Group #28298750 — subgroups

Subgroups — #28298750

Order 13568.

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

normal subgroups only

This group has 184 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	#20291075	C₂	1	1	yes	maximal, characteristic
2	#20290276	C₂	1	1	yes	maximal, characteristic
3	#20273460	C₂	1	1	yes	maximal, characteristic
4	#16811645	—	2	1	no	—
5	C₄⋊#11719910	—	2	1	no	—
6	#16811518	C₄	1	2	yes	—
7	#16811509	—	2	1	no	—
8	#16811486	C₄	1	2	yes	—
9	#16811484	—	2	1	no	—
10	#16810675	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810375	—	2	1	no	—
13	#16810374	C₄	1	1	yes	characteristic
14	C₅₃×#334	C₄	1	1	yes	characteristic
15	C₂×#11719910	—	2	2	no	—
16	C₄⋊#88324	—	2	2	no	—
17	#13133274	—	4	1	no	—
18	#13133272	—	4	1	no	—
19	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
21	C₄²×D₅₃	D₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	1	no	—
23	#13133129	—	4	1	no	—
24	C₄×#88322	D₄	1	1	yes	characteristic
25	C₈×Dic₅₃	—	4	1	no	—
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
32	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
33	C₂×#88324	—	4	2	no	—
34	C₂×C₈×C₅₃	—	4	1	no	—
35	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×Dic₅₃	D₈	1	1	yes	characteristic
40	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
41	C₂×#88322	—	4	1	no	—
42	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
44	C₈×D₅₃	—	2	1	no	—
45	C₈×D₅₃	—	2	1	no	—
46	C₈×D₅₃	—	4	1	no	—
47	#11719910	—	4	2	no	—
48	C₂²×D₅₃	#106	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
52	C₂×Dic₅₃	#102	1	1	yes	characteristic
53	C₂×Dic₅₃	#103	1	1	yes	characteristic
54	C₂×Dic₅₃	#107	1	1	yes	characteristic
55	C₄×D₅₃	—	2	1	no	—
56	C₄×D₅₃	—	2	1	no	—
57	C₄×D₅₃	—	2	2	no	—
58	C₄×D₅₃	—	4	1	no	—
59	#88324	—	8	2	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	—	8	1	no	—
62	C₈×C₅₃	#108	1	1	yes	characteristic
63	C₈×C₅₃	#108	1	1	yes	characteristic
64	#88322	—	2	1	no	—
65	#88322	—	2	1	no	—
66	#88322	—	8	1	no	—
67	#7289	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	#340	1	1	yes	characteristic
69	C₂×D₅₃	—	2	1	no	—
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	#341	1	1	yes	characteristic
75	C₄×C₅₃	#344	1	1	yes	characteristic
76	Dic₅₃	—	2	1	no	—
77	Dic₅₃	—	2	1	no	—
78	Dic₅₃	#359	1	2	yes	—
79	#2162	—	53	1	no	—
80	#1568	—	53	1	no	—
81	#1564	—	53	1	no	—
82	C₂×C₅₃	#1301	1	1	yes	characteristic
83	C₂×C₅₃	#1382	1	1	yes	characteristic
84	C₂×C₅₃	#1389	1	1	yes	characteristic
85	D₅₃	—	4	1	no	—
86	#432	—	106	1	no	—
87	#431	—	106	1	no	—
88	C₂×C₄×C₈	—	53	1	no	—
89	C₄⋊C₁₆	—	106	1	no	—
90	C₂²⋊C₁₆	—	53	2	no	—
91	C₄×C₁₆	—	106	1	no	—
92	#335	—	53	2	no	—
93	#335	—	106	1	no	—
94	#334	—	53	1	no	—
95	#334	—	106	1	no	—
96	#334	#6157	1	1	yes	characteristic
97	C₅₃	#7289	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	53	1	no	—
99	C₂²×C₈	—	53	1	no	—
100	C₂×C₄²	—	53	1	no	—
101	C₂×C₁₆	—	106	2	no	—
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	—	106	2	no	—
104	C₄⋊C₈	—	106	1	no	—
105	C₂²⋊C₈	—	212	1	no	—
106	C₂²⋊C₈	—	212	1	no	—
107	#97	—	212	1	no	—
108	C₄×C₈	—	53	1	no	—
109	C₄×C₈	—	53	1	no	—
110	C₄×C₈	—	53	1	no	—
111	C₄×C₈	—	212	1	no	—
112	C₄×C₈	C₂×#6157	1	1	yes	characteristic
113	C₂²×C₄	—	53	1	no	—
114	C₂²×C₄	—	53	1	no	—
115	C₂²×C₄	—	53	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	53	1	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	106	1	no	—
120	C₂×C₈	—	106	1	no	—
121	C₂×C₈	—	212	1	no	—
122	C₂×C₈	—	212	2	no	—
123	C₂×C₈	—	212	1	no	—
124	C₂×C₈	C₄×#6157	1	1	yes	characteristic
125	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
126	C₄²	—	53	1	no	—
127	C₄²	—	53	1	no	—
128	C₄²	—	53	1	no	—
129	C₄²	C₂²⋊#6157	1	1	yes	characteristic
130	C₁₆	—	212	2	no	—
131	C₂³	—	53	1	no	—
132	C₂×C₄	—	53	1	no	—
133	C₂×C₄	—	53	1	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	106	2	no	—
136	C₂×C₄	—	106	1	no	—
137	C₂×C₄	—	106	1	no	—
138	C₂×C₄	—	212	1	no	—
139	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	#13133279	1	1	yes	characteristic
141	C₂×C₄	#13133280	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	8	1	no	—
144	C₈	—	106	1	no	—
145	C₈	—	106	1	no	—
146	C₈	—	424	1	no	—
147	C₈	—	424	2	no	—
148	C₈	#13133287	1	1	yes	characteristic
149	C₈	#13133288	1	1	yes	characteristic
150	C₂²	—	106	1	no	—
151	C₂²	—	106	1	no	—
152	C₂²	—	106	1	no	—
153	C₂²	#16811496	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	53	2	no	—
157	C₄	—	106	1	no	—
158	C₄	—	106	1	no	—
159	C₄	#16811499	1	1	yes	characteristic
160	C₄	#16811506	1	1	yes	characteristic
161	C₂	—	212	1	no	—
162	C₂	#20289607	1	1	yes	characteristic
163	C₂	#20289778	1	1	yes	characteristic
164	C₂	#20289797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291075

