Group #28298772 — subgroups

Subgroups — #28298772

Order 13568.

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

normal subgroups only

This group has 208 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	#20291074	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133288	C₂	1	1	yes	maximal, characteristic
3	#20273477	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133288	—	2	1	no	—
5	C₄×#11719936	—	2	1	no	—
6	#16811544	C₄	1	2	yes	—
7	#16811518	—	2	1	no	—
8	C₄×#11719910	—	2	1	no	—
9	#16811499	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	—	2	2	no	—
16	C₂×#11719910	—	2	2	no	—
17	#13133288	—	2	4	no	—
18	#13133272	—	4	1	no	—
19	C₄×#88324	—	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	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
34	C₂×#88324	—	4	2	no	—
35	#11719936	—	2	4	no	—
36	#11719936	—	4	2	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	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	D₈	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	2	no	—
48	C₈×D₅₃	—	4	1	no	—
49	#11719910	—	4	2	no	—
50	C₂²×D₅₃	#103	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
54	C₂×Dic₅₃	#102	1	1	yes	characteristic
55	C₂×Dic₅₃	#106	1	1	yes	characteristic
56	C₂×Dic₅₃	#107	1	1	yes	characteristic
57	C₄×D₅₃	—	2	1	no	—
58	C₄×D₅₃	—	2	1	no	—
59	C₄×D₅₃	—	2	1	no	—
60	C₄×D₅₃	—	2	1	no	—
61	C₄×D₅₃	—	4	1	no	—
62	#88324	—	4	4	no	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	8	1	no	—
66	#88322	—	2	1	no	—
67	#88322	—	8	1	no	—
68	#88322	#104	1	2	yes	—
69	#7297	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	#340	1	1	yes	characteristic
71	C₂×D₅₃	—	2	1	no	—
72	C₂×D₅₃	—	2	1	no	—
73	C₂×D₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#339	1	1	yes	characteristic
76	C₄×C₅₃	#344	1	1	yes	characteristic
77	C₄×C₅₃	#359	1	1	yes	characteristic
78	C₄×C₅₃	#359	1	1	yes	characteristic
79	Dic₅₃	—	2	1	no	—
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	—	2	1	no	—
82	#2162	—	53	1	no	—
83	C₄×#108	—	53	1	no	—
84	#1568	—	53	1	no	—
85	C₂×C₅₃	#1303	1	1	yes	characteristic
86	C₂×C₅₃	#1382	1	1	yes	characteristic
87	C₂×C₅₃	#1388	1	1	yes	characteristic
88	D₅₃	—	4	1	no	—
89	#432	—	106	1	no	—
90	C₂×#108	—	106	1	no	—
91	C₄×M₁₆	—	106	1	no	—
92	C₂×C₄×C₈	—	53	1	no	—
93	C₄⋊C₁₆	—	53	2	no	—
94	C₂²⋊C₁₆	—	106	1	no	—
95	C₄×C₁₆	—	106	1	no	—
96	#341	—	53	2	no	—
97	#335	—	53	1	no	—
98	#335	#6157	1	1	yes	characteristic
99	#334	—	106	1	no	—
100	C₅₃	#7297	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	106	2	no	—
102	C₂²×C₈	—	53	1	no	—
103	C₂²×C₈	—	53	1	no	—
104	C₂×C₄²	—	53	1	no	—
105	C₂×C₁₆	—	106	2	no	—
106	#108	—	106	4	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	106	1	no	—
109	C₂²⋊C₈	—	212	1	no	—
110	#97	—	212	1	no	—
111	#97	—	212	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	—	53	1	no	—
115	C₄×C₈	—	212	1	no	—
116	C₄×C₈	C₂×#6157	1	1	yes	characteristic
117	C₂²×C₄	—	53	1	no	—
118	C₂²×C₄	—	53	1	no	—
119	C₂²×C₄	—	53	1	no	—
120	M₁₆	—	106	4	no	—
121	M₁₆	—	212	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	53	1	no	—
125	C₂×C₈	—	106	2	no	—
126	C₂×C₈	—	212	1	no	—
127	C₂×C₈	—	212	1	no	—
128	C₂×C₈	—	212	2	no	—
129	C₂×C₈	C₄×#6157	1	1	yes	characteristic
130	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
131	C₄²	—	53	1	no	—
132	C₄²	—	53	1	no	—
133	C₄²	—	53	1	no	—
134	C₄²	C₂²⋊#6157	1	1	yes	characteristic
135	C₁₆	—	212	2	no	—
136	C₂³	—	53	1	no	—
137	C₂×C₄	—	53	1	no	—
138	C₂×C₄	—	53	1	no	—
139	C₂×C₄	—	53	1	no	—
140	C₂×C₄	—	106	1	no	—
141	C₂×C₄	—	106	1	no	—
142	C₂×C₄	—	106	1	no	—
143	C₂×C₄	—	106	1	no	—
144	C₂×C₄	—	212	1	no	—
145	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
146	C₂×C₄	#13133279	1	1	yes	characteristic
147	C₂×C₄	#13133280	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	8	1	no	—
151	C₈	—	53	2	no	—
152	C₈	—	106	1	no	—
153	C₈	—	424	1	no	—
154	C₈	—	212	4	no	—
155	C₂²	—	106	1	no	—
156	C₂²	—	106	1	no	—
157	C₂²	—	106	1	no	—
158	C₂²	#16811496	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	106	1	no	—
161	C₄	—	106	1	no	—
162	C₄	—	106	1	no	—
163	C₄	#16811495	1	1	yes	characteristic
164	C₄	#16811505	1	1	yes	characteristic
165	C₄	#16811537	1	1	yes	characteristic
166	C₄	#16811538	1	1	yes	characteristic
167	C₂	—	212	1	no	—
168	C₂	#20289611	1	1	yes	characteristic
169	C₂	#20289776	1	1	yes	characteristic
170	C₂	#20289792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291074

