Group #28298316 — subgroups

Subgroups — #28298316

Order 13568.

214 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 96 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	#20290262	C₂	1	2	yes	maximal
2	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
3	#16811639	—	2	2	no	—
4	#16811486	—	2	2	no	—
5	#16811484	C₄	1	4	yes	—
6	C₂²×#11719908	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
8	#16810398	—	2	2	no	—
9	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
10	C₄⋊#88324	—	2	4	no	—
11	C₂²⋊#88324	—	4	2	no	—
12	C₄×#88324	—	4	2	no	—
13	C₂²×#88322	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
15	C₂×#11719908	—	2	1	no	—
16	C₂×#11719908	—	2	1	no	—
17	C₂×#11719908	C₈	1	2	yes	complemented
18	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
19	C₈×Dic₅₃	D₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	—	2	1	no	—
22	C₂×C₁₆×C₅₃	—	2	1	no	—
23	C₂×C₁₆×C₅₃	C₈	1	2	yes	complemented
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
26	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
27	C₂×#88324	—	4	4	no	—
28	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
34	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
35	C₄×Dic₅₃	SD₁₆	1	2	yes	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₅₃	—	2	2	no	—
40	#11719908	—	2	2	no	—
41	C₂³×C₅₃	#107	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
48	C₂×Dic₅₃	—	4	1	no	—
49	C₂×Dic₅₃	—	4	1	no	—
50	C₂×Dic₅₃	#102	1	2	yes	—
51	C₂×Dic₅₃	#103	1	2	yes	—
52	#88324	—	8	4	no	—
53	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
54	C₈×C₅₃	#97	1	1	yes	characteristic
55	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
57	#88322	—	2	2	no	—
58	#7157	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	#335	1	1	yes	characteristic
62	C₂²×C₅₃	#335	1	1	yes	characteristic
63	C₂²×C₅₃	#340	1	1	yes	characteristic
64	C₄×C₅₃	#334	1	1	yes	characteristic
65	C₄×C₅₃	#334	1	1	yes	characteristic
66	C₄×C₅₃	#336	1	1	yes	characteristic
67	C₄×C₅₃	#341	1	1	yes	characteristic
68	Dic₅₃	—	2	2	no	—
69	Dic₅₃	—	2	2	no	—
70	C₂×#346	—	53	1	no	—
71	#1564	—	53	2	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	#1277	1	1	yes	characteristic
75	C₂×C₅₃	#1384	1	1	yes	characteristic
76	C₂×C₅₃	#1385	1	1	yes	characteristic
77	C₂²×C₁₆	—	53	1	no	—
78	C₂²×C₁₆	#6157	1	1	yes	characteristic
79	#431	—	106	2	no	—
80	C₂×C₄×C₈	—	53	1	no	—
81	#346	—	106	2	no	—
82	#335	—	106	2	no	—
83	#334	—	53	4	no	—
84	C₅₃	#7157	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	53	1	no	—
86	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
87	C₂×C₄²	—	53	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	—	53	2	no	—
91	C₂×C₁₆	—	106	1	no	—
92	C₂×C₁₆	—	106	1	no	—
93	C₂×C₁₆	#88324	1	2	yes	complemented
94	C₄⋊C₈	—	106	4	no	—
95	C₂²⋊C₈	—	212	2	no	—
96	C₄×C₈	—	53	2	no	—
97	C₄×C₈	—	53	2	no	—
98	C₄×C₈	—	212	2	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
102	C₂×C₈	—	53	2	no	—
103	C₂×C₈	—	106	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	—	212	4	no	—
106	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
107	C₂×C₈	#11719936	1	1	yes	characteristic
108	C₂×C₈	C₄×#6157	1	1	yes	characteristic
109	C₂×C₈	C₂×#88324	1	1	yes	characteristic
110	C₂×C₈	#11719940	1	1	yes	characteristic
111	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
112	C₄²	—	53	2	no	—
113	C₄²	—	53	2	no	—
114	C₁₆	—	2	2	no	—
115	C₁₆	—	106	2	no	—
116	C₂³	#13133286	1	1	yes	characteristic
117	C₂×C₄	—	53	2	no	—
118	C₂×C₄	—	53	2	no	—
119	C₂×C₄	—	212	1	no	—
120	C₂×C₄	—	212	1	no	—
121	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
123	C₂×C₄	#13133284	1	1	yes	characteristic
124	C₂×C₄	#13133285	1	1	yes	characteristic
125	C₂×C₄	#13133287	1	1	yes	characteristic
126	C₂×C₄	#13133288	1	1	yes	characteristic
127	C₈	—	106	2	no	—
128	C₈	—	424	4	no	—
129	C₈	C₄×#88324	1	1	yes	characteristic
130	C₈	#13133272	1	1	yes	characteristic
131	C₈	C₂²⋊#88324	1	1	yes	characteristic
132	C₈	#13133274	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#16811486	1	1	yes	characteristic
136	C₂²	#16811487	1	1	yes	characteristic
137	C₂²	#16811497	1	1	yes	characteristic
138	C₄	—	106	2	no	—
139	C₄	—	106	2	no	—
140	C₄	#16811484	1	1	yes	characteristic
141	C₄	#16811485	1	1	yes	characteristic
142	C₄	#16811489	1	1	yes	characteristic
143	C₄	#16811499	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#20289545	1	1	yes	characteristic
147	C₂	#20289783	1	1	yes	characteristic
148	C₂	#20289785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290262

