Group #28298784 — subgroups

Subgroups — #28298784

Order 13568.

230 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 62 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	#20291078	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133286	C₂	1	1	yes	maximal, characteristic
3	#20273475	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133286	—	2	1	no	—
5	C₄×C₄⋊#6157	—	2	1	no	—
6	#16811544	C₄	1	2	yes	complemented
7	#16811518	—	2	1	no	—
8	#16811511	—	2	1	no	—
9	#16811497	C₄	1	2	yes	—
10	#16810679	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810375	—	2	1	no	—
13	#16810374	C₄	1	1	yes	characteristic, complemented
14	C₅₃×#334	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#6157	—	2	2	no	—
16	C₂×C₄×#6157	—	4	1	no	—
17	C₂×#11719910	—	2	2	no	—
18	#13133286	—	2	4	no	—
19	(C₂×#6157)⋊C₄	—	4	1	no	—
20	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₅₃	D₄	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	Q₈	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	1	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×#6157	—	4	2	no	—
32	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
34	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
35	C₄⋊#6157	—	2	4	no	—
36	C₄⋊#6157	—	4	2	no	—
37	C₄×#6157	—	8	2	no	—
38	C₂×C₈×C₅₃	—	4	1	no	—
39	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
44	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
45	C₂×#88322	—	4	1	no	—
46	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
48	C₈×D₅₃	—	2	2	no	—
49	C₈×D₅₃	—	4	1	no	—
50	#11719910	—	4	2	no	—
51	C₂²×D₅₃	#102	1	1	yes	characteristic
52	C₂×#6157	—	4	4	no	—
53	C₂×#6157	—	8	2	no	—
54	C₂×#6157	—	8	2	no	—
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	#103	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	—	2	1	no	—
64	C₄×D₅₃	—	2	1	no	—
65	C₄×D₅₃	—	4	1	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	—	8	1	no	—
69	#88322	—	2	1	no	—
70	#88322	—	8	1	no	—
71	#88322	#104	1	2	yes	—
72	#7301	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	#340	1	1	yes	characteristic
74	C₂×D₅₃	—	2	1	no	—
75	C₂×D₅₃	—	2	1	no	—
76	C₂×D₅₃	#361	1	1	yes	characteristic
77	C₂×D₅₃	#362	1	1	yes	characteristic
78	#6157	—	8	4	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	#339	1	1	yes	characteristic
82	C₄×C₅₃	#344	1	1	yes	characteristic
83	Dic₅₃	—	2	1	no	—
84	Dic₅₃	—	2	1	no	—
85	Dic₅₃	—	2	1	no	—
86	#2163	—	53	1	no	—
87	C₄×#107	—	53	1	no	—
88	#1568	—	53	1	no	—
89	C₂×C₅₃	#1303	1	1	yes	characteristic
90	C₂×C₅₃	#1384	1	1	yes	characteristic
91	C₂×C₅₃	#1387	1	1	yes	characteristic
92	D₅₃	—	2	1	no	—
93	D₅₃	—	2	1	no	—
94	#432	—	106	1	no	—
95	C₂×#107	—	106	1	no	—
96	C₂×C₄×C₈	—	53	1	no	—
97	C₄×C₄⋊C₄	—	106	1	no	—
98	C₄⋊C₁₆	—	53	2	no	—
99	C₂²⋊C₁₆	—	106	1	no	—
100	#346	—	106	1	no	—
101	#340	—	53	2	no	—
102	#335	—	106	1	no	—
103	#334	—	53	1	no	—
104	#334	#6157	1	1	yes	characteristic, complemented
105	C₅₃	#7301	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	53	1	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂×C₄⋊C₄	—	106	2	no	—
109	C₂×C₄²	—	53	1	no	—
110	C₂×C₄²	—	212	1	no	—
111	C₂×C₁₆	—	106	2	no	—
112	#107	—	106	4	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	106	1	no	—
115	C₂²⋊C₈	—	212	1	no	—
116	#97	—	212	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	C₂×#6157	1	1	yes	characteristic
121	#95	—	212	1	no	—
122	C₂²×C₄	—	53	1	no	—
123	C₂²×C₄	—	53	1	no	—
124	C₂²×C₄	—	53	1	no	—
125	C₂²×C₄	—	212	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	53	1	no	—
128	C₂×C₈	—	53	1	no	—
129	C₂×C₈	—	106	2	no	—
130	C₂×C₈	—	212	1	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	C₄⋊C₄	—	106	4	no	—
135	C₄⋊C₄	—	212	2	no	—
136	C₄²	—	53	1	no	—
137	C₄²	—	53	1	no	—
138	C₄²	—	53	1	no	—
139	C₄²	—	424	2	no	—
140	C₄²	C₂²⋊#6157	1	1	yes	characteristic
141	C₁₆	—	212	2	no	—
142	C₂³	—	53	1	no	—
143	C₂×C₄	—	53	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	106	1	no	—
147	C₂×C₄	—	106	1	no	—
148	C₂×C₄	—	106	1	no	—
149	C₂×C₄	—	106	1	no	—
150	C₂×C₄	—	424	2	no	—
151	C₂×C₄	—	424	2	no	—
152	C₂×C₄	—	212	1	no	—
153	C₂×C₄	—	212	4	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	—	2	1	no	—
159	C₈	—	8	1	no	—
160	C₈	—	53	2	no	—
161	C₈	—	106	1	no	—
162	C₈	—	424	1	no	—
163	C₂²	—	53	1	no	—
164	C₂²	—	53	1	no	—
165	C₂²	—	106	1	no	—
166	C₂²	—	106	1	no	—
167	C₂²	#16811496	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	106	1	no	—
171	C₄	—	106	1	no	—
172	C₄	—	106	1	no	—
173	C₄	—	424	4	no	—
174	C₄	#16811495	1	1	yes	characteristic
175	C₄	#16811505	1	1	yes	characteristic
176	C₂	—	106	1	no	—
177	C₂	—	106	1	no	—
178	C₂	#20289611	1	1	yes	characteristic
179	C₂	#20289782	1	1	yes	characteristic
180	C₂	#20289791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291078

