Group #27277796 — subgroups

Subgroups — #27277796

Order 13568.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20274879	C₂	1	1	yes	maximal, characteristic
2	#20273007	C₂	1	1	yes	maximal, characteristic
3	#20272575	C₂	1	2	yes	maximal
4	#20272575	C₂	1	2	yes	maximal
5	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133149	—	2	1	no	—
9	#16810489	C₂²	1	2	yes	—
10	#16810345	C₄	1	4	yes	complemented
11	C₅₃×#423	C₂²	1	1	yes	characteristic
12	C₅₃×#423	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#341	—	2	1	no	—
15	C₅₃×#341	—	2	1	no	—
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	#13133151	—	4	2	no	—
18	#13133149	—	4	2	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₂²⋊#88322	—	2	2	no	—
21	C₄×#88322	C₂×C₄	1	4	yes	—
22	#13133124	—	2	4	no	—
23	C₂×M₁₆×C₅₃	—	2	1	no	—
24	C₂×M₁₆×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	#41	1	2	yes	—
34	C₂²×Dic₅₃	—	4	2	no	—
35	M₁₆×C₅₃	—	4	1	no	—
36	M₁₆×C₅₃	—	4	1	no	—
37	M₁₆×C₅₃	—	4	1	no	—
38	M₁₆×C₅₃	—	4	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	4	1	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₃	#41	1	2	yes	—
50	C₄⋊Dic₅₃	—	4	2	no	—
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₂×#88322	—	2	4	no	—
53	C₂³×C₅₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	4	2	no	—
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#117	1	2	yes	—
57	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#125	1	2	yes	—
59	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
63	C₂×Dic₅₃	—	8	4	no	—
64	C₂×Dic₅₃	—	8	2	no	—
65	C₈×C₅₃	—	2	2	no	—
66	C₈×C₅₃	—	2	2	no	—
67	C₈×C₅₃	—	8	1	no	—
68	C₈×C₅₃	—	8	1	no	—
69	#88322	—	4	4	no	—
70	#8795	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	#382	1	1	yes	characteristic
73	C₂²×C₅₃	#490	1	1	yes	characteristic
74	C₂²×C₅₃	#491	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	#428	1	1	yes	characteristic
78	C₄×C₅₃	#428	1	1	yes	characteristic
79	C₄×C₅₃	#489	1	1	yes	characteristic
80	C₄×C₅₃	#489	1	1	yes	characteristic
81	Dic₅₃	—	16	2	no	—
82	#1846	—	53	1	no	—
83	#1837	D₅₃	1	1	yes	characteristic
84	#1452	—	53	1	no	—
85	#1286	—	53	2	no	—
86	#1286	—	53	2	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	#1833	1	2	yes	—
90	C₂×C₅₃	#1836	1	1	yes	characteristic
91	#431	—	53	2	no	—
92	C₂×#107	—	106	1	no	—
93	C₂×#106	—	106	1	no	—
94	#423	C₂×D₅₃	1	1	yes	characteristic
95	#423	C₂×D₅₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
97	#382	—	53	2	no	—
98	#341	—	2	1	no	—
99	#341	—	2	1	no	—
100	#322	—	53	4	no	—
101	C₅₃	#8795	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	106	2	no	—
107	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
108	#107	—	212	2	no	—
109	#106	—	212	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	106	2	no	—
113	C₂²⋊C₈	—	106	2	no	—
114	C₄×C₈	—	53	4	no	—
115	C₄×C₈	Dic₁₀₆	1	2	yes	—
116	C₄×C₈	C₄×D₅₃	1	2	yes	—
117	#95	—	106	4	no	—
118	C₂²×C₄	—	212	2	no	—
119	C₂²×C₄	#11719949	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	M₁₆	—	4	1	no	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	4	1	no	—
124	M₁₆	—	4	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	106	4	no	—
132	C₂×C₈	#11719919	1	2	yes	—
133	C₂×C₈	#11719919	1	2	yes	—
134	C₄⋊C₄	—	212	2	no	—
135	C₄⋊C₄	—	212	2	no	—
136	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₄²	#11719947	1	2	yes	—
139	C₂³	#13133206	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	424	4	no	—
142	C₂×C₄	—	424	2	no	—
143	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
144	C₂×C₄	#13133175	1	2	yes	—
145	C₂×C₄	#13133204	1	1	yes	characteristic
146	C₂×C₄	#13133208	1	2	yes	—
147	C₂×C₄	#13133242	1	1	yes	characteristic
148	C₂×C₄	#13133242	1	1	yes	characteristic
149	C₂×C₄	#13133247	1	1	yes	characteristic
150	C₂×C₄	#13133247	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	8	1	no	—
154	C₈	—	8	1	no	—
155	C₈	—	212	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#16810488	1	1	yes	characteristic
158	C₂²	#16810989	1	1	yes	characteristic
159	C₂²	#16810993	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	848	2	no	—
163	C₄	#16810666	1	1	yes	characteristic
164	C₄	#16810666	1	1	yes	characteristic
165	C₄	#16810982	1	1	yes	characteristic
166	C₄	#16810982	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#20274840	1	2	yes	—
170	C₂	#20274848	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274879

