Group #28303613 — subgroups

Subgroups — #28303613

Order 13568.

322 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 124 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	#20290639	C₂	1	2	yes	maximal
2	#20290273	C₂	1	2	yes	maximal
3	#20290270	C₂	1	2	yes	maximal
4	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811627	C₂²	1	4	yes	—
6	#16811499	—	2	4	no	—
7	#16811486	—	2	4	no	—
8	#16811484	—	2	4	no	—
9	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₈×#88322	C₄	1	1	yes	characteristic
13	C₈×#88322	C₄	1	1	yes	characteristic
14	C₈×#88322	C₄	1	1	yes	characteristic
15	C₈²×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#11719936	—	2	4	no	—
17	C₄⋊#88324	—	2	4	no	—
18	C₄⋊#88324	—	2	4	no	—
19	C₂×C₈×D₅₃	D₄	1	2	yes	—
20	C₂×C₈×D₅₃	Q₈	1	2	yes	—
21	C₂×C₈×D₅₃	Q₈	1	2	yes	—
22	C₄²×D₅₃	C₂³	1	1	yes	characteristic
23	C₄×#88322	C₂×C₄	1	1	yes	characteristic
24	C₄×#88322	C₂×C₄	1	1	yes	characteristic
25	C₄×#88322	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	D₄	1	2	yes	—
27	C₈×Dic₅₃	D₄	1	2	yes	—
28	C₈×Dic₅₃	Q₈	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×#88324	—	4	4	no	—
36	#11719936	—	4	8	no	—
37	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
43	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×#88322	C₄⋊C₄	1	2	yes	—
45	C₂×#88322	C₄⋊C₄	1	2	yes	—
46	C₂×#88322	C₄⋊C₄	1	2	yes	—
47	C₈×D₅₃	—	2	4	no	—
48	C₈×D₅₃	—	2	4	no	—
49	C₈×D₅₃	—	2	4	no	—
50	C₂²×D₅₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	#127	1	1	yes	characteristic
55	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×D₅₃	—	2	2	no	—
58	C₄×D₅₃	—	2	2	no	—
59	C₄×D₅₃	#135	1	4	yes	—
60	#88324	—	8	4	no	—
61	C₈×C₅₃	—	2	2	no	—
62	C₈×C₅₃	#108	1	4	yes	—
63	C₈×C₅₃	#108	1	4	yes	—
64	#88322	—	2	2	no	—
65	#88322	—	2	2	no	—
66	#88322	—	2	2	no	—
67	#8824	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	#384	1	1	yes	characteristic
69	C₂×D₅₃	—	2	1	no	—
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	#499	1	2	yes	—
72	C₄×C₅₃	#427	1	2	yes	—
73	C₄×C₅₃	C₂×#108	1	2	yes	—
74	C₄×C₅₃	C₂×#108	1	2	yes	—
75	Dic₅₃	—	2	1	no	—
76	Dic₅₃	—	2	1	no	—
77	Dic₅₃	#495	1	2	yes	—
78	#1837	—	53	2	no	—
79	#1567	—	53	2	no	—
80	#1566	—	53	2	no	—
81	C₂×C₈²	—	53	1	no	—
82	C₂×C₅₃	#1846	1	1	yes	characteristic
83	C₂×C₅₃	#1848	1	1	yes	characteristic
84	C₂×C₅₃	#1848	1	1	yes	characteristic
85	D₅₃	—	2	2	no	—
86	#423	—	53	4	no	—
87	C₂×C₄×C₈	—	53	1	no	—
88	C₂×C₄×C₈	—	53	1	no	—
89	C₂×C₄×C₈	—	53	1	no	—
90	#341	—	106	4	no	—
91	#335	—	106	4	no	—
92	#334	—	106	4	no	—
93	C₈²	—	53	1	no	—
94	C₈²	—	53	1	no	—
95	C₈²	—	53	1	no	—
96	C₈²	#6157	1	1	yes	characteristic
97	C₅₃	#8824	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	106	4	no	—
99	C₂²×C₈	—	53	2	no	—
100	C₂²×C₈	—	53	2	no	—
101	C₂²×C₈	—	53	2	no	—
102	C₂×C₄²	—	53	1	no	—
103	C₄⋊C₈	—	106	4	no	—
104	C₄⋊C₈	—	106	4	no	—
105	C₄×C₈	—	53	2	no	—
106	C₄×C₈	—	53	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	—	53	1	no	—
109	C₄×C₈	—	53	1	no	—
110	C₄×C₈	—	53	1	no	—
111	C₄×C₈	C₂×#6157	1	1	yes	characteristic
112	C₄×C₈	C₂×#6157	1	1	yes	characteristic
113	C₄×C₈	C₂×#6157	1	1	yes	characteristic
114	C₂²×C₄	—	53	1	no	—
115	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	53	1	no	—
117	M₁₆	—	212	8	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	—	53	2	no	—
120	C₂×C₈	—	53	2	no	—
121	C₂×C₈	—	106	4	no	—
122	C₂×C₈	—	106	4	no	—
123	C₂×C₈	—	106	4	no	—
124	C₂×C₈	—	212	4	no	—
125	C₂×C₈	C₄⋊#6157	1	2	yes	—
126	C₂×C₈	C₄⋊#6157	1	2	yes	—
127	C₂×C₈	C₄⋊#6157	1	2	yes	—
128	C₄²	—	53	1	no	—
129	C₄²	—	53	1	no	—
130	C₄²	—	53	1	no	—
131	C₄²	C₂²×#6157	1	1	yes	characteristic
132	C₂³	—	53	1	no	—
133	C₂×C₄	—	53	1	no	—
134	C₂×C₄	—	53	4	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	106	2	no	—
144	C₈	—	106	2	no	—
145	C₈	—	106	2	no	—
146	C₈	—	424	4	no	—
147	C₈	#13133287	1	4	yes	—
148	C₈	#13133288	1	4	yes	—
149	C₂²	—	53	2	no	—
150	C₂²	—	106	1	no	—
151	C₂²	—	106	1	no	—
152	C₂²	#16811569	1	1	yes	characteristic
153	C₄	—	53	2	no	—
154	C₄	—	106	1	no	—
155	C₄	—	106	1	no	—
156	C₄	#16811633	1	2	yes	—
157	C₄	C₂×#13133287	1	2	yes	—
158	C₄	C₂×#13133288	1	2	yes	—
159	C₂	—	106	2	no	—
160	C₂	#20290652	1	1	yes	characteristic
161	C₂	#20290654	1	1	yes	characteristic
162	C₂	#20290656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290639

