Group #27274539 — subgroups

Subgroups — #27274539

Order 13568.

330 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 139 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	#20274401	C₂	1	1	yes	maximal, characteristic
2	#20273007	C₂	1	1	yes	maximal, characteristic
3	#20272545	C₂	1	2	yes	maximal
4	#20272543	C₂	1	2	yes	maximal
5	C₅₃×#1452	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133128	C₂²	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
10	#16810469	C₂²	1	1	yes	characteristic
11	#16810345	C₄	1	4	yes	complemented
12	C₅₃×#431	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#322	C₄	1	4	yes	complemented
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
17	C₄⋊#88322	—	2	2	no	—
18	C₂²⋊#88322	—	2	2	no	—
19	#13133128	—	2	2	no	—
20	#13133127	—	2	4	no	—
21	C₄×#88322	C₂×C₄	1	4	yes	—
22	C₈×Dic₅₃	—	2	4	no	—
23	#13133124	—	2	2	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₂²⋊C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	#41	1	2	yes	—
34	C₂²×Dic₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	4	no	—
38	C₂×C₈×C₅₃	C₄²	1	2	yes	—
39	C₂×C₈×C₅₃	C₄²	1	2	yes	—
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	M₁₆	1	4	yes	—
49	C₂³×C₅₃	#117	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#117	1	2	yes	—
53	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#117	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
57	C₂×Dic₅₃	—	2	4	no	—
58	C₂×Dic₅₃	—	4	4	no	—
59	C₈×C₅₃	—	2	2	no	—
60	C₈×C₅₃	—	2	2	no	—
61	C₈×C₅₃	—	4	4	no	—
62	#88322	—	2	4	no	—
63	#88322	—	4	4	no	—
64	#8139	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	—	2	2	no	—
66	C₂²×C₅₃	#376	1	1	yes	characteristic
67	C₂²×C₅₃	#432	1	2	yes	—
68	C₄×C₅₃	—	2	2	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
71	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
72	C₄×C₅₃	#432	1	2	yes	—
73	Dic₅₃	—	4	4	no	—
74	#1726	—	53	1	no	—
75	#1452	—	53	1	no	—
76	#1452	D₅₃	1	1	yes	characteristic
77	#1275	—	53	2	no	—
78	#1274	—	53	2	no	—
79	C₂×C₅₃	—	2	2	no	—
80	C₂×C₅₃	#1745	1	1	yes	characteristic
81	C₂×C₅₃	#1747	1	2	yes	—
82	#431	—	53	2	no	—
83	#431	C₂×D₅₃	1	2	yes	—
84	C₂×#97	—	53	2	no	—
85	C₂×#97	—	53	1	no	—
86	C₂×C₄×C₈	—	53	2	no	—
87	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	#376	—	53	1	no	—
89	#322	—	53	4	no	—
90	#322	Dic₅₃	1	4	yes	complemented
91	C₅₃	#8139	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	53	2	no	—
93	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
95	C₂×C₄²	—	53	2	no	—
96	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	106	2	no	—
99	C₂²⋊C₈	—	2	2	no	—
100	C₂²⋊C₈	—	106	2	no	—
101	#97	—	106	4	no	—
102	#97	—	106	2	no	—
103	C₄×C₈	—	53	4	no	—
104	C₄×C₈	—	106	4	no	—
105	C₄×C₈	C₄×D₅₃	1	2	yes	—
106	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
107	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
108	#95	—	106	2	no	—
109	C₂²×C₄	—	106	2	no	—
110	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
111	C₂²×C₄	#11719945	1	2	yes	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	53	4	no	—
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	—	106	4	no	—
119	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
120	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
121	C₄²	—	53	4	no	—
122	C₄²	—	106	2	no	—
123	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
124	C₄²	#11719945	1	2	yes	—
125	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
126	C₂³	#13133174	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	106	4	no	—
129	C₂×C₄	—	212	4	no	—
130	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
131	C₂×C₄	#13133174	1	2	yes	—
132	C₂×C₄	#13133174	1	1	yes	characteristic
133	C₂×C₄	#13133176	1	2	yes	—
134	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
135	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	4	4	no	—
139	C₈	—	106	4	no	—
140	C₈	—	212	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#16810469	1	1	yes	characteristic
143	C₂²	#16810678	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	212	4	no	—
147	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
148	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
149	C₄	#16810678	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#20274463	1	1	yes	characteristic
152	C₂	#20274470	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274401