C₂

yes

maximal, characteristic

C₄×#13133286

C₂

yes

maximal, characteristic

#20273475

C₂

yes

maximal, characteristic

C₂×#13133286

—

C₄×C₄⋊#6157

—

#16811544

C₄

yes

—

—

C₄

yes

—

#16810679

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810375

—

#16810374

C₄

yes

characteristic, complemented

C₅₃×#334

C₄

yes

characteristic, complemented

C₂×C₄⋊#6157

—

C₂×C₄×#6157

—

C₂×#11719910

—

#13133286

—

(C₂×#6157)⋊C₄

—

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

C₂²×#6157

—

C₂×C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₅₃

D₈

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

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

SD₁₆

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

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

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2163

—

C₄×#107

—

#1568

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

#432

—

106

—

C₂×#107

—

106

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

106

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

100

#346

—

106

—

101

#340

—

102

#335

—

106

—

103

#334

—

104

#334

#6157

yes

characteristic, complemented

105

C₅₃

#7301

yes

characteristic, complemented, Sylow

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄⋊C₄

—

106

—

109

C₂×C₄²

—

110

C₂×C₄²

—

212

—

111

C₂×C₁₆

—

106

—

112

#107

—

106

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

106

—

115

C₂²⋊C₈

—

212

—

116

#97

—

212

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₂×#6157

yes

characteristic

121

#95

—

212

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

212

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

106

—

130

C₂×C₈

—

212

—

131

C₂×C₈

—

212

—

132

C₂×C₈

C₄×#6157

yes

characteristic

133

C₂×C₈

C₄⋊#6157

yes

characteristic

134

C₄⋊C₄

—

106

—

135

C₄⋊C₄

—

212

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

424

—

140

C₄²

C₂²⋊#6157

yes

characteristic

141

C₁₆

—

212

—

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

424

—

151

C₂×C₄

—

424

—

152

C₂×C₄

—

212

—

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

—

161

C₈

—

106

—

162

C₈

—

424

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

106

—

166

C₂²

—

106

—

167

C₂²

#16811496

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

106

—

171

C₄

—

106

—

172

C₄

—

106

—

173

C₄

—

424

—

174

C₄

#16811495

yes

characteristic

175

C₄

#16811505

yes

characteristic

176

C₂

—

106

—

177

C₂

—

106

—

178

C₂

#20289611

yes

characteristic

179

C₂

#20289782

yes

characteristic

180

C₂

#20289791

yes

characteristic

	subgroup	G/H
center	C₂²	#16811496
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#334	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂²×D₅₃	#102
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#334	#6157
hypocenter	C₅₃	#7301
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#16811496

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×#334

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄×C₈×D₅₃

C₂²

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
6. #16811544	C₄	semidirect
13. #16810374	C₄	semidirect
14. C₅₃×#334	C₄	semidirect
104. #334	#6157	semidirect
105. C₅₃	#7301	semidirect

H ≅ G/N

type

6. #16811544

C₄

semidirect

13. #16810374

C₄

semidirect

14. C₅₃×#334

C₄

semidirect

104. #334

#6157

semidirect

105. C₅₃

#7301

semidirect

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

subgroup	index
#20291078	2
C₄×#13133286	2
#20273475	2
#7301	53

subgroup

index

Subgroups — #28298784

Conjugacy classes of subgroups

Complements

Maximal subgroups