Group #28298380 — subgroups

Subgroups — #28298380

Order 13568.

246 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₄×#13133292	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133287	C₂	1	1	yes	maximal, characteristic
3	#335×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133292	—	2	1	no	—
5	C₂×#13133287	C₄	1	2	yes	—
6	C₄×#11719936	—	2	1	no	—
7	#16811512	C₄	1	2	yes	—
8	C₄×#11719910	—	2	1	no	—
9	#16811499	—	2	1	no	—
10	C₄⋊C₈×D₅₃	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810376	C₄	1	1	yes	characteristic
13	#16810375	—	2	1	no	—
14	C₅₃×#335	C₄	1	1	yes	characteristic
15	C₂×#11719936	—	2	2	no	—
16	#13133292	—	2	4	no	—
17	C₂×#11719910	—	2	2	no	—
18	#13133287	—	2	2	no	—
19	#13133287	C₈	1	4	yes	complemented
20	#13133272	—	4	1	no	—
21	C₄×#88324	—	4	1	no	—
22	C₂×C₈×D₅₃	—	4	1	no	—
23	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
25	C₄²×D₅₃	D₄	1	1	yes	characteristic
26	C₄⋊#88322	—	2	1	no	—
27	#13133146	—	4	1	no	—
28	C₄×#88322	D₄	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
36	C₂×#88324	—	4	2	no	—
37	#11719936	—	2	4	no	—
38	#11719936	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	1	no	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	D₈	1	1	yes	characteristic
45	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×#88322	—	4	1	no	—
47	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
49	C₈×D₅₃	—	2	2	no	—
50	C₈×D₅₃	—	8	2	no	—
51	C₈×D₅₃	C₂×C₈	1	2	yes	—
52	C₈×D₅₃	M₁₆	1	2	yes	—
53	#11719910	—	2	4	no	—
54	C₂²×D₅₃	#103	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
58	C₂×Dic₅₃	#102	1	1	yes	characteristic
59	C₂×Dic₅₃	#106	1	1	yes	characteristic
60	C₂×Dic₅₃	#107	1	1	yes	characteristic
61	C₄×D₅₃	—	2	1	no	—
62	C₄×D₅₃	—	2	1	no	—
63	C₄×D₅₃	—	4	1	no	—
64	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
65	C₄×D₅₃	#104	1	2	yes	—
66	#88324	—	4	4	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	—	8	1	no	—
69	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
70	C₈×C₅₃	#97	1	1	yes	characteristic
71	#88322	—	2	1	no	—
72	#88322	—	8	1	no	—
73	#88322	C₄⋊C₈	1	2	yes	—
74	#7176	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	#340	1	1	yes	characteristic
76	C₂×D₅₃	—	2	1	no	—
77	C₂×D₅₃	—	2	1	no	—
78	C₂×D₅₃	Q₈⋊C₈	1	2	yes	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	#336	1	1	yes	characteristic
81	C₄×C₅₃	#339	1	1	yes	characteristic
82	C₄×C₅₃	#361	1	1	yes	characteristic
83	C₄×C₅₃	#362	1	1	yes	characteristic
84	Dic₅₃	—	2	1	no	—
85	Dic₅₃	—	2	1	no	—
86	Dic₅₃	D₄⋊C₈	1	2	yes	—
87	C₄×M₃₂	—	53	1	no	—
88	C₄×#108	—	53	1	no	—
89	C₂×#335	—	53	1	no	—
90	C₂×C₅₃	#1279	1	1	yes	characteristic
91	C₂×C₅₃	#1387	1	1	yes	characteristic
92	C₂×C₅₃	#1388	1	1	yes	characteristic
93	D₅₃	—	2	2	no	—
94	C₂×M₃₂	—	106	1	no	—
95	C₂×#108	—	53	2	no	—
96	C₂×C₄⋊C₈	—	106	1	no	—
97	C₄×M₁₆	—	106	1	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	#346	—	53	2	no	—
100	C₄×C₁₆	—	106	1	no	—
101	#341	—	106	1	no	—
102	#335	—	53	1	no	—
103	#335	—	106	1	no	—
104	#335	#6157	1	1	yes	characteristic
105	C₅₃	#7176	1	1	yes	characteristic, complemented, Sylow
106	C₂×M₁₆	—	106	2	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂²×C₈	—	212	1	no	—
110	C₂×C₄²	—	53	1	no	—
111	M₃₂	—	106	4	no	—
112	C₂×C₁₆	—	106	2	no	—
113	#108	—	53	4	no	—
114	#108	—	106	2	no	—
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	106	1	no	—
117	C₄⋊C₈	—	212	1	no	—
118	#97	—	212	1	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₄×C₈	—	53	1	no	—
122	C₄×C₈	—	212	1	no	—
123	C₄×C₈	C₂×#6157	1	1	yes	characteristic
124	C₂²×C₄	—	53	1	no	—
125	C₂²×C₄	—	53	1	no	—
126	C₂²×C₄	—	53	1	no	—
127	M₁₆	—	106	4	no	—
128	M₁₆	—	212	2	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	53	1	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	53	2	no	—
133	C₂×C₈	—	53	1	no	—
134	C₂×C₈	—	106	2	no	—
135	C₂×C₈	—	424	2	no	—
136	C₂×C₈	—	212	1	no	—
137	C₂×C₈	—	212	2	no	—
138	C₂×C₈	C₄×#6157	1	1	yes	characteristic
139	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
140	C₄²	—	53	1	no	—
141	C₄²	—	53	1	no	—
142	C₄²	—	53	1	no	—
143	C₄²	C₂²⋊#6157	1	1	yes	characteristic
144	C₁₆	—	106	4	no	—
145	C₂³	—	53	1	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	53	2	no	—
148	C₂×C₄	—	53	2	no	—
149	C₂×C₄	—	53	1	no	—
150	C₂×C₄	—	53	1	no	—
151	C₂×C₄	—	106	1	no	—
152	C₂×C₄	—	106	1	no	—
153	C₂×C₄	—	212	1	no	—
154	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#13133279	1	1	yes	characteristic
156	C₂×C₄	#13133280	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	8	1	no	—
159	C₈	—	53	2	no	—
160	C₈	—	106	1	no	—
161	C₈	—	424	1	no	—
162	C₈	—	212	4	no	—
163	C₈	C₈×#6157	1	1	yes	characteristic
164	C₈	#13133271	1	1	yes	characteristic
165	C₂²	—	53	2	no	—
166	C₂²	—	106	1	no	—
167	C₂²	—	106	1	no	—
168	C₂²	#16811496	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	53	2	no	—
171	C₄	—	106	1	no	—
172	C₄	—	106	1	no	—
173	C₄	(C₂×#6157)⋊C₈	1	1	yes	characteristic
174	C₄	#16811495	1	1	yes	characteristic
175	C₄	#16811541	1	1	yes	characteristic
176	C₄	#16811542	1	1	yes	characteristic
177	C₂	—	106	2	no	—
178	C₂	#20289549	1	1	yes	characteristic
179	C₂	#20289789	1	1	yes	characteristic
180	C₂	#20289792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133292

