Group #27274544 — subgroups

Subgroups — #27274544

Order 13568.

330 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20274403	C₂	1	1	yes	maximal, characteristic
2	#20273008	C₂	1	2	yes	maximal
3	#20272546	C₂	1	1	yes	maximal, characteristic
4	#20272543	C₂	1	2	yes	maximal
5	C₅₃×#1275	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
10	#16810469	C₂²	1	1	yes	characteristic
11	#16810345	C₄	1	4	yes	complemented
12	#16810344	C₄	1	4	yes	complemented
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
18	C₄⋊#88322	—	2	2	no	—
19	C₂²⋊#88322	—	2	2	no	—
20	#13133127	—	2	4	no	—
21	C₄×#88322	C₂×C₄	1	4	yes	—
22	C₄×#88322	C₂×C₄	1	2	yes	—
23	C₄×#88322	C₂×C₄	1	2	yes	—
24	C₈×Dic₅₃	—	2	4	no	—
25	#13133124	—	2	1	no	—
26	#13133124	—	2	1	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₂²⋊C₈×C₅₃	—	2	2	no	—
31	#97×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
41	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
42	C₄²×C₅₃	#41	1	2	yes	—
43	C₄×Dic₅₃	—	2	2	no	—
44	C₄×Dic₅₃	M₁₆	1	4	yes	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	—	2	2	no	—
47	C₂×#88322	—	2	4	no	—
48	C₂×#88322	C₄²	1	1	yes	characteristic
49	C₂×#88322	C₄²	1	2	yes	—
50	C₂×#88322	C₄²	1	1	yes	characteristic
51	C₂³×C₅₃	#117	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#117	1	2	yes	—
56	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#117	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₅₃	—	2	4	no	—
61	C₂×Dic₅₃	—	4	2	no	—
62	C₂×Dic₅₃	—	4	2	no	—
63	C₈×C₅₃	—	2	4	no	—
64	C₈×C₅₃	—	4	4	no	—
65	#88322	—	2	1	no	—
66	#88322	—	2	1	no	—
67	#88322	—	2	2	no	—
68	#88322	—	4	4	no	—
69	#8140	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	—	2	1	no	—
71	C₂²×C₅₃	—	2	1	no	—
72	C₂²×C₅₃	#376	1	1	yes	characteristic
73	C₂²×C₅₃	#432	1	2	yes	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
77	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
78	C₄×C₅₃	#432	1	2	yes	—
79	Dic₅₃	—	4	4	no	—
80	#1726	—	53	1	no	—
81	#1452	—	53	2	no	—
82	#1275	—	53	1	no	—
83	#1275	D₅₃	1	1	yes	characteristic
84	#1274	—	53	2	no	—
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	#1745	1	1	yes	characteristic
87	C₂×C₅₃	#1747	1	1	yes	characteristic
88	C₂×C₅₃	#1747	1	1	yes	characteristic
89	#431	—	53	2	no	—
90	#431	C₂×D₅₃	1	2	yes	—
91	C₂×#97	—	53	2	no	—
92	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
93	C₂×C₄×C₈	—	53	1	no	—
94	C₂×C₄×C₈	—	53	2	no	—
95	#376	—	53	1	no	—
96	#322	—	53	4	no	—
97	#322	—	53	4	no	—
98	C₅₃	#8140	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	53	1	no	—
100	C₂²×C₈	—	53	1	no	—
101	C₂²×C₈	C₄×D₅₃	1	2	yes	—
102	C₂×C₄²	—	53	2	no	—
103	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	106	2	no	—
106	C₂²⋊C₈	—	2	2	no	—
107	C₂²⋊C₈	—	106	2	no	—
108	#97	—	2	2	no	—
109	#97	—	106	4	no	—
110	C₄×C₈	—	53	2	no	—
111	C₄×C₈	—	53	2	no	—
112	C₄×C₈	—	53	4	no	—
113	C₄×C₈	—	106	4	no	—
114	C₄×C₈	C₄×D₅₃	1	4	yes	—
115	#95	—	106	1	no	—
116	#95	—	106	1	no	—
117	C₂²×C₄	—	106	2	no	—
118	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₄	#11719947	1	1	yes	characteristic
120	C₂²×C₄	#11719949	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	53	1	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	53	2	no	—
127	C₂×C₈	—	106	2	no	—
128	C₂×C₈	—	106	2	no	—
129	C₂×C₈	—	106	4	no	—
130	C₂×C₈	#11719912	1	4	yes	—
131	C₄²	—	53	4	no	—
132	C₄²	—	106	2	no	—
133	C₄²	C₂×C₄×D₅₃	1	2	yes	—
134	C₄²	#11719947	1	2	yes	—
135	C₂³	#13133175	1	1	yes	characteristic
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	106	4	no	—
139	C₂×C₄	—	212	2	no	—
140	C₂×C₄	—	212	2	no	—
141	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
142	C₂×C₄	#13133173	1	2	yes	—
143	C₂×C₄	#13133173	1	1	yes	characteristic
144	C₂×C₄	#13133175	1	2	yes	—
145	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
146	C₂×C₄	C₂×#11719912	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₈	—	106	1	no	—
150	C₈	—	106	1	no	—
151	C₈	—	106	2	no	—
152	C₈	—	212	4	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#16810468	1	1	yes	characteristic
156	C₂²	#16810679	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	212	4	no	—
160	C₄	C₄×#11719912	1	1	yes	characteristic
161	C₄	#16810575	1	1	yes	characteristic
162	C₄	#16810677	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#20274462	1	1	yes	characteristic
165	C₂	#20274469	1	1	yes	characteristic
166	C₂	#20274471	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274403

