Group #27277902 — subgroups

Subgroups — #27277902

Order 13568.

338 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20274886	C₂	1	2	yes	maximal
2	#20273471	C₂	1	1	yes	maximal, characteristic
3	#20273470	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133152	—	2	4	no	—
7	#16810647	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810374	—	2	2	no	—
11	#16810373	—	2	4	no	—
12	#16810345	C₄	1	4	yes	—
13	C₅₃×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#334	—	2	2	no	—
16	C₂×#11719917	—	2	2	no	—
17	C₂²×#88322	D₄	1	2	yes	—
18	C₂²×#88322	Q₈	1	2	yes	—
19	#13133152	—	4	8	no	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄×#88322	C₂×C₄	1	2	yes	—
23	C₄×#88322	C₂×C₄	1	2	yes	—
24	C₄×#88322	D₄	1	2	yes	—
25	C₄×#88322	Q₈	1	2	yes	—
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	D₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	M₁₆×C₅₃	—	4	4	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	#11719917	—	4	4	no	—
45	C₂×#88322	—	4	2	no	—
46	C₂×#88322	—	4	2	no	—
47	C₂×#88322	C₄⋊C₄	1	4	yes	—
48	C₂×#88322	C₄⋊C₄	1	4	yes	—
49	C₂×#88322	SD₁₆	1	8	yes	—
50	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
60	C₈×C₅₃	—	2	4	no	—
61	C₈×C₅₃	—	8	2	no	—
62	#88322	—	2	4	no	—
63	#88322	—	8	2	no	—
64	#88322	#106	1	8	yes	—
65	#8825	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	—	2	1	no	—
67	C₂²×C₅₃	—	2	1	no	—
68	C₂²×C₅₃	#384	1	1	yes	characteristic
69	C₂²×C₅₃	#498	1	2	yes	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	C₂×#106	1	2	yes	—
73	C₄×C₅₃	#430	1	2	yes	—
74	C₄×C₅₃	#430	1	2	yes	—
75	C₄×C₅₃	#492	1	2	yes	—
76	#1848	—	53	2	no	—
77	#1566	—	53	2	no	—
78	#1566	—	53	1	no	—
79	#1566	D₅₃	1	1	yes	characteristic
80	C₂×#322	—	53	1	no	—
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	#1844	1	1	yes	characteristic
83	C₂×C₅₃	#1849	1	1	yes	characteristic
84	C₂×C₅₃	#1849	1	1	yes	characteristic
85	C₂×#108	—	106	4	no	—
86	#423	—	53	2	no	—
87	#423	C₂×D₅₃	1	2	yes	—
88	C₂×C₄×C₈	—	53	1	no	—
89	C₂×C₄×C₈	—	53	1	no	—
90	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
91	#334	—	2	2	no	—
92	#334	—	106	4	no	—
93	#334	—	106	2	no	—
94	#322	—	53	4	no	—
95	C₅₃	#8825	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	106	2	no	—
98	C₂²×C₈	—	53	2	no	—
99	C₂²×C₈	—	53	2	no	—
100	C₂²×C₈	Dic₁₀₆	1	2	yes	—
101	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
102	#108	—	212	8	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	106	2	no	—
106	C₄⋊C₈	—	106	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	—	53	2	no	—
109	C₄×C₈	—	53	2	no	—
110	C₄×C₈	—	53	2	no	—
111	C₄×C₈	C₄×D₅₃	1	2	yes	—
112	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
113	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
114	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
115	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
116	M₁₆	—	4	4	no	—
117	M₁₆	—	212	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	53	4	no	—
121	C₂×C₈	—	53	8	no	—
122	C₂×C₈	—	53	4	no	—
123	C₂×C₈	—	212	2	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	#11719919	1	4	yes	—
126	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
127	C₄²	D₄×D₅₃	1	1	yes	characteristic
128	C₄²	Q₈×D₅₃	1	1	yes	characteristic
129	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂³	#13133216	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
136	C₂×C₄	#13133213	1	1	yes	characteristic
137	C₂×C₄	#13133215	1	1	yes	characteristic
138	C₂×C₄	#13133216	1	1	yes	characteristic
139	C₂×C₄	SD₁₆×D₅₃	1	4	yes	—
140	C₈	—	2	4	no	—
141	C₈	—	8	2	no	—
142	C₈	—	53	8	no	—
143	C₈	—	106	4	no	—
144	C₈	—	424	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#16810492	1	1	yes	characteristic
148	C₂²	#16811020	1	2	yes	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#106×D₅₃	1	2	yes	—
152	C₄	#16810670	1	2	yes	—
153	C₄	#16810671	1	2	yes	—
154	C₄	#16810994	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	#20274870	1	1	yes	characteristic
157	C₂	#20274888	1	1	yes	characteristic
158	C₂	#20274889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274886

C₂

yes

maximal

#20273471

C₂

yes

maximal, characteristic

#20273470

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₅₃×#1566

C₂

yes

maximal, characteristic

C₂×#13133152

—

#16810647

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

—

C₄

yes

—

C₅₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#334

—

C₂×#11719917

—

C₂²×#88322

D₄

yes

—

C₂²×#88322

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

SD₁₆

yes

—

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

yes

—

#8825

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#498

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#106

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

#492

yes

—

—

—

—

D₅₃

yes

characteristic

C₂×#322

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

106

—

#423

—

#423

C₂×D₅₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#334

—

#334

—

106

—

#334

—

106

—

#322

—

C₅₃

#8825

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

Dic₁₀₆

yes

—

101

C₂×C₄²

C₂²×D₅₃

yes

characteristic

102

#108

—

212

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

—

106

C₄⋊C₈

—

106

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₄×D₅₃

yes

—

112

C₄×C₈

C₂²⋊D₅₃

yes

—

113

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

114

C₂²×C₄

D₄×D₅₃

yes

characteristic

115

C₂²×C₄

Q₈×D₅₃

yes

characteristic

116

M₁₆

—

117

M₁₆

—

212

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

212

—

124

C₂×C₈

—

212

—

125

C₂×C₈

#11719919

yes

—

126

C₄²

C₂×C₄×D₅₃

yes

characteristic

127

C₄²

D₄×D₅₃

yes

characteristic

128

C₄²

Q₈×D₅₃

yes

characteristic

129

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₂³

#13133216

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

134

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

135

C₂×C₄

C₂×#11719919

yes

characteristic

136

C₂×C₄

#13133213

yes

characteristic

137

C₂×C₄

#13133215

yes

characteristic

138

C₂×C₄

#13133216

yes

characteristic

139

C₂×C₄

SD₁₆×D₅₃

yes

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

106

—

144

C₈

—

424

—

145

C₂²

—

146

C₂²

—

147

C₂²

#16810492

yes

characteristic

148

C₂²

#16811020

yes

—

149

C₄

—

150

C₄

—

151

C₄

#106×D₅₃

yes

—

152

C₄

#16810670

yes

—

153

C₄

#16810671

yes

—

154

C₄

#16810994

yes

—

155

C₂

—

156

C₂

#20274870

yes

characteristic

157

C₂

#20274888

yes

characteristic

158

C₂

#20274889

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#8825
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1566

C₂

socle

C₂²×C₅₃

#384

prime radical

C₂³×C₅₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
95. C₅₃	#8825	semidirect

H ≅ G/N

type

95. C₅₃

#8825

semidirect

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

subgroup	index
#20274886	2
#20273471	2
#20273470	2
C₂×#16810345	2
C₅₃×#1566	2
#8825	53

subgroup

index

C₅₃×#1566

Subgroups — #27277902

Conjugacy classes of subgroups

Complements

Maximal subgroups