Group #28298064 — subgroups

Subgroups — #28298064

Order 13568.

292 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 102 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	#20290632	C₂	1	2	yes	maximal
2	#20274866	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88324	—	2	2	no	—
4	#16811489	—	2	2	no	—
5	#16811489	C₄	1	4	yes	—
6	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
7	C₂×#13133146	—	2	2	no	—
8	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×#88324	—	2	4	no	—
11	C₄⋊#88324	—	4	2	no	—
12	C₄⋊#88324	—	4	2	no	—
13	C₂²×#88322	—	2	1	no	—
14	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
17	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
18	C₄⋊#88322	—	4	1	no	—
19	#13133146	—	4	2	no	—
20	#13133146	—	4	2	no	—
21	C₂²×C₈×C₅₃	—	2	1	no	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	—	4	1	no	—
24	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×Dic₅₃	C₂²⋊C₄	1	2	yes	—
28	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×#88324	—	4	4	no	—
30	C₂×#88324	—	4	4	no	—
31	C₂×#88324	—	4	4	no	—
32	C₂×C₈×C₅₃	—	4	1	no	—
33	C₂×C₈×C₅₃	—	4	2	no	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	2	1	no	—
36	C₄×Dic₅₃	—	2	1	no	—
37	C₄×Dic₅₃	—	2	1	no	—
38	C₄×Dic₅₃	—	4	2	no	—
39	C₄×Dic₅₃	—	4	1	no	—
40	C₄×Dic₅₃	D₈	1	2	yes	—
41	C₄×Dic₅₃	SD₁₆	1	2	yes	—
42	C₄×Dic₅₃	SD₁₆	1	2	yes	—
43	C₄×Dic₅₃	Q₁₆	1	2	yes	—
44	C₂×#88322	—	4	1	no	—
45	C₂×#88322	—	4	2	no	—
46	C₂³×C₅₃	#95	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#102	1	2	yes	—
54	C₂×C₄×C₅₃	#103	1	2	yes	—
55	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	—	2	2	no	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	#102	1	2	yes	—
62	C₂×Dic₅₃	#103	1	2	yes	—
63	C₂×Dic₅₃	#106	1	2	yes	—
64	C₂×Dic₅₃	#107	1	2	yes	—
65	#88324	—	8	4	no	—
66	C₈×C₅₃	—	8	1	no	—
67	#88322	—	8	1	no	—
68	#7040	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
70	C₂²×C₅₃	#340	1	2	yes	—
71	C₂²×C₅₃	#341	1	1	yes	characteristic
72	C₂²×C₅₃	#343	1	1	yes	characteristic
73	C₂²×C₅₃	#344	1	2	yes	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	4	1	no	—
77	C₄×C₅₃	#331	1	1	yes	characteristic
78	C₄×C₅₃	#332	1	2	yes	—
79	C₄×C₅₃	#333	1	1	yes	characteristic
80	Dic₅₃	—	2	2	no	—
81	Dic₅₃	—	2	2	no	—
82	Dic₅₃	—	2	2	no	—
83	Dic₅₃	—	2	2	no	—
84	#1842	—	53	1	no	—
85	#1831	—	53	2	no	—
86	C₂×C₅₃	#1286	1	2	yes	—
87	C₂×C₅₃	#1290	1	1	yes	characteristic
88	C₂×C₅₃	#1296	1	1	yes	characteristic
89	C₂×C₅₃	#1301	1	2	yes	—
90	C₂×C₅₃	#1307	1	1	yes	characteristic
91	C₂×C₄⋊C₈	—	53	1	no	—
92	C₂×C₄⋊C₈	—	106	2	no	—
93	C₂×C₄⋊C₈	—	106	2	no	—
94	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
95	C₄³	—	53	1	no	—
96	#336	—	53	4	no	—
97	#336	—	106	2	no	—
98	C₅₃	#7040	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	106	4	no	—
101	C₂²×C₈	—	106	1	no	—
102	C₂×C₄²	—	53	2	no	—
103	C₂×C₄²	—	53	1	no	—
104	C₂×C₄²	—	53	2	no	—
105	C₂×C₄²	—	53	1	no	—
106	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	212	1	no	—
109	C₄⋊C₈	—	212	2	no	—
110	C₄⋊C₈	—	212	2	no	—
111	C₄⋊C₈	—	212	2	no	—
112	C₄⋊C₈	—	212	2	no	—
113	C₂²×C₄	—	53	2	no	—
114	C₂²×C₄	—	53	2	no	—
115	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
116	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	212	4	no	—
121	C₂×C₈	—	212	4	no	—
122	C₂×C₈	—	212	4	no	—
123	C₂×C₈	—	212	1	no	—
124	C₂×C₈	—	212	2	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	53	2	no	—
128	C₄²	—	53	2	no	—
129	C₄²	—	53	2	no	—
130	C₄²	—	53	2	no	—
131	C₄²	—	106	1	no	—
132	C₄²	—	106	1	no	—
133	C₄²	—	212	1	no	—
134	C₄²	—	212	2	no	—
135	C₂³	(C₂×#6157)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	53	2	no	—
141	C₂×C₄	—	53	2	no	—
142	C₂×C₄	—	53	2	no	—
143	C₂×C₄	—	53	2	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	—	106	2	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	2	no	—
148	C₂×C₄	#13133277	1	1	yes	characteristic
149	C₂×C₄	#13133278	1	1	yes	characteristic
150	C₂×C₄	#13133279	1	2	yes	—
151	C₂×C₄	#13133280	1	2	yes	—
152	C₂×C₄	#13133287	1	1	yes	characteristic
153	C₂×C₄	#13133288	1	1	yes	characteristic
154	C₈	—	8	1	no	—
155	C₈	—	424	4	no	—
156	C₈	—	424	1	no	—
157	C₂²	#16811491	1	1	yes	characteristic
158	C₂²	#16811496	1	2	yes	—
159	C₂²	#16811499	1	1	yes	characteristic
160	C₂²	#16811503	1	1	yes	characteristic
161	C₂²	#16811506	1	2	yes	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	106	2	no	—
166	C₄	—	106	2	no	—
167	C₄	—	106	2	no	—
168	C₄	—	106	2	no	—
169	C₄	#16811481	1	1	yes	characteristic
170	C₄	#16811482	1	2	yes	—
171	C₄	#16811483	1	1	yes	characteristic
172	C₂	#20289570	1	2	yes	—
173	C₂	#20289580	1	1	yes	characteristic
174	C₂	#20289593	1	1	yes	characteristic
175	C₂	#20289607	1	2	yes	—
176	C₂	#20289621	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290632