C₂

yes

maximal

C₂×#16810398

C₂

yes

maximal, characteristic

#16811639

—

#16811486

—

#16811484

C₄

yes

—

C₂²×#11719908

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810398

—

C₂²×C₁₆×C₅₃

C₄

yes

—

—

—

C₂²×#88322

Q₈

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

—

—

C₈

yes

complemented

C₈×Dic₅₃

C₂×C₄

yes

—

C₈×Dic₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₅₃

D₈

yes

characteristic

C₂²×Dic₅₃

Q₁₆

yes

characteristic

C₂×#88324

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

—

C₄×Dic₅₃

SD₁₆

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

#102

yes

—

C₂×Dic₅₃

#103

yes

—

#88324

—

C₈×C₅₃

C₄×C₈

yes

characteristic

C₈×C₅₃

#97

yes

characteristic

C₈×C₅₃

C₂²⋊C₈

yes

characteristic

C₈×C₅₃

C₂²⋊C₈

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₂×#346

—

#1564

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

—

C₂²×C₁₆

#6157

yes

characteristic

#431

—

106

—

C₂×C₄×C₈

—

#346

—

106

—

#335

—

106

—

#334

—

C₅₃

#7157

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

#88324

yes

complemented

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

212

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄⋊#6157

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

212

—

106

C₂×C₈

D₅₃⋊C₈

yes

characteristic

107

C₂×C₈

#11719936

yes

characteristic

108

C₂×C₈

C₄×#6157

yes

characteristic

109

C₂×C₈

C₂×#88324

yes

characteristic

110

C₂×C₈

#11719940

yes

characteristic

111

C₂×C₈

C₂²⋊#6157

yes

characteristic

112

C₄²

—

113

C₄²

—

114

C₁₆

—

115

C₁₆

—

106

—

116

C₂³

#13133286

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

212

—

120

C₂×C₄

—

212

—

121

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

122

C₂×C₄

C₄⋊#88324

yes

characteristic

123

C₂×C₄

#13133284

yes

characteristic

124

C₂×C₄

#13133285

yes

characteristic

125

C₂×C₄

#13133287

yes

characteristic

126

C₂×C₄

#13133288

yes

characteristic

127

C₈

—

106

—

128

C₈

—

424

—

129

C₈

C₄×#88324

yes

characteristic

130

C₈

#13133272

yes

characteristic

131

C₈

C₂²⋊#88324

yes

characteristic

132

C₈

#13133274

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

#16811486

yes

characteristic

136

C₂²

#16811487

yes

characteristic

137

C₂²

#16811497

yes

characteristic

138

C₄

—

106

—

139

C₄

—

106

—

140

C₄

#16811484

yes

characteristic

141

C₄

#16811485

yes

characteristic

142

C₄

#16811489

yes

characteristic

143

C₄

#16811499

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#20289545

yes

characteristic

147

C₂

#20289783

yes

characteristic

148

C₂

#20289785

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133285
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#107
cosocle	C₂×C₈×Dic₅₃	C₂²
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#7157
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

#13133285

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₂²×C₈

C₂×#6157

Fitting

C₂²×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂×C₈×Dic₅₃

C₂²

C₂²×C₁₆

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
17. C₂×#11719908	C₈	semidirect
23. C₂×C₁₆×C₅₃	C₈	semidirect
84. C₅₃	#7157	semidirect
93. C₂×C₁₆	#88324	semidirect

H ≅ G/N

type

17. C₂×#11719908

C₈

semidirect

23. C₂×C₁₆×C₅₃

C₈

semidirect

84. C₅₃

#7157

semidirect

93. C₂×C₁₆

#88324

semidirect

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

subgroup	index
#20290262	2
C₂×#16810398	2
#7157	53

subgroup

index

#20290262

C₂×#16810398

#7157

Subgroups — #28298316

Conjugacy classes of subgroups

Complements

Maximal subgroups