Group #28311405 — subgroups

Subgroups — #28311405

Order 13568.

282 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 108 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	#20290656	C₂	1	2	yes	maximal
2	#20289775	C₂	1	4	yes	maximal
3	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133288	—	2	2	no	—
5	C₂×#13133288	C₂²	1	4	yes	—
6	#16811627	—	2	2	no	—
7	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
8	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₄×#11719908	C₄	1	1	yes	characteristic
10	C₁₆×Dic₅₃	C₄	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719936	—	2	4	no	—
13	#13133288	—	2	8	no	—
14	#13133288	—	4	4	no	—
15	C₄⋊#88324	—	4	2	no	—
16	C₄⋊#88324	—	4	2	no	—
17	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
18	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
19	C₄²×D₅₃	D₄	1	1	yes	characteristic
20	C₂×#11719908	C₂×C₄	1	2	yes	—
21	C₁₆×D₅₃	—	2	4	no	—
22	C₄×#88322	Q₈	1	1	yes	characteristic
23	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
29	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×#88324	—	4	4	no	—
31	#11719936	—	4	4	no	—
32	#11719936	—	4	4	no	—
33	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄×Dic₅₃	D₈	1	1	yes	characteristic
38	C₄×Dic₅₃	D₈	1	1	yes	characteristic
39	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
41	C₈×D₅₃	SD₁₆	1	4	yes	—
42	C₈×D₅₃	#41	1	4	yes	—
43	C₁₆×C₅₃	—	2	2	no	—
44	#11719908	—	2	2	no	—
45	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×Dic₅₃	#107	1	1	yes	characteristic
50	C₂×Dic₅₃	#107	1	1	yes	characteristic
51	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
52	C₄×D₅₃	—	2	2	no	—
53	C₄×D₅₃	—	2	2	no	—
54	C₄×D₅₃	#124	1	2	yes	—
55	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
56	#88324	—	8	4	no	—
57	C₈×C₅₃	#106	1	2	yes	—
58	C₈×C₅₃	#117	1	2	yes	—
59	#88322	#106	1	2	yes	—
60	#88322	#125	1	2	yes	—
61	#12891	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
63	C₂×D₅₃	—	2	1	no	—
64	C₂×D₅₃	—	2	1	no	—
65	C₂×D₅₃	#487	1	2	yes	—
66	C₄×C₅₃	#368	1	2	yes	—
67	C₄×C₅₃	#368	1	2	yes	—
68	C₄×C₅₃	#382	1	1	yes	characteristic
69	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
70	Dic₅₃	—	2	1	no	—
71	Dic₅₃	—	2	1	no	—
72	Dic₅₃	#486	1	2	yes	—
73	C₂×C₄×C₁₆	—	53	1	no	—
74	#1848	—	53	2	no	—
75	#1381	—	53	4	no	—
76	C₂×C₅₃	#1834	1	1	yes	characteristic
77	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
78	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
79	D₅₃	—	2	2	no	—
80	C₂²×C₁₆	—	53	2	no	—
81	C₂×#108	—	53	4	no	—
82	C₂×#108	—	106	2	no	—
83	#423	—	106	2	no	—
84	C₂×C₄×C₈	—	53	1	no	—
85	C₄×C₁₆	—	53	2	no	—
86	C₄×C₁₆	—	53	1	no	—
87	C₄×C₁₆	#6157	1	1	yes	characteristic
88	C₅₃	#12891	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₁₆	—	106	4	no	—
90	C₂²×C₈	—	53	1	no	—
91	C₂²×C₈	—	53	1	no	—
92	C₂×C₄²	—	53	1	no	—
93	C₂×C₁₆	—	53	2	no	—
94	C₂×C₁₆	—	106	4	no	—
95	C₂×C₁₆	C₂×#6157	1	2	yes	—
96	#108	—	106	8	no	—
97	#108	—	212	4	no	—
98	C₄⋊C₈	—	212	2	no	—
99	C₄⋊C₈	—	212	2	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	—	53	1	no	—
103	C₄×C₈	C₂×#6157	1	1	yes	characteristic
104	C₂²×C₄	—	53	1	no	—
105	C₂²×C₄	—	53	1	no	—
106	C₂²×C₄	—	53	1	no	—
107	M₁₆	—	212	4	no	—
108	M₁₆	—	212	4	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	4	no	—
112	C₂×C₈	—	53	4	no	—
113	C₂×C₈	—	212	4	no	—
114	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
115	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
116	C₄²	—	53	1	no	—
117	C₄²	—	53	1	no	—
118	C₄²	—	53	1	no	—
119	C₄²	C₄⋊#6157	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	106	2	no	—
122	C₂³	—	53	1	no	—
123	C₂×C₄	—	53	1	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	53	2	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	—	106	2	no	—
130	C₂×C₄	#13133286	1	1	yes	characteristic
131	C₂×C₄	#13133286	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
133	C₈	—	53	2	no	—
134	C₈	—	53	2	no	—
135	C₈	—	424	4	no	—
136	C₈	#13133285	1	2	yes	—
137	C₈	#13133296	1	2	yes	—
138	C₂²	—	53	2	no	—
139	C₂²	—	106	1	no	—
140	C₂²	—	106	1	no	—
141	C₂²	C₂×#13133286	1	1	yes	characteristic
142	C₄	—	53	2	no	—
143	C₄	—	106	1	no	—
144	C₄	—	106	1	no	—
145	C₄	#16811551	1	2	yes	—
146	C₄	#16811552	1	2	yes	—
147	C₄	#16811567	1	1	yes	characteristic
148	C₄	C₂×#13133285	1	1	yes	characteristic
149	C₂	—	106	2	no	—
150	C₂	#20290636	1	1	yes	characteristic
151	C₂	C₂×#16811551	1	1	yes	characteristic
152	C₂	C₂×#16811552	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290656

