Group #27300477 — subgroups

Subgroups — #27300477

Order 13568.

318 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	4	yes	maximal
4	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	#16810493	—	2	1	no	—
9	#16810399	C₄	1	4	yes	complemented
10	C₂×#107×C₅₃	C₂²	1	2	yes	—
11	C₂×#106×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#382	—	2	1	no	—
14	C₂×C₄⋊Dic₅₃	—	2	2	no	—
15	C₂×C₄⋊Dic₅₃	—	4	2	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	C₂×C₄	1	4	yes	—
19	#13133151	—	2	4	no	—
20	#13133151	—	4	2	no	—
21	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	#107×C₅₃	—	2	4	no	—
26	#106×C₅₃	—	4	2	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₃	Q₈	1	2	yes	—
29	#95×C₅₃	—	4	2	no	—
30	C₂²×C₄×C₅₃	—	4	2	no	—
31	C₂²×C₄×C₅₃	D₈	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
36	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	#41	1	2	yes	—
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₄⋊C₄×C₅₃	—	4	2	no	—
42	C₄⋊C₄×C₅₃	—	4	2	no	—
43	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₅₃	Q₁₆	1	2	yes	—
45	C₄⋊Dic₅₃	—	4	2	no	—
46	C₄⋊Dic₅₃	—	4	2	no	—
47	C₄⋊Dic₅₃	—	8	4	no	—
48	#11719908	—	2	4	no	—
49	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	8	2	no	—
53	C₂×C₄×C₅₃	—	8	2	no	—
54	C₂×C₄×C₅₃	—	8	2	no	—
55	C₂×C₄×C₅₃	#107	1	4	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×Dic₅₃	—	8	2	no	—
63	C₂×Dic₅₃	—	8	4	no	—
64	C₈×C₅₃	#106	1	4	yes	—
65	C₈×C₅₃	#117	1	2	yes	—
66	C₈×C₅₃	#125	1	2	yes	—
67	#12890	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
69	C₂²×C₅₃	#486	1	1	yes	characteristic
70	C₂²×C₅₃	#486	1	1	yes	characteristic
71	C₂²×C₅₃	#509	1	2	yes	—
72	C₂²×C₅₃	#510	1	2	yes	—
73	C₄×C₅₃	—	2	4	no	—
74	C₄×C₅₃	—	16	2	no	—
75	C₄×C₅₃	#382	1	1	yes	characteristic
76	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
77	C₄×C₅₃	#487	1	1	yes	characteristic
78	C₄×C₅₃	#487	1	1	yes	characteristic
79	Dic₅₃	—	16	2	no	—
80	C₂×#346	—	53	1	no	—
81	#1846	D₅₃	1	1	yes	characteristic
82	#1845	—	53	1	no	—
83	#1380	—	53	4	no	—
84	C₂×C₅₃	#1834	1	1	yes	characteristic
85	C₂×C₅₃	#2159	1	2	yes	—
86	C₂×C₅₃	#2243	1	2	yes	—
87	C₂×C₅₃	#2244	1	1	yes	characteristic
88	C₂×C₅₃	#2245	1	1	yes	characteristic
89	C₂²×C₁₆	—	53	2	no	—
90	C₂×#107	—	53	2	no	—
91	C₂×#107	—	106	1	no	—
92	C₂×#107	C₂×D₅₃	1	2	yes	—
93	C₂×#106	—	2	1	no	—
94	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
95	#384	—	106	1	no	—
96	#382	—	2	1	no	—
97	#346	—	53	4	no	—
98	C₅₃	#12890	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	106	2	no	—
103	C₂×C₄⋊C₄	—	212	2	no	—
104	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂×C₁₆	—	53	4	no	—
106	C₂×C₁₆	—	106	2	no	—
107	C₂×C₁₆	—	106	2	no	—
108	#107	—	2	4	no	—
109	#107	—	106	4	no	—
110	#107	—	212	2	no	—
111	#106	—	4	2	no	—
112	C₄×C₈	Dic₁₀₆	1	2	yes	—
113	C₄×C₈	C₄×D₅₃	1	2	yes	—
114	#95	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	212	2	no	—
117	C₂²×C₄	#11719922	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
119	C₂×C₈	#11719919	1	2	yes	—
120	C₂×C₈	#11719923	1	2	yes	—
121	C₂×C₈	#11719924	1	2	yes	—
122	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
124	C₂×C₈	#11719947	1	2	yes	—
125	C₂×C₈	#11719949	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	424	4	no	—
129	C₄⋊C₄	—	212	2	no	—
130	C₄⋊C₄	—	212	2	no	—
131	C₄²	#11719919	1	2	yes	—
132	C₄²	#11719925	1	2	yes	—
133	C₁₆	—	106	4	no	—
134	C₂³	C₂×#11719922	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	424	4	no	—
141	C₂×C₄	—	424	2	no	—
142	C₂×C₄	#13133150	1	4	yes	—
143	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
144	C₂×C₄	#13133204	1	1	yes	characteristic
145	C₂×C₄	#13133206	1	1	yes	characteristic
146	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
147	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
148	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
149	C₈	#13133148	1	4	yes	—
150	C₈	#13133175	1	2	yes	—
151	C₈	#13133208	1	2	yes	—
152	C₂²	C₂×#13133150	1	1	yes	characteristic
153	C₂²	#16810969	1	1	yes	characteristic
154	C₂²	#16810971	1	1	yes	characteristic
155	C₂²	#16811058	1	2	yes	—
156	C₂²	#16811063	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	848	2	no	—
160	C₄	#16810488	1	1	yes	characteristic
161	C₄	C₂×#13133148	1	1	yes	characteristic
162	C₄	#16810973	1	1	yes	characteristic
163	C₄	#16810975	1	1	yes	characteristic
164	C₂	#20274842	1	1	yes	characteristic
165	C₂	#20276504	1	2	yes	—
166	C₂	#20276951	1	2	yes	—
167	C₂	#20276958	1	1	yes	characteristic
168	C₂	#20276962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810399

