Group #27277905 — subgroups

Subgroups — #27277905

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	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20273473	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1567	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	#16810376	—	2	2	no	—
11	#16810375	—	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×#335	—	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	D₈	1	4	yes	—
50	C₂×#88322	Q₁₆	1	4	yes	—
51	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
62	C₈×C₅₃	—	2	4	no	—
63	C₈×C₅₃	—	8	2	no	—
64	#88322	—	2	4	no	—
65	#88322	—	8	2	no	—
66	#88322	#107	1	8	yes	—
67	#8826	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	#384	1	1	yes	characteristic
71	C₂²×C₅₃	#500	1	2	yes	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	C₂×#107	1	2	yes	—
75	C₄×C₅₃	#430	1	2	yes	—
76	C₄×C₅₃	#430	1	2	yes	—
77	C₄×C₅₃	#493	1	1	yes	characteristic
78	C₄×C₅₃	#494	1	1	yes	characteristic
79	#1848	—	53	2	no	—
80	#1567	—	53	2	no	—
81	#1567	—	53	1	no	—
82	#1567	D₅₃	1	1	yes	characteristic
83	C₂×#322	—	53	1	no	—
84	C₂×C₅₃	—	2	2	no	—
85	C₂×C₅₃	#1845	1	1	yes	characteristic
86	C₂×C₅₃	#1849	1	1	yes	characteristic
87	C₂×C₅₃	#1849	1	1	yes	characteristic
88	C₂×#108	—	106	4	no	—
89	#423	—	53	2	no	—
90	#423	C₂×D₅₃	1	2	yes	—
91	C₂×C₄×C₈	—	53	1	no	—
92	C₂×C₄×C₈	—	53	1	no	—
93	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
94	#335	—	2	2	no	—
95	#335	—	106	4	no	—
96	#335	—	106	2	no	—
97	#322	—	53	4	no	—
98	C₅₃	#8826	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	106	2	no	—
101	C₂²×C₈	—	53	2	no	—
102	C₂²×C₈	—	53	2	no	—
103	C₂²×C₈	Dic₁₀₆	1	2	yes	—
104	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
105	#108	—	212	8	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	106	2	no	—
109	C₄⋊C₈	—	106	2	no	—
110	C₄×C₈	—	53	2	no	—
111	C₄×C₈	—	53	2	no	—
112	C₄×C₈	—	53	2	no	—
113	C₄×C₈	—	53	2	no	—
114	C₄×C₈	C₄×D₅₃	1	2	yes	—
115	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
116	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
117	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
118	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
119	M₁₆	—	4	4	no	—
120	M₁₆	—	212	4	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	53	4	no	—
124	C₂×C₈	—	53	4	no	—
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	—	53	4	no	—
127	C₂×C₈	—	212	2	no	—
128	C₂×C₈	—	212	2	no	—
129	C₂×C₈	#11719919	1	4	yes	—
130	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₄²	D₄×D₅₃	1	1	yes	characteristic
132	C₄²	Q₈×D₅₃	1	1	yes	characteristic
133	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂³	#13133216	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
140	C₂×C₄	#13133213	1	1	yes	characteristic
141	C₂×C₄	#13133215	1	1	yes	characteristic
142	C₂×C₄	#13133216	1	1	yes	characteristic
143	C₂×C₄	D₈×D₅₃	1	2	yes	—
144	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
145	C₈	—	2	4	no	—
146	C₈	—	8	2	no	—
147	C₈	—	53	8	no	—
148	C₈	—	106	4	no	—
149	C₈	—	424	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#16810492	1	1	yes	characteristic
153	C₂²	#16811027	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#107×D₅₃	1	2	yes	—
157	C₄	#16810670	1	2	yes	—
158	C₄	#16810671	1	2	yes	—
159	C₄	#16810998	1	1	yes	characteristic
160	C₄	#16811001	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#20274873	1	1	yes	characteristic
163	C₂	#20274888	1	1	yes	characteristic
164	C₂	#20274889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274886

C₂

yes

maximal

#20273474

C₂

yes

maximal, characteristic

#20273473

C₂

yes

maximal

C₂×#16810345

C₂

yes

maximal, characteristic

C₅₃×#1567

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×#335

—

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

D₈

yes

—

C₂×#88322

Q₁₆

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×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

yes

—

#8826

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#500

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

#430

yes

—

C₄×C₅₃

yes

C₄×C₅₃

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

#335

—

#335

—

106

—

#335

—

106

—

#322

—

C₅₃

#8826

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

100

C₂×M₁₆

—

106

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

Dic₁₀₆

yes

—

104

C₂×C₄²

C₂²×D₅₃

yes

characteristic

105

#108

—

212

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

106

—

109

C₄⋊C₈

—

106

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₄×D₅₃

yes

—

115

C₄×C₈

C₂²⋊D₅₃

yes

—

116

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

117

C₂²×C₄

D₄×D₅₃

yes

characteristic

118

C₂²×C₄

Q₈×D₅₃

yes

characteristic

119

M₁₆

—

120

M₁₆

—

212

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

212

—

128

C₂×C₈

—

212

—

129

C₂×C₈

#11719919

yes

—

130

C₄²

C₂×C₄×D₅₃

yes

characteristic

131

C₄²

D₄×D₅₃

yes

characteristic

132

C₄²

Q₈×D₅₃

yes

characteristic

133

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

134

C₂³

#13133216

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

138

C₂×C₄

C₄⋊C₄×D₅₃

yes

characteristic

139

C₂×C₄

C₂×#11719919

yes

characteristic

140

C₂×C₄

#13133213

yes

characteristic

141

C₂×C₄

#13133215

yes

characteristic

142

C₂×C₄

#13133216

yes

characteristic

143

C₂×C₄

D₈×D₅₃

yes

—

144

C₂×C₄

Q₁₆×D₅₃

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

106

—

149

C₈

—

424

—

150

C₂²

—

151

C₂²

—

152

C₂²

#16810492

yes

characteristic

153

C₂²

#16811027

yes

—

154

C₄

—

155

C₄

—

156

C₄

#107×D₅₃

yes

—

157

C₄

#16810670

yes

—

158

C₄

#16810671

yes

—

159

C₄

#16810998

yes

characteristic

160

C₄

#16811001

yes

characteristic

161

C₂

—

162

C₂

#20274873

yes

characteristic

163

C₂

#20274888

yes

characteristic

164

C₂

#20274889

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1567	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1567	D₅₃
hypocenter	C₅₃	#8826
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1567

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 ◁ #1567

N	H ≅ G/N	type
98. C₅₃	#8826	semidirect

H ≅ G/N

type

98. C₅₃

#8826

semidirect

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

subgroup	index
#20274886	2
#20273474	2
#20273473	2
C₂×#16810345	2
C₅₃×#1567	2
#8826	53

subgroup

index

C₅₃×#1567

Subgroups — #27277905

Conjugacy classes of subgroups

Complements

Maximal subgroups