C₂

yes

maximal

#20274866

C₂

yes

maximal, characteristic

C₂×C₄⋊#88324

—

#16811489

—

#16811489

C₄

yes

—

C₂×C₄⋊#88322

C₄

yes

characteristic

C₂×#13133146

—

C₄²×Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂²×#88324

—

C₄⋊#88324

—

C₄⋊#88324

—

C₂²×#88322

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

—

C₂×C₄×Dic₅₃

Q₈

yes

—

—

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₅₃

C₂²⋊C₄

yes

—

C₂²×Dic₅₃

C₄⋊C₄

yes

—

C₂×#88324

—

C₂×#88324

—

C₂×#88324

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₈

yes

—

C₄×Dic₅₃

SD₁₆

yes

—

C₄×Dic₅₃

SD₁₆

yes

—

C₄×Dic₅₃

Q₁₆

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#95

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#103

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#102

yes

—

C₂×Dic₅₃

#103

yes

—

C₂×Dic₅₃

#106

yes

—

C₂×Dic₅₃

#107

yes

—

#88324

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#331

yes

characteristic

C₄×C₅₃

#332

yes

—

C₄×C₅₃

#333

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1842

—

#1831

—

C₂×C₅₃

#1286

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1307

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄⋊C₈

#6157

yes

characteristic

C₄³

—

#336

—

#336

—

106

—

C₅₃

#7040

yes

characteristic, complemented, Sylow

C₂²×C₈

—

100

C₂²×C₈

—

106

—

101

C₂²×C₈

—

106

—

102

C₂×C₄²

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×#6157

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

212

—

109

C₄⋊C₈

—

212

—

110

C₄⋊C₈

—

212

—

111

C₄⋊C₈

—

212

—

112

C₄⋊C₈

—

212

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₄×#6157

yes

characteristic

116

C₂²×C₄

C₄⋊#6157

yes

characteristic

117

C₂²×C₄

C₂²⋊#6157

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

212

—

121

C₂×C₈

—

212

—

122

C₂×C₈

—

212

—

123

C₂×C₈

—

212

—

124

C₂×C₈

—

212

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

106

—

132

C₄²

—

106

—

133

C₄²

—

212

—

134

C₄²

—

212

—

135

C₂³

(C₂×#6157)⋊C₄

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

#13133277

yes

characteristic

149

C₂×C₄

#13133278

yes

characteristic

150

C₂×C₄

#13133279

yes

—

151

C₂×C₄

#13133280

yes

—

152

C₂×C₄

#13133287

yes

characteristic

153

C₂×C₄

#13133288

yes

characteristic

154

C₈

—

155

C₈

—

424

—

156

C₈

—

424

—

157

C₂²

#16811491

yes

characteristic

158

C₂²

#16811496

yes

—

159

C₂²

#16811499

yes

characteristic

160

C₂²

#16811503

yes

characteristic

161

C₂²

#16811506

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

106

—

166

C₄

—

106

—

167

C₄

—

106

—

168

C₄

—

106

—

169

C₄

#16811481

yes

characteristic

170

C₄

#16811482

yes

—

171

C₄

#16811483

yes

characteristic

172

C₂

#20289570

yes

—

173

C₂

#20289580

yes

characteristic

174

C₂

#20289593

yes

characteristic

175

C₂

#20289607

yes

—

176

C₂

#20289621

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×#6157)⋊C₄
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄²×Dic₅₃	C₂²
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	#7040
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

(C₂×#6157)⋊C₄

commutator

C₂²×C₄×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×#6157

Fitting

C₂×C₄⋊C₈×C₅₃

C₄

socle, prime radical

C₂³×C₅₃

#95

cosocle

C₄²×Dic₅₃

C₂²

C₂×C₄⋊C₈

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
98. C₅₃	#7040	semidirect

H ≅ G/N

type

98. C₅₃

#7040

semidirect

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

subgroup	index
#20290632	2
#20274866	2
#7040	53

subgroup

index

#20290632

#20274866

#7040

Subgroups — #28298064

Conjugacy classes of subgroups

Complements

Maximal subgroups