C₂

yes

maximal, characteristic

#20274875

C₂

yes

maximal, characteristic

#20272806

C₂

yes

maximal

C₅₃×#1846

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

—

C₂×#13133151

—

C₂×#13133151

C₂²

yes

—

#16810493

—

#16810399

C₄

yes

complemented

C₂×#107×C₅₃

C₂²

yes

—

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#382

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

C₂×#11719908

—

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

—

#13133151

—

#13133151

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#107×C₅₃

—

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719908

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#117

yes

—

C₈×C₅₃

#125

yes

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#510

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

C₂×#346

—

#1846

D₅₃

yes

characteristic

#1845

—

#1380

—

C₂×C₅₃

#1834

yes

characteristic

C₂×C₅₃

#2159

yes

—

C₂×C₅₃

#2243

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

—

106

—

C₂×#107

C₂×D₅₃

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#384

—

106

—

#382

—

#346

—

C₅₃

#12890

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

100

C₂²×C₈

C₂²×D₅₃

yes

characteristic

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

106

—

103

C₂×C₄⋊C₄

—

212

—

104

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

105

C₂×C₁₆

—

106

C₂×C₁₆

—

106

—

107

C₂×C₁₆

—

106

—

108

#107

—

109

#107

—

106

—

110

#107

—

212

—

111

#106

—

112

C₄×C₈

Dic₁₀₆

yes

—

113

C₄×C₈

C₄×D₅₃

yes

—

114

#95

—

115

C₂²×C₄

—

116

C₂²×C₄

—

212

—

117

C₂²×C₄

#11719922

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

119

C₂×C₈

#11719919

yes

—

120

C₂×C₈

#11719923

yes

—

121

C₂×C₈

#11719924

yes

—

122

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

123

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

124

C₂×C₈

#11719947

yes

—

125

C₂×C₈

#11719949

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

424

—

129

C₄⋊C₄

—

212

—

130

C₄⋊C₄

—

212

—

131

C₄²

#11719919

yes

—

132

C₄²

#11719925

yes

—

133

C₁₆

—

106

—

134

C₂³

C₂×#11719922

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

424

—

141

C₂×C₄

—

424

—

142

C₂×C₄

#13133150

yes

—

143

C₂×C₄

C₂×#11719919

yes

characteristic

144

C₂×C₄

#13133204

yes

characteristic

145

C₂×C₄

#13133206

yes

characteristic

146

C₂×C₄

C₂×#11719924

yes

characteristic

147

C₂×C₄

C₂×#11719923

yes

characteristic

148

C₂×C₄

C₂×#11719925

yes

characteristic

149

C₈

#13133148

yes

—

150

C₈

#13133175

yes

—

151

C₈

#13133208

yes

—

152

C₂²

C₂×#13133150

yes

characteristic

153

C₂²

#16810969

yes

characteristic

154

C₂²

#16810971

yes

characteristic

155

C₂²

#16811058

yes

—

156

C₂²

#16811063

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

848

—

160

C₄

#16810488

yes

characteristic

161

C₄

C₂×#13133148

yes

characteristic

162

C₄

#16810973

yes

characteristic

163

C₄

#16810975

yes

characteristic

164

C₂

#20274842

yes

characteristic

165

C₂

#20276504

yes

—

166

C₂

#20276951

yes

—

167

C₂

#20276958

yes

characteristic

168

C₂

#20276962

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1846	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1846	D₅₃
hypocenter	C₅₃	#12890
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1846

C₂

socle, prime radical

C₂³×C₅₃

C₂×D₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. #16810399	C₄	semidirect
98. C₅₃	#12890	semidirect

H ≅ G/N

type

9. #16810399

C₄

semidirect

98. C₅₃

#12890

semidirect

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

subgroup	index
C₂×#16810399	2
#20274875	2
#20272806	2
C₅₃×#1846	2
#12890	53

subgroup

index

C₅₃×#1846

Subgroups — #27300477

Conjugacy classes of subgroups

Complements

Maximal subgroups