Group #28298314 — subgroups

Subgroups — #28298314

Order 13568.

234 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 112 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	C₂×#16811484	C₂	1	2	yes	maximal
2	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88324	—	2	2	no	—
4	#16811484	—	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	C₂²×#88324	—	4	2	no	—
11	C₄⋊#88324	—	2	4	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	SD₁₆	1	2	yes	—
26	C₂×#88324	—	4	4	no	—
27	C₂×#88324	—	8	2	no	—
28	C₂×#88324	—	8	2	no	—
29	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
35	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
36	C₄×Dic₅₃	D₈	1	1	yes	characteristic
37	C₄×Dic₅₃	Q₁₆	1	1	yes	characteristic
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	C₄⋊C₄	1	2	yes	—
41	C₁₆×C₅₃	—	2	2	no	—
42	#11719908	—	2	2	no	—
43	C₂³×C₅₃	#106	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	2	no	—
51	C₂×Dic₅₃	—	2	2	no	—
52	C₂×Dic₅₃	#102	1	2	yes	—
53	C₂×Dic₅₃	#103	1	2	yes	—
54	#88324	—	8	4	no	—
55	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
56	C₈×C₅₃	#97	1	1	yes	characteristic
57	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
59	#88322	—	2	2	no	—
60	#7156	—	53	1	no	maximal, Sylow
61	C₂²×C₅₃	#334	1	1	yes	characteristic
62	C₂²×C₅₃	#334	1	1	yes	characteristic
63	C₂²×C₅₃	#340	1	1	yes	characteristic
64	C₂²×C₅₃	#365	1	2	yes	—
65	C₂²×C₅₃	#365	1	2	yes	—
66	C₄×C₅₃	#335	1	1	yes	characteristic
67	C₄×C₅₃	#335	1	1	yes	characteristic
68	C₄×C₅₃	#336	1	1	yes	characteristic
69	C₄×C₅₃	#341	1	1	yes	characteristic
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	—	2	2	no	—
72	C₂×#346	—	53	1	no	—
73	C₂×#334	—	53	2	no	—
74	C₂×C₅₃	#1277	1	1	yes	characteristic
75	C₂×C₅₃	#1368	1	1	yes	characteristic
76	C₂×C₅₃	#1368	1	1	yes	characteristic
77	C₂×C₅₃	#1368	1	2	yes	—
78	C₂×C₅₃	#1383	1	2	yes	—
79	C₂²×C₁₆	—	53	1	no	—
80	C₂²×C₁₆	#6157	1	1	yes	characteristic
81	C₂×C₄⋊C₈	—	106	2	no	—
82	C₂×C₄×C₈	—	53	1	no	—
83	#346	—	106	2	no	—
84	#334	—	53	4	no	—
85	#334	—	106	2	no	—
86	C₅₃	#7156	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₈	—	53	1	no	—
88	C₂²×C₈	—	212	2	no	—
89	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
90	C₂×C₄²	—	53	1	no	—
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	53	2	no	—
94	C₂×C₁₆	—	106	1	no	—
95	C₂×C₁₆	—	106	1	no	—
96	C₂×C₁₆	#88324	1	2	yes	complemented
97	C₄⋊C₈	—	106	4	no	—
98	C₄⋊C₈	—	212	2	no	—
99	C₄×C₈	—	53	2	no	—
100	C₄×C₈	—	53	2	no	—
101	C₂²×C₄	—	53	2	no	—
102	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
103	C₂×C₈	—	53	2	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	—	106	1	no	—
106	C₂×C₈	—	424	2	no	—
107	C₂×C₈	—	424	2	no	—
108	C₂×C₈	—	212	4	no	—
109	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
110	C₂×C₈	#11719936	1	1	yes	characteristic
111	C₂×C₈	C₄×#6157	1	1	yes	characteristic
112	C₂×C₈	C₂×#88324	1	1	yes	characteristic
113	C₂×C₈	#11719940	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
115	C₄²	—	53	2	no	—
116	C₄²	—	53	1	no	—
117	C₄²	—	53	1	no	—
118	C₁₆	—	2	2	no	—
119	C₁₆	—	106	2	no	—
120	C₂³	#13133285	1	1	yes	characteristic
121	C₂×C₄	—	53	2	no	—
122	C₂×C₄	—	53	2	no	—
123	C₂×C₄	—	106	2	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
126	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
127	C₂×C₄	#13133284	1	1	yes	characteristic
128	C₂×C₄	#13133286	1	1	yes	characteristic
129	C₂×C₄	#13133287	1	1	yes	characteristic
130	C₂×C₄	#13133288	1	1	yes	characteristic
131	C₈	—	106	2	no	—
132	C₈	—	424	4	no	—
133	C₈	C₄×#88324	1	1	yes	characteristic
134	C₈	#13133272	1	1	yes	characteristic
135	C₈	C₂²⋊#88324	1	1	yes	characteristic
136	C₈	#13133274	1	1	yes	characteristic
137	C₂²	#16811484	1	1	yes	characteristic
138	C₂²	#16811485	1	1	yes	characteristic
139	C₂²	#16811497	1	1	yes	characteristic
140	C₂²	#16811547	1	2	yes	—
141	C₂²	#16811548	1	2	yes	—
142	C₄	—	106	2	no	—
143	C₄	—	106	2	no	—
144	C₄	#16811486	1	1	yes	characteristic
145	C₄	#16811487	1	1	yes	characteristic
146	C₄	#16811489	1	1	yes	characteristic
147	C₄	#16811499	1	1	yes	characteristic
148	C₂	#20289545	1	1	yes	characteristic
149	C₂	#20289739	1	1	yes	characteristic
150	C₂	#20289740	1	1	yes	characteristic
151	C₂	#20289741	1	2	yes	—
152	C₂	#20289780	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811484