C₂

yes

maximal

#20289775

C₂

yes

maximal

C₄×C₁₆×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133288

—

C₂×#13133288

C₂²

yes

—

#16811627

—

C₂×C₁₆×D₅₃

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₄×#11719908

C₄

yes

characteristic

C₁₆×Dic₅₃

C₄

yes

—

C₄×C₁₆×C₅₃

C₄

yes

—

—

—

—

—

C₂×C₈×D₅₃

D₄

yes

characteristic

C₂×C₈×D₅₃

C₂³

yes

characteristic

C₄²×D₅₃

D₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₁₆×D₅₃

—

C₄×#88322

Q₈

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

Q₁₆

yes

characteristic

C₂×C₄×D₅₃

Q₁₆

yes

characteristic

C₂×C₄×D₅₃

C₂×D₄

yes

—

—

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

D₈

yes

characteristic

C₄×Dic₅₃

D₈

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₈×D₅₃

SD₁₆

yes

—

C₈×D₅₃

#41

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

C₂×Q₁₆

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

C₂×D₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

#124

yes

—

C₄×D₅₃

C₂×SD₁₆

yes

—

#88324

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#117

yes

—

#88322

#106

yes

—

#88322

#125

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#487

yes

—

C₄×C₅₃

#368

yes

—

C₄×C₅₃

#368

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#486

yes

—

C₂×C₄×C₁₆

—

#1848

—

#1381

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

106

—

#423

—

106

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

#6157

yes

characteristic

C₅₃

#12891

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×#6157

yes

—

#108

—

106

—

#108

—

212

—

C₄⋊C₈

—

212

—

C₄⋊C₈

—

212

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×#6157

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

M₁₆

—

212

—

108

M₁₆

—

212

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

212

—

114

C₂×C₈

C₄⋊#6157

yes

characteristic

115

C₂×C₈

C₂²×#6157

yes

characteristic

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄⋊#6157

yes

characteristic

120

C₁₆

—

121

C₁₆

—

106

—

122

C₂³

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

106

—

130

C₂×C₄

#13133286

yes

characteristic

131

C₂×C₄

#13133286

yes

characteristic

132

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

424

—

136

C₈

#13133285

yes

—

137

C₈

#13133296

yes

—

138

C₂²

—

139

C₂²

—

106

—

140

C₂²

—

106

—

141

C₂²

C₂×#13133286

yes

characteristic

142

C₄

—

143

C₄

—

106

—

144

C₄

—

106

—

145

C₄

#16811551

yes

—

146

C₄

#16811552

yes

—

147

C₄

#16811567

yes

characteristic

148

C₄

C₂×#13133285

yes

characteristic

149

C₂

—

106

—

150

C₂

#20290636

yes

characteristic

151

C₂

C₂×#16811551

yes

characteristic

152

C₂

C₂×#16811552

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13133286
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂²×D₅₃	C₂×Q₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#12891
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#13133286

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₄×C₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂²×D₅₃

C₂×Q₁₆

cosocle

C₂×C₈×D₅₃

C₂³

C₄×C₁₆

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
88. C₅₃	#12891	semidirect

H ≅ G/N

type

88. C₅₃

#12891

semidirect

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

subgroup	index
#20290656	2
#20289775	2
C₄×C₁₆×D₅₃	2
#12891	53

subgroup

index

#20290656

#20289775

C₄×C₁₆×D₅₃

#12891

Subgroups — #28311405

Conjugacy classes of subgroups

Complements

Maximal subgroups