C₂

yes

maximal, characteristic

C₄×#13133287

C₂

yes

maximal, characteristic

#335×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133292

—

C₂×#13133287

C₄

yes

—

C₄×#11719936

—

#16811512

C₄

yes

—

C₄×#11719910

—

#16811499

—

C₄⋊C₈×D₅₃

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810376

C₄

yes

characteristic

#16810375

—

C₅₃×#335

C₄

yes

—

—

—

—

C₈

yes

complemented

#13133272

—

C₄×#88324

—

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

D₄

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

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

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

D₈

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

C₂×C₈

yes

—

C₈×D₅₃

M₁₆

yes

—

#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

C₂²⋊C₈

yes

—

C₄×D₅₃

#104

yes

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

C₈×C₅₃

yes

—

—

C₄⋊C₈

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

Q₈⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

D₄⋊C₈

yes

—

C₄×M₃₂

—

C₄×#108

—

C₂×#335

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×M₃₂

—

106

—

C₂×#108

—

C₂×C₄⋊C₈

—

106

—

C₄×M₁₆

—

106

—

C₂×C₄×C₈

—

#346

—

100

C₄×C₁₆

—

106

—

101

#341

—

106

—

102

#335

—

103

#335

—

106

—

104

#335

#6157

yes

characteristic

105

C₅₃

#7176

yes

characteristic, complemented, Sylow

106

C₂×M₁₆

—

106

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

—

212

—

110

C₂×C₄²

—

111

M₃₂

—

106

—

112

C₂×C₁₆

—

106

—

113

#108

—

114

#108

—

106

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

106

—

117

C₄⋊C₈

—

212

—

118

#97

—

212

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

212

—

123

C₄×C₈

C₂×#6157

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

M₁₆

—

106

—

128

M₁₆

—

212

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

106

—

135

C₂×C₈

—

424

—

136

C₂×C₈

—

212

—

137

C₂×C₈

—

212

—

138

C₂×C₈

C₄×#6157

yes

characteristic

139

C₂×C₈

C₄⋊#6157

yes

characteristic

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

C₂²⋊#6157

yes

characteristic

144

C₁₆

—

106

—

145

C₂³

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

106

—

152

C₂×C₄

—

106

—

153

C₂×C₄

—

212

—

154

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

155

C₂×C₄

#13133279

yes

characteristic

156

C₂×C₄

#13133280

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

106

—

161

C₈

—

424

—

162

C₈

—

212

—

163

C₈

C₈×#6157

yes

characteristic

164

C₈

#13133271

yes

characteristic

165

C₂²

—

166

C₂²

—

106

—

167

C₂²

—

106

—

168

C₂²

#16811496

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

106

—

172

C₄

—

106

—

173

C₄

(C₂×#6157)⋊C₈

yes

characteristic

174

C₄

#16811495

yes

characteristic

175

C₄

#16811541

yes

characteristic

176

C₄

#16811542

yes

characteristic

177

C₂

—

106

—

178

C₂

#20289549

yes

characteristic

179

C₂

#20289789

yes

characteristic

180

C₂

#20289792

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133279
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#103
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#335	#6157
hypocenter	C₅₃	#7176
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂×C₄

#13133279

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#335

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
19. #13133287	C₈	semidirect
105. C₅₃	#7176	semidirect

H ≅ G/N

type

19. #13133287

C₈

semidirect

105. C₅₃

#7176

semidirect

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

subgroup	index
C₄×#13133292	2
C₄×#13133287	2
#335×D₅₃	2
#7176	53

subgroup

index

C₄×#13133292

C₄×#13133287

#335×D₅₃

#7176

Subgroups — #28298380

Conjugacy classes of subgroups

Complements

Maximal subgroups