C₂

yes

maximal, characteristic

#20290276

C₂

yes

maximal, characteristic

#20273460

C₂

yes

maximal, characteristic

#16811645

—

C₄⋊#11719910

—

#16811518

C₄

yes

—

#16811509

—

#16811486

C₄

yes

—

#16811484

—

#16810675

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810375

—

#16810374

C₄

yes

characteristic

C₅₃×#334

C₄

yes

—

—

—

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₅₃

—

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

SD₁₆

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

characteristic

C₂×#88324

—

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

D₈

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

—

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

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#359

yes

—

#2162

—

#1568

—

#1564

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#432

—

106

—

#431

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

106

—

#335

—

#335

—

106

—

#334

—

#334

—

106

—

yes

C₅₃

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

—

101

C₂×C₁₆

—

106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

106

—

104

C₄⋊C₈

—

106

—

105

C₂²⋊C₈

—

212

—

106

C₂²⋊C₈

—

212

—

107

#97

—

212

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

212

—

112

C₄×C₈

C₂×#6157

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

106

—

121

C₂×C₈

—

212

—

122

C₂×C₈

—

212

—

123

C₂×C₈

—

212

—

124

C₂×C₈

C₄×#6157

yes

characteristic

125

C₂×C₈

C₄⋊#6157

yes

characteristic

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

C₂²⋊#6157

yes

characteristic

130

C₁₆

—

212

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

212

—

139

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

140

C₂×C₄

#13133279

yes

characteristic

141

C₂×C₄

#13133280

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

106

—

145

C₈

—

106

—

146

C₈

—

424

—

147

C₈

—

424

—

148

C₈

#13133287

yes

characteristic

149

C₈

#13133288

yes

characteristic

150

C₂²

—

106

—

151

C₂²

—

106

—

152

C₂²

—

106

—

153

C₂²

#16811496

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

106

—

158

C₄

—

106

—

159

C₄

#16811499

yes

characteristic

160

C₄

#16811506

yes

characteristic

161

C₂

—

212

—

162

C₂

#20289607

yes

characteristic

163

C₂

#20289778

yes

characteristic

164

C₂

#20289797

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	#106
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#334	#6157
hypocenter	C₅₃	#7289
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
97. C₅₃	#7289	semidirect

H ≅ G/N

type

97. C₅₃

#7289

semidirect

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

subgroup	index
#20291075	2
#20290276	2
#20273460	2
#7289	53

subgroup

index

Subgroups — #28298750

Conjugacy classes of subgroups

Complements

Maximal subgroups