C₂

yes

maximal, characteristic

#20273007

C₂

yes

maximal, characteristic

#20272545

C₂

yes

maximal

#20272543

C₂

yes

maximal

C₅₃×#1452

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133128

C₂²

yes

characteristic

C₂×#13133127

C₄

yes

—

C₂×C₈×Dic₅₃

C₄

yes

—

#16810469

C₂²

yes

characteristic

#16810345

C₄

yes

complemented

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#322

C₄

yes

complemented

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₈×Dic₅₃

—

#13133124

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₄²

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

M₁₆

yes

—

—

—

—

M₁₆

yes

—

C₂³×C₅₃

#117

yes

characteristic

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

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×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

—

#1726

—

#1452

—

#1452

D₅₃

yes

characteristic

#1275

—

#1274

—

C₂×C₅₃

—

C₂×C₅₃

#1745

yes

characteristic

C₂×C₅₃

#1747

yes

—

#431

—

#431

C₂×D₅₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

—

Dic₅₃

yes

complemented

C₅₃

#8139

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

106

—

101

#97

—

106

—

102

#97

—

106

—

103

C₄×C₈

—

104

C₄×C₈

—

106

—

105

C₄×C₈

C₄×D₅₃

yes

—

106

C₄×C₈

C₂×Dic₅₃

yes

—

107

C₄×C₈

C₂×Dic₅₃

yes

—

108

#95

—

106

—

109

C₂²×C₄

—

106

—

110

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

111

C₂²×C₄

#11719945

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

119

C₂×C₈

C₄×Dic₅₃

yes

—

120

C₂×C₈

C₄×Dic₅₃

yes

—

121

C₄²

—

122

C₄²

—

106

—

123

C₄²

C₂×C₄×D₅₃

yes

characteristic

124

C₄²

#11719945

yes

—

125

C₄²

C₂²×Dic₅₃

yes

characteristic

126

C₂³

#13133174

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

212

—

130

C₂×C₄

C₂×C₄×Dic₅₃

yes

characteristic

131

C₂×C₄

#13133174

yes

—

132

C₂×C₄

#13133174

yes

characteristic

133

C₂×C₄

#13133176

yes

—

134

C₂×C₄

M₁₆×D₅₃

yes

—

135

C₂×C₄

M₁₆×D₅₃

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

106

—

140

C₈

—

212

—

141

C₂²

—

142

C₂²

#16810469

yes

characteristic

143

C₂²

#16810678

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

212

—

147

C₄

M₁₆×Dic₅₃

yes

characteristic

148

C₄

M₁₆×Dic₅₃

yes

characteristic

149

C₄

#16810678

yes

—

150

C₂

—

151

C₂

#20274463

yes

characteristic

152

C₂

#20274470

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1452	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1452	D₅₃
hypocenter	C₅₃	#8139
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #1452

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₅₃

commutator

C₂×C₄×C₅₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1452

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #1452

N	H ≅ G/N	type
11. #16810345	C₄	semidirect
14. C₅₃×#322	C₄	semidirect
90. #322	Dic₅₃	semidirect
91. C₅₃	#8139	semidirect

H ≅ G/N

type

11. #16810345

C₄

semidirect

14. C₅₃×#322

C₄

semidirect

90. #322

Dic₅₃

semidirect

91. C₅₃

#8139

semidirect

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

subgroup	index
#20274401	2
#20273007	2
#20272545	2
#20272543	2
C₅₃×#1452	2
#8139	53

subgroup

index

C₅₃×#1452

Subgroups — #27274539

Conjugacy classes of subgroups

Complements

Maximal subgroups