C₂

yes

maximal

C₂×#16810398

C₂

yes

maximal, characteristic

C₂×C₄⋊#88324

—

#16811484

—

#16811484

C₄

yes

—

C₂²×#11719908

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810398

—

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

C₂²×#88324

—

C₄⋊#88324

—

C₄⋊#88324

—

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

SD₁₆

yes

—

C₂×#88324

—

C₂×#88324

—

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

D₈

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

—

—

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

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#365

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₂×#346

—

C₂×#334

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#1383

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

#6157

yes

characteristic

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

—

#346

—

106

—

#334

—

#334

—

106

—

C₅₃

#7156

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

212

—

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

—

C₄⋊C₈

—

212

—

C₄×C₈

—

100

C₄×C₈

—

101

C₂²×C₄

—

102

C₂²×C₄

C₄⋊#6157

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

106

—

106

C₂×C₈

—

424

—

107

C₂×C₈

—

424

—

108

C₂×C₈

—

212

—

109

C₂×C₈

D₅₃⋊C₈

yes

characteristic

110

C₂×C₈

#11719936

yes

characteristic

111

C₂×C₈

C₄×#6157

yes

characteristic

112

C₂×C₈

C₂×#88324

yes

characteristic

113

C₂×C₈

#11719940

yes

characteristic

114

C₂×C₈

C₂²⋊#6157

yes

characteristic

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₁₆

—

119

C₁₆

—

106

—

120

C₂³

#13133285

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

106

—

125

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

126

C₂×C₄

C₄⋊#88324

yes

characteristic

127

C₂×C₄

#13133284

yes

characteristic

128

C₂×C₄

#13133286

yes

characteristic

129

C₂×C₄

#13133287

yes

characteristic

130

C₂×C₄

#13133288

yes

characteristic

131

C₈

—

106

—

132

C₈

—

424

—

133

C₈

C₄×#88324

yes

characteristic

134

C₈

#13133272

yes

characteristic

135

C₈

C₂²⋊#88324

yes

characteristic

136

C₈

#13133274

yes

characteristic

137

C₂²

#16811484

yes

characteristic

138

C₂²

#16811485

yes

characteristic

139

C₂²

#16811497

yes

characteristic

140

C₂²

#16811547

yes

—

141

C₂²

#16811548

yes

—

142

C₄

—

106

—

143

C₄

—

106

—

144

C₄

#16811486

yes

characteristic

145

C₄

#16811487

yes

characteristic

146

C₄

#16811489

yes

characteristic

147

C₄

#16811499

yes

characteristic

148

C₂

#20289545

yes

characteristic

149

C₂

#20289739

yes

characteristic

150

C₂

#20289740

yes

characteristic

151

C₂

#20289741

yes

—

152

C₂

#20289780

yes

—

	subgroup	G/H
center	C₂³	#13133285
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	#106
cosocle	C₂×C₈×Dic₅₃	C₂²
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#7156
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂³

#13133285

commutator

C₈×C₅₃

C₄×C₈

Frattini

C₂²×C₈

C₂×#6157

Fitting

C₂²×C₁₆×C₅₃

C₄

socle, prime radical

C₂³×C₅₃

#106

cosocle

C₂×C₈×Dic₅₃

C₂²

C₂²×C₁₆

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
17. C₂×#11719908	C₈	semidirect
23. C₂×C₁₆×C₅₃	C₈	semidirect
86. C₅₃	#7156	semidirect
96. C₂×C₁₆	#88324	semidirect

H ≅ G/N

type

17. C₂×#11719908

C₈

semidirect

23. C₂×C₁₆×C₅₃

C₈

semidirect

86. C₅₃

#7156

semidirect

96. C₂×C₁₆

#88324

semidirect

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

subgroup	index
C₂×#16811484	2
C₂×#16810398	2
#7156	53

subgroup

index

C₂×#16811484

C₂×#16810398

#7156

Subgroups — #28298314

Conjugacy classes of subgroups

Complements

Maximal subgroups