Group #27283942 — subgroups

Subgroups — #27283942

Order 13568.

542 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 199 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	#20275677	C₂	1	4	yes	maximal
2	#20274872	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133149	—	2	2	no	—
6	C₂×C₄×#88322	C₂²	1	2	yes	—
7	#16810493	C₂²	1	2	yes	—
8	#16810385	—	2	8	no	—
9	#16810344	—	2	2	no	—
10	C₂²×Q₁₆×C₅₃	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#494	—	2	2	no	—
13	C₅₃×#494	C₄	1	4	yes	complemented
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₅₃	—	2	4	no	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	#13133149	—	4	4	no	—
19	C₄×#88322	—	2	2	no	—
20	C₄×#88322	—	2	2	no	—
21	C₂²×Q₈×C₅₃	—	2	2	no	—
22	C₂×Q₁₆×C₅₃	—	2	8	no	—
23	C₂×Q₁₆×C₅₃	—	4	4	no	—
24	C₂×Q₁₆×C₅₃	—	4	4	no	—
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	4	no	—
28	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	D₄	1	2	yes	—
33	C₂×Q₈×C₅₃	—	2	4	no	—
34	C₂×Q₈×C₅₃	—	4	2	no	—
35	C₂×Q₈×C₅₃	—	4	4	no	—
36	C₂×Q₈×C₅₃	—	4	4	no	—
37	C₂²×C₄×C₅₃	—	4	2	no	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
40	C₂²×Dic₅₃	—	4	2	no	—
41	Q₁₆×C₅₃	—	4	16	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂×C₈×C₅₃	D₈	1	4	yes	—
44	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
45	C₄⋊C₄×C₅₃	—	2	8	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₃	#41	1	2	yes	—
50	C₄⋊Dic₅₃	—	4	4	no	—
51	C₄⋊Dic₅₃	—	4	8	no	—
52	C₂×#88322	—	2	4	no	—
53	C₂×#88322	—	2	8	no	—
54	C₂³×C₅₃	#127	1	1	yes	characteristic
55	Q₈×C₅₃	—	4	8	no	—
56	Q₈×C₅₃	—	8	4	no	—
57	C₂×C₄×C₅₃	—	4	4	no	—
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	—	8	2	no	—
60	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₅₃	#121	1	2	yes	—
63	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	#137	1	8	yes	—
67	C₂×Dic₅₃	—	8	2	no	—
68	C₂×Dic₅₃	—	8	4	no	—
69	C₈×C₅₃	#102	1	8	yes	—
70	#88322	—	4	4	no	—
71	#9839	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	#390	1	1	yes	characteristic
73	C₂²×C₅₃	#492	1	1	yes	characteristic
74	C₂²×C₅₃	#493	1	1	yes	characteristic
75	C₂²×C₅₃	#497	1	4	yes	—
76	C₄×C₅₃	—	8	4	no	—
77	C₄×C₅₃	C₂×#102	1	2	yes	—
78	C₄×C₅₃	#441	1	4	yes	—
79	C₄×C₅₃	#470	1	4	yes	—
80	C₄×C₅₃	#486	1	2	yes	—
81	Dic₅₃	—	16	2	no	—
82	C₂×#494	D₅₃	1	1	yes	characteristic
83	#1990	—	53	4	no	—
84	#1844	—	53	1	no	—
85	C₂×#322	—	53	1	no	—
86	C₂×C₅₃	#1716	1	2	yes	—
87	C₂×C₅₃	#1722	1	2	yes	—
88	C₂×C₅₃	#1957	1	1	yes	characteristic
89	C₂×C₅₃	#1971	1	2	yes	—
90	C₂²×Q₁₆	—	2	2	no	—
91	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
92	#494	—	2	2	no	—
93	#494	Dic₅₃	1	4	yes	complemented
94	C₂×#106	—	106	2	no	—
95	C₂×C₄×C₈	—	53	2	no	—
96	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
97	#384	—	53	2	no	—
98	#340	—	106	8	no	—
99	#322	—	106	2	no	—
100	C₅₃	#9839	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	2	2	no	—
102	C₂×Q₁₆	—	2	8	no	—
103	C₂×Q₁₆	—	4	4	no	—
104	C₂×Q₁₆	—	4	4	no	—
105	C₂²×C₈	—	53	4	no	—
106	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
107	C₄⋊Q₈	—	2	4	no	—
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	C₂×Dic₅₃	1	4	yes	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	—	106	4	no	—
112	C₂×C₄⋊C₄	—	106	2	no	—
113	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
114	#106	—	212	4	no	—
115	C₄×C₈	—	106	2	no	—
116	C₄×C₈	—	106	2	no	—
117	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
118	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂×Q₈	—	4	4	no	—
122	C₂×Q₈	—	4	4	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	212	2	no	—
125	C₂²×C₄	D₄×D₅₃	1	2	yes	—
126	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
127	Q₁₆	—	4	16	no	—
128	C₂×C₈	—	106	4	no	—
129	C₂×C₈	—	106	8	no	—
130	C₂×C₈	#11719922	1	4	yes	—
131	C₂×C₈	#11719923	1	4	yes	—
132	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	212	4	no	—
136	C₄⋊C₄	—	212	8	no	—
137	C₄²	#11719947	1	2	yes	—
138	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
139	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂³	#13133213	1	1	yes	characteristic
141	Q₈	—	4	8	no	—
142	Q₈	—	8	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	424	4	no	—
147	C₂×C₄	—	424	2	no	—
148	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
149	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
150	C₂×C₄	#13133188	1	2	yes	—
151	C₂×C₄	#13133204	1	1	yes	characteristic
152	C₂×C₄	C₂×#11719922	1	2	yes	—
153	C₂×C₄	C₂×#11719923	1	2	yes	—
154	C₂×C₄	#13133249	1	8	yes	—
155	C₈	—	212	4	no	—
156	C₈	#13133138	1	8	yes	—
157	C₂²	#16810526	1	1	yes	characteristic
158	C₂²	#16810995	1	1	yes	characteristic
159	C₂²	#16810999	1	1	yes	characteristic
160	C₂²	#16811016	1	4	yes	—
161	C₄	—	8	4	no	—
162	C₄	—	848	2	no	—
163	C₄	C₂×#13133138	1	2	yes	—
164	C₄	#16810730	1	4	yes	—
165	C₄	#16810878	1	4	yes	—
166	C₄	#16810969	1	2	yes	—
167	C₂	#20274346	1	2	yes	—
168	C₂	#20274387	1	2	yes	—
169	C₂	#20275522	1	1	yes	characteristic
170	C₂	#20275587	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20275677