C₂

yes

maximal, characteristic

#20273008

C₂

yes

maximal

#20272546

C₂

yes

maximal, characteristic

#20272543

C₂

yes

maximal

C₅₃×#1275

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133127

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

—

#16810469

C₂²

yes

characteristic

#16810345

C₄

yes

complemented

#16810344

C₄

yes

complemented

C₅₃×#431

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₈×Dic₅₃

—

#13133124

—

#13133124

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

C₂²×C₄×C₅₃

yes

C₂²×C₄×C₅₃

yes

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

#41

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

M₁₆

yes

—

—

—

—

C₄²

yes

characteristic

C₂×#88322

C₄²

yes

—

C₂×#88322

C₄²

yes

characteristic

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#376

yes

characteristic

C₂²×C₅₃

#432

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×M₁₆

yes

characteristic

C₄×C₅₃

C₄×M₁₆

yes

characteristic

C₄×C₅₃

#432

yes

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic

#1274

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×D₅₃

yes

—

C₂×#97

—

C₂×#97

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#322

—

#322

—

C₅₃

#8140

yes

characteristic, complemented, Sylow

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

C₄×D₅₃

yes

—

102

C₂×C₄²

—

103

C₂×C₄²

C₂²×D₅₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

—

106

C₂²⋊C₈

—

107

C₂²⋊C₈

—

106

—

108

#97

—

109

#97

—

106

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

106

—

114

C₄×C₈

C₄×D₅₃

yes

—

115

#95

—

106

—

116

#95

—

106

—

117

C₂²×C₄

—

106

—

118

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

119

C₂²×C₄

#11719947

yes

characteristic

120

C₂²×C₄

#11719949

yes

characteristic

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

106

—

130

C₂×C₈

#11719912

yes

—

131

C₄²

—

132

C₄²

—

106

—

133

C₄²

C₂×C₄×D₅₃

yes

—

134

C₄²

#11719947

yes

—

135

C₂³

#13133175

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

212

—

140

C₂×C₄

—

212

—

141

C₂×C₄

C₄²×D₅₃

yes

characteristic

142

C₂×C₄

#13133173

yes

—

143

C₂×C₄

#13133173

yes

characteristic

144

C₂×C₄

#13133175

yes

—

145

C₂×C₄

M₁₆×D₅₃

yes

—

146

C₂×C₄

C₂×#11719912

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

106

—

150

C₈

—

106

—

151

C₈

—

106

—

152

C₈

—

212

—

153

C₂²

—

154

C₂²

—

155

C₂²

#16810468

yes

characteristic

156

C₂²

#16810679

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

212

—

160

C₄

C₄×#11719912

yes

characteristic

161

C₄

#16810575

yes

characteristic

162

C₄

#16810677

yes

—

163

C₂

—

164

C₂

#20274462

yes

characteristic

165

C₂

#20274469

yes

characteristic

166

C₂

#20274471

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1275	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1275	D₅₃
hypocenter	C₅₃	#8140
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #1275

subgroup

G/H

center

C₂×C₄

C₄²×D₅₃

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1275

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #1275

N	H ≅ G/N	type
11. #16810345	C₄	semidirect
12. #16810344	C₄	semidirect
98. C₅₃	#8140	semidirect

H ≅ G/N

type

11. #16810345

C₄

semidirect

12. #16810344

C₄

semidirect

98. C₅₃

#8140

semidirect

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

subgroup	index
#20274403	2
#20273008	2
#20272546	2
#20272543	2
C₅₃×#1275	2
#8140	53

subgroup

index

C₅₃×#1275

Subgroups — #27274544

Conjugacy classes of subgroups

Complements

Maximal subgroups