C₂

yes

maximal, characteristic

C₄×#13133288

C₂

yes

maximal, characteristic

#20273477

C₂

yes

maximal, characteristic

C₂×#13133288

—

C₄×#11719936

—

#16811544

C₄

yes

—

#16811518

—

C₄×#11719910

—

#16811499

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

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

—

#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

—

—

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

—

#2162

—

C₄×#108

—

#1568

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#432

—

106

—

C₂×#108

—

106

—

C₄×M₁₆

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

C₄×C₁₆

—

106

—

—

—

yes

—

106

—

100

C₅₃

#7297

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

106

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

106

—

106

#108

—

106

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

106

—

109

C₂²⋊C₈

—

212

—

110

#97

—

212

—

111

#97

—

212

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

212

—

116

C₄×C₈

C₂×#6157

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

M₁₆

—

106

—

121

M₁₆

—

212

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

212

—

127

C₂×C₈

—

212

—

128

C₂×C₈

—

212

—

129

C₂×C₈

C₄×#6157

yes

characteristic

130

C₂×C₈

C₄⋊#6157

yes

characteristic

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₂²⋊#6157

yes

characteristic

135

C₁₆

—

212

—

136

C₂³

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

106

—

141

C₂×C₄

—

106

—

142

C₂×C₄

—

106

—

143

C₂×C₄

—

106

—

144

C₂×C₄

—

212

—

145

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

146

C₂×C₄

#13133279

yes

characteristic

147

C₂×C₄

#13133280

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

106

—

153

C₈

—

424

—

154

C₈

—

212

—

155

C₂²

—

106

—

156

C₂²

—

106

—

157

C₂²

—

106

—

158

C₂²

#16811496

yes

characteristic

159

C₄

—

160

C₄

—

106

—

161

C₄

—

106

—

162

C₄

—

106

—

163

C₄

#16811495

yes

characteristic

164

C₄

#16811505

yes

characteristic

165

C₄

#16811537

yes

characteristic

166

C₄

#16811538

yes

characteristic

167

C₂

—

212

—

168

C₂

#20289611

yes

characteristic

169

C₂

#20289776

yes

characteristic

170

C₂

#20289792

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133279
commutator	C₂×C₈×C₅₃	C₄²
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	#7297
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂×C₄

#13133279

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 ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
100. C₅₃	#7297	semidirect

H ≅ G/N

type

100. C₅₃

#7297

semidirect

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

subgroup	index
#20291074	2
C₄×#13133288	2
#20273477	2
#7297	53

subgroup

index

Subgroups — #28298772

Conjugacy classes of subgroups

Complements

Maximal subgroups