈×C₅₃	—	4	1	no	—
17	C₄×Q₈×C₅₃	—	4	1	no	—
18	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	2	yes	—
22	C₄⋊C₈×C₅₃	—	4	1	no	—
23	C₄×C₈×C₅₃	—	4	1	no	—
24	C₂×Q₈×C₅₃	—	2	2	no	—
25	C₂×Q₈×C₅₃	—	4	1	no	—
26	C₂×Q₈×C₅₃	—	4	1	no	—
27	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	4	1	no	—
34	C₂×C₈×C₅₃	—	4	1	no	—
35	C₂×C₈×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	4	1	no	—
37	C₄⋊C₄×C₅₃	—	4	2	no	—
38	C₄⋊C₄×C₅₃	—	4	1	no	—
39	C₄²×C₅₃	—	2	1	no	—
40	C₄²×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₂×#88322	—	4	4	no	—
44	C₂³×C₅₃	#95	1	1	yes	characteristic
45	Q₈×C₅₃	—	4	4	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	4	1	no	—
56	C₂×C₄×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#108	1	2	yes	—
60	C₈×C₅₃	—	4	2	no	—
61	C₈×C₅₃	—	8	1	no	—
62	#88322	—	8	4	no	—
63	#7248	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	#338	1	1	yes	characteristic
67	C₂²×C₅₃	#341	1	1	yes	characteristic
68	C₂²×C₅₃	#342	1	1	yes	characteristic
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	4	2	no	—
72	C₄×C₅₃	—	4	2	no	—
73	C₄×C₅₃	—	4	1	no	—
74	C₄×C₅₃	#355	1	1	yes	characteristic
75	C₄×C₅₃	#356	1	1	yes	characteristic
76	#1984	D₅₃	1	1	yes	characteristic
77	#1467	—	53	2	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	#1293	1	1	yes	characteristic
81	C₂×C₅₃	#1391	1	1	yes	characteristic
82	C₂×C₅₃	#1394	1	1	yes	characteristic
83	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
84	#432	—	106	2	no	—
85	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
86	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
87	#336	—	2	2	no	—
88	#324	—	53	4	no	—
89	#324	—	106	2	no	—
90	C₅₃	#7248	1	1	yes	characteristic, complemented, Sylow
91	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₄×Q₈	—	4	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
97	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
98	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
100	C₄⋊C₈	—	4	1	no	—
101	C₄⋊C₈	—	212	2	no	—
102	C₂²⋊C₈	—	106	4	no	—
103	#97	—	212	2	no	—
104	C₄×C₈	—	4	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
109	C₂²×C₄	#11719919	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
111	C₂²×C₄	#11719921	1	2	yes	—
112	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	212	4	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	2	no	—
125	C₂³	#13133124	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	#13133132	1	1	yes	characteristic
139	C₂×C₄	#13133132	1	1	yes	characteristic
140	C₂×C₄	#13133153	1	2	yes	—
141	C₈	—	4	2	no	—
142	C₈	—	8	1	no	—
143	C₈	—	424	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#16810381	1	1	yes	characteristic
147	C₂²	#16810388	1	1	yes	characteristic
148	C₂²	#16810390	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	1	no	—
153	C₄	—	4	2	no	—
154	C₄	#16810420	1	1	yes	characteristic
155	C₄	#16810422	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	#20272592	1	1	yes	characteristic
159	C₂	#20272836	1	1	yes	characteristic
160	C₂	#20272842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20273059

C₂

yes

maximal

C₅₃×#1984

C₂

yes

maximal, characteristic

#16810680

—

#16810349

—

#16810349

C₄

yes

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₅₃×#336

—

C₄⋊#88322

—

C₂²⋊#88322

—

#13133128

—

C₂²×Q₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₂×#88322

—

C₂³×C₅₃

#95

yes

characteristic

Q₈×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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#7248

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

D₅₃

yes

characteristic

#1467

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

#432

—

106

—

C₂×C₄⋊C₈

Dic₅₃

yes

characteristic

C₂×C₄×C₈

Dic₅₃

yes

—

—

—

106

—

C₅₃

#7248

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₁₀₆

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×D₅₃

yes

—

C₂×C₄⋊C₄

D₂₁₂

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

C₂×C₄²

C₂²⋊D₅₃

yes

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

212

—

102

C₂²⋊C₈

—

106

—

103

#97

—

212

—

104

C₄×C₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

C₄×Dic₅₃

yes

characteristic

109

C₂²×C₄

#11719919

yes

—

110

C₂²×C₄

C₄⋊Dic₅₃

yes

characteristic

111

C₂²×C₄

#11719921

yes

—

112

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

212

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₂³

#13133124

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#13133132

yes

characteristic

139

C₂×C₄

#13133132

yes

characteristic

140

C₂×C₄

#13133153

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

424

—

144

C₂²

—

145

C₂²

—

146

C₂²

#16810381

yes

characteristic

147

C₂²

#16810388

yes

characteristic

148

C₂²

#16810390

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#16810420

yes

characteristic

155

C₄

#16810422

yes

characteristic

156

C₂

—

157

C₂

—

158

C₂

#20272592

yes

characteristic

159

C₂

#20272836

yes

characteristic

160

C₂

#20272842

yes

characteristic

	subgroup	G/H
center	C₂²	#16810390
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₅₃
Fitting	C₅₃×#1984	C₂
socle	C₂²×C₅₃	#342
prime radical	C₂³×C₅₃	#95
cosocle	C₂×C₄×Q₈×C₅₃	C₂²
hypercenter	#1984	D₅₃
hypocenter	C₅₃	#7248
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1984

subgroup

G/H

center

C₂²

#16810390

commutator

C₂²×C₄×C₅₃

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₅₃

Fitting

C₅₃×#1984

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×Q₈×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1984

N	H ≅ G/N	type
90. C₅₃	#7248	semidirect

H ≅ G/N

type

90. C₅₃

#7248

semidirect

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

subgroup	index
#20273059	2
C₅₃×#1984	2
#7248	53

subgroup

index

#20273059

C₅₃×#1984

#7248

Subgroups — #27271208

Conjugacy classes of subgroups

Complements

Maximal subgroups