C₂

yes

maximal, characteristic

#20273007

C₂

yes

maximal, characteristic

#20272575

C₂

yes

maximal

#20272575

C₂

yes

maximal

C₅₃×#1837

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

C₂×#13133151

—

C₂×#13133149

—

#16810489

C₂²

yes

—

#16810345

C₄

yes

complemented

C₅₃×#423

C₂²

yes

characteristic

C₅₃×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#341

—

C₅₃×#341

—

C₂×C₄⋊Dic₅₃

—

—

—

—

—

C₂×C₄

yes

—

#13133124

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#8795

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1846

—

#1837

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

yes

—

—

106

—

C₂×#106

—

106

—

#423

C₂×D₅₃

yes

characteristic

#423

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

—

—

—

100

#322

—

101

C₅₃

#8795

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

105

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

106

C₂×C₄⋊C₄

—

106

—

107

C₂×C₄²

C₂²×D₅₃

yes

characteristic

108

#107

—

212

—

109

#106

—

212

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

106

—

113

C₂²⋊C₈

—

106

—

114

C₄×C₈

—

115

C₄×C₈

Dic₁₀₆

yes

—

116

C₄×C₈

C₄×D₅₃

yes

—

117

#95

—

106

—

118

C₂²×C₄

—

212

—

119

C₂²×C₄

#11719949

yes

—

120

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

106

—

132

C₂×C₈

#11719919

yes

—

133

C₂×C₈

#11719919

yes

—

134

C₄⋊C₄

—

212

—

135

C₄⋊C₄

—

212

—

136

C₄²

C₂×Dic₁₀₆

yes

characteristic

137

C₄²

C₂×C₄×D₅₃

yes

characteristic

138

C₄²

#11719947

yes

—

139

C₂³

#13133206

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

424

—

142

C₂×C₄

—

424

—

143

C₂×C₄

C₂×#11719919

yes

characteristic

144

C₂×C₄

#13133175

yes

—

145

C₂×C₄

#13133204

yes

characteristic

146

C₂×C₄

#13133208

yes

—

147

C₂×C₄

#13133242

yes

characteristic

148

C₂×C₄

#13133242

yes

characteristic

149

C₂×C₄

#13133247

yes

characteristic

150

C₂×C₄

#13133247

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

212

—

156

C₂²

—

157

C₂²

#16810488

yes

characteristic

158

C₂²

#16810989

yes

characteristic

159

C₂²

#16810993

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

848

—

163

C₄

#16810666

yes

characteristic

164

C₄

#16810666

yes

characteristic

165

C₄

#16810982

yes

characteristic

166

C₄

#16810982

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#20274840

yes

—

170

C₂

#20274848

yes

characteristic

	subgroup	G/H
center	C₂²	#16810488
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1837	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#124
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1837	D₅₃
hypocenter	C₅₃	#8795
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#16810488

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1837

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
10. #16810345	C₄	semidirect
101. C₅₃	#8795	semidirect

H ≅ G/N

type

10. #16810345

C₄

semidirect

101. C₅₃

#8795

semidirect

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

subgroup	index
#20274879	2
#20273007	2
#20272575	2
C₅₃×#1837	2
#8795	53

subgroup

index

C₅₃×#1837

Subgroups — #27277796

Conjugacy classes of subgroups

Complements

Maximal subgroups