C₂

yes

maximal

#20290273

C₂

yes

maximal

#20290270

C₂

yes

maximal

C₈²×D₅₃

C₂

yes

maximal, characteristic

#16811627

C₂²

yes

—

#16811499

—

#16811486

—

#16811484

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₈×#88322

C₄

yes

characteristic

C₈×#88322

C₄

yes

characteristic

C₈×#88322

C₄

yes

characteristic

C₈²×C₅₃

C₄

yes

—

—

—

C₂×C₈×D₅₃

D₄

yes

—

C₂×C₈×D₅₃

Q₈

yes

—

C₂×C₈×D₅₃

Q₈

yes

—

C₄²×D₅₃

C₂³

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₅₃

C₂×Q₈

yes

characteristic

C₂×#88324

—

#11719936

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₂²×D₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

#127

yes

characteristic

C₂×Dic₅₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₅₃

C₄⋊Q₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

#135

yes

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

#108

yes

—

C₈×C₅₃

#108

yes

—

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#499

yes

—

C₄×C₅₃

#427

yes

—

C₄×C₅₃

C₂×#108

yes

—

C₄×C₅₃

C₂×#108

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#495

yes

—

#1837

—

#1567

—

#1566

—

C₂×C₈²

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#341

—

106

—

#335

—

106

—

#334

—

106

—

C₈²

—

C₈²

—

C₈²

—

C₈²

#6157

yes

characteristic

C₅₃

#8824

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

106

—

104

C₄⋊C₈

—

106

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₂×#6157

yes

characteristic

112

C₄×C₈

C₂×#6157

yes

characteristic

113

C₄×C₈

C₂×#6157

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

M₁₆

—

212

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

106

—

124

C₂×C₈

—

212

—

125

C₂×C₈

C₄⋊#6157

yes

—

126

C₂×C₈

C₄⋊#6157

yes

—

127

C₂×C₈

C₄⋊#6157

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×#6157

yes

characteristic

132

C₂³

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

140

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

141

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

142

C₈

—

143

C₈

—

106

—

144

C₈

—

106

—

145

C₈

—

106

—

146

C₈

—

424

—

147

C₈

#13133287

yes

—

148

C₈

#13133288

yes

—

149

C₂²

—

150

C₂²

—

106

—

151

C₂²

—

106

—

152

C₂²

#16811569

yes

characteristic

153

C₄

—

154

C₄

—

106

—

155

C₄

—

106

—

156

C₄

#16811633

yes

—

157

C₄

C₂×#13133287

yes

—

158

C₄

C₂×#13133288

yes

—

159

C₂

—

106

—

160

C₂

#20290652

yes

characteristic

161

C₂

#20290654

yes

characteristic

162

C₂

#20290656

yes

characteristic

	subgroup	G/H
center	C₂²	#16811569
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂²×C₅₃	#384
prime radical	C₂²×D₅₃	C₄⋊Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	C₈²	#6157
hypocenter	C₅₃	#8824
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#16811569

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₈²×C₅₃

C₄

socle

C₂²×C₅₃

#384

prime radical

C₂²×D₅₃

C₄⋊Q₈

cosocle

C₄²×D₅₃

C₂³

C₈²

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
97. C₅₃	#8824	semidirect

H ≅ G/N

type

97. C₅₃

#8824

semidirect

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

subgroup	index
#20290639	2
#20290273	2
#20290270	2
C₈²×D₅₃	2
#8824	53

subgroup

index

C₈²×D₅₃

Subgroups — #28303613

Conjugacy classes of subgroups

Complements

Maximal subgroups