Group #28298606 — subgroups

Subgroups — #28298606

Order 13568.

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20291048	C₂	1	1	yes	maximal, characteristic
2	#20290271	C₂	1	1	yes	maximal, characteristic
3	#20273477	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133292	C₄	1	2	yes	—
5	#16811627	—	2	1	no	—
6	#16811544	—	2	1	no	—
7	C₄⋊#11719910	—	2	1	no	—
8	#16811485	—	2	1	no	—
9	#16811485	C₄	1	2	yes	—
10	#16810679	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810376	C₄	1	1	yes	characteristic
13	#16810373	—	2	1	no	—
14	C₅₃×#335	C₄	1	1	yes	characteristic
15	C₂×#11719936	—	4	1	no	—
16	#13133292	—	4	2	no	—
17	C₂×#11719910	—	2	2	no	—
18	C₄⋊#88324	—	2	2	no	—
19	C₄⋊#88324	—	4	1	no	—
20	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
22	C₄²×D₅₃	D₄	1	1	yes	characteristic
23	C₄⋊#88322	—	2	1	no	—
24	#13133129	—	4	1	no	—
25	#13133127	—	4	1	no	—
26	C₄×#88322	D₄	1	1	yes	characteristic
27	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₅₃	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	1	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
33	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
34	C₂×#88324	—	4	2	no	—
35	#11719936	—	8	1	no	—
36	#11719936	—	8	1	no	—
37	C₂×C₈×C₅₃	—	4	1	no	—
38	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
43	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
44	C₂×#88322	—	4	1	no	—
45	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₅₃	—	2	1	no	—
48	C₈×D₅₃	—	2	1	no	—
49	C₈×D₅₃	—	4	1	no	—
50	#11719910	—	4	2	no	—
51	C₂²×D₅₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
55	C₂×Dic₅₃	#102	1	1	yes	characteristic
56	C₂×Dic₅₃	#103	1	1	yes	characteristic
57	C₂×Dic₅₃	#106	1	1	yes	characteristic
58	C₄×D₅₃	—	2	1	no	—
59	C₄×D₅₃	—	2	1	no	—
60	C₄×D₅₃	—	2	1	no	—
61	C₄×D₅₃	—	2	1	no	—
62	C₄×D₅₃	—	4	1	no	—
63	#88324	—	8	2	no	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	—	8	1	no	—
67	#88322	—	2	1	no	—
68	#88322	—	8	1	no	—
69	#88322	#100	1	1	yes	characteristic
70	#88322	#101	1	1	yes	characteristic
71	#7244	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	#340	1	1	yes	characteristic
73	C₂×D₅₃	—	2	1	no	—
74	C₂×D₅₃	—	2	1	no	—
75	C₂×D₅₃	—	2	1	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	#338	1	1	yes	characteristic
78	C₄×C₅₃	#343	1	1	yes	characteristic
79	C₄×C₅₃	#360	1	1	yes	characteristic
80	C₄×C₅₃	#360	1	1	yes	characteristic
81	Dic₅₃	—	2	1	no	—
82	Dic₅₃	—	2	1	no	—
83	Dic₅₃	—	2	1	no	—
84	#2150	—	53	1	no	—
85	#1568	—	53	1	no	—
86	#1566	—	53	1	no	—
87	C₂×C₅₃	#1292	1	1	yes	characteristic
88	C₂×C₅₃	#1386	1	1	yes	characteristic
89	C₂×C₅₃	#1388	1	1	yes	characteristic
90	D₅₃	—	4	1	no	—
91	C₂×M₃₂	—	53	2	no	—
92	#432	—	106	1	no	—
93	#423	—	106	1	no	—
94	C₂×C₄×C₈	—	53	1	no	—
95	C₄⋊C₁₆	—	106	1	no	—
96	C₄⋊C₁₆	—	106	1	no	—
97	#335	—	53	1	no	—
98	#335	#6157	1	1	yes	characteristic
99	#334	—	53	2	no	—
100	#334	—	106	1	no	—
101	#334	—	106	1	no	—
102	C₅₃	#7244	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	212	1	no	—
104	C₂²×C₈	—	53	1	no	—
105	C₂²×C₈	—	53	1	no	—
106	C₂×C₄²	—	53	1	no	—
107	M₃₂	—	212	2	no	—
108	C₂×C₁₆	—	106	2	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	106	2	no	—
111	C₄⋊C₈	—	106	1	no	—
112	C₄⋊C₈	—	212	1	no	—
113	C₂²⋊C₈	—	212	1	no	—
114	#97	—	212	1	no	—
115	C₄×C₈	—	53	1	no	—
116	C₄×C₈	—	53	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	C₂×#6157	1	1	yes	characteristic
119	C₂²×C₄	—	53	1	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	53	1	no	—
122	M₁₆	—	424	1	no	—
123	M₁₆	—	424	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	53	1	no	—
127	C₂×C₈	—	106	1	no	—
128	C₂×C₈	—	106	1	no	—
129	C₂×C₈	—	212	1	no	—
130	C₂×C₈	—	212	2	no	—
131	C₂×C₈	—	212	1	no	—
132	C₂×C₈	C₄×#6157	1	1	yes	characteristic
133	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
134	C₄²	—	53	1	no	—
135	C₄²	—	53	1	no	—
136	C₄²	—	53	1	no	—
137	C₄²	C₂²⋊#6157	1	1	yes	characteristic
138	C₁₆	—	212	2	no	—
139	C₂³	—	53	1	no	—
140	C₂×C₄	—	53	1	no	—
141	C₂×C₄	—	53	1	no	—
142	C₂×C₄	—	53	1	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	106	1	no	—
145	C₂×C₄	—	106	1	no	—
146	C₂×C₄	—	106	1	no	—
147	C₂×C₄	—	212	1	no	—
148	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#13133279	1	1	yes	characteristic
150	C₂×C₄	#13133280	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	8	1	no	—
154	C₈	—	53	1	no	—
155	C₈	—	53	1	no	—
156	C₈	—	106	1	no	—
157	C₈	—	424	2	no	—
158	C₈	—	424	1	no	—
159	C₂²	—	106	1	no	—
160	C₂²	—	106	1	no	—
161	C₂²	—	106	1	no	—
162	C₂²	#16811496	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	106	1	no	—
165	C₄	—	106	1	no	—
166	C₄	—	106	1	no	—
167	C₄	#16811492	1	1	yes	characteristic
168	C₄	#16811502	1	1	yes	characteristic
169	C₄	#16811539	1	1	yes	characteristic
170	C₄	#16811540	1	1	yes	characteristic
171	C₂	—	212	1	no	—
172	C₂	#20289584	1	1	yes	characteristic
173	C₂	#20289786	1	1	yes	characteristic
174	C₂	#20289794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291048