C₂

yes

maximal

#20274872

C₂

yes

maximal, characteristic

C₂×#16810344

C₂

yes

maximal, characteristic

C₂×C₅₃×#494

C₂

yes

maximal, characteristic

C₂×#13133149

—

C₂×C₄×#88322

C₂²

yes

—

#16810493

C₂²

yes

—

#16810385

—

#16810344

—

C₂²×Q₁₆×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#494

—

C₅₃×#494

C₄

yes

complemented

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133149

—

C₄×#88322

—

C₄×#88322

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#127

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#102

yes

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#102

yes

—

C₄×C₅₃

#441

yes

—

C₄×C₅₃

#470

yes

—

C₄×C₅₃

#486

yes

—

Dic₅₃

—

C₂×#494

D₅₃

yes

—

—

—

C₂×C₅₃

#1716

yes

—

C₂×C₅₃

#1722

yes

—

C₂×C₅₃

#1957

yes

characteristic

C₂×C₅₃

#1971

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₅₃

yes

—

#494

—

#494

Dic₅₃

yes

complemented

C₂×#106

—

106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

—

#340

—

106

—

#322

—

106

—

100

C₅₃

#9839

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²⋊D₅₃

yes

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

C₂×Dic₅₃

yes

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

106

—

112

C₂×C₄⋊C₄

—

106

—

113

C₂×C₄²

C₂²×D₅₃

yes

characteristic

114

#106

—

212

—

115

C₄×C₈

—

106

—

116

C₄×C₈

—

106

—

117

C₄×C₈

C₂×Dic₅₃

yes

—

118

C₄×C₈

C₂²⋊D₅₃

yes

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

212

—

125

C₂²×C₄

D₄×D₅₃

yes

—

126

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

127

Q₁₆

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

106

—

130

C₂×C₈

#11719922

yes

—

131

C₂×C₈

#11719923

yes

—

132

C₂×C₈

C₂²⋊Dic₅₃

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

212

—

136

C₄⋊C₄

—

212

—

137

C₄²

#11719947

yes

—

138

C₄²

C₂²×Dic₅₃

yes

characteristic

139

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

140

C₂³

#13133213

yes

characteristic

141

Q₈

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

424

—

147

C₂×C₄

—

424

—

148

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

149

C₂×C₄

D₄×Dic₅₃

yes

—

150

C₂×C₄

#13133188

yes

—

151

C₂×C₄

#13133204

yes

characteristic

152

C₂×C₄

C₂×#11719922

yes

—

153

C₂×C₄

C₂×#11719923

yes

—

154

C₂×C₄

#13133249

yes

—

155

C₈

—

212

—

156

C₈

#13133138

yes

—

157

C₂²

#16810526

yes

characteristic

158

C₂²

#16810995

yes

characteristic

159

C₂²

#16810999

yes

characteristic

160

C₂²

#16811016

yes

—

161

C₄

—

162

C₄

—

848

—

163

C₄

C₂×#13133138

yes

—

164

C₄

#16810730

yes

—

165

C₄

#16810878

yes

—

166

C₄

#16810969

yes

—

167

C₂

#20274346

yes

—

168

C₂

#20274387

yes

—

169

C₂

#20275522

yes

characteristic

170

C₂

#20275587

yes

—

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#9839
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂³

#13133213

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₅₃×#494

C₂

socle, prime radical

C₂³×C₅₃

#127

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
13. C₅₃×#494	C₄	semidirect
93. #494	Dic₅₃	semidirect
100. C₅₃	#9839	semidirect

H ≅ G/N

type

13. C₅₃×#494

C₄

semidirect

93. #494

Dic₅₃

semidirect

100. C₅₃

#9839

semidirect

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

subgroup	index
#20275677	2
#20274872	2
C₂×#16810344	2
C₂×C₅₃×#494	2
#9839	53

subgroup

index

C₂×C₅₃×#494

Subgroups — #27283942

Conjugacy classes of subgroups

Complements

Maximal subgroups