C₂

yes

maximal, characteristic

#20290271

C₂

yes

maximal, characteristic

#20273477

C₂

yes

maximal, characteristic

C₂×#13133292

C₄

yes

—

—

—

—

—

C₄

yes

—

#16810679

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic

#16810373

—

C₅₃×#335

C₄

yes

—

—

—

—

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

—

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅₃

D₈

yes

characteristic

C₂×C₄×D₅₃

Q₁₆

yes

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

#11719910

—

C₂²×D₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

yes

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2150

—

#1568

—

#1566

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×M₃₂

—

#432

—

106

—

#423

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

C₄⋊C₁₆

—

106

—

—

yes

—

100

#334

—

106

—

101

#334

—

106

—

102

C₅₃

#7244

yes

characteristic, complemented, Sylow

103

C₂×M₁₆

—

212

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

M₃₂

—

212

—

108

C₂×C₁₆

—

106

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

106

—

111

C₄⋊C₈

—

106

—

112

C₄⋊C₈

—

212

—

113

C₂²⋊C₈

—

212

—

114

#97

—

212

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×#6157

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

M₁₆

—

424

—

123

M₁₆

—

424

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

212

—

130

C₂×C₈

—

212

—

131

C₂×C₈

—

212

—

132

C₂×C₈

C₄×#6157

yes

characteristic

133

C₂×C₈

C₄⋊#6157

yes

characteristic

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²⋊#6157

yes

characteristic

138

C₁₆

—

212

—

139

C₂³

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

212

—

148

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

149

C₂×C₄

#13133279

yes

characteristic

150

C₂×C₄

#13133280

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

—

106

—

157

C₈

—

424

—

158

C₈

—

424

—

159

C₂²

—

106

—

160

C₂²

—

106

—

161

C₂²

—

106

—

162

C₂²

#16811496

yes

characteristic

163

C₄

—

164

C₄

—

106

—

165

C₄

—

106

—

166

C₄

—

106

—

167

C₄

#16811492

yes

characteristic

168

C₄

#16811502

yes

characteristic

169

C₄

#16811539

yes

characteristic

170

C₄

#16811540

yes

characteristic

171

C₂

—

212

—

172

C₂

#20289584

yes

characteristic

173

C₂

#20289786

yes

characteristic

174

C₂

#20289794

yes

characteristic

	subgroup	G/H
center	C₂²	#16811496
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#107
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#335	#6157
hypocenter	C₅₃	#7244
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂²

#16811496

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#335

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
102. C₅₃	#7244	semidirect

H ≅ G/N

type

102. C₅₃

#7244

semidirect

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

subgroup	index
#20291048	2
#20290271	2
#20273477	2
#7244	53

subgroup

index

Subgroups — #28298606

Conjugacy classes of subgroups

Complements

Maximal subgroups