Group #28309901 — subgroups

Subgroups — #28309901

Order 13568.

226 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 86 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	#20290276	C₂	1	1	yes	maximal, characteristic
2	#20290270	C₂	1	1	yes	maximal, characteristic
3	#20289741	C₂	1	4	yes	maximal, complemented
4	#346×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811645	—	2	1	no	—
6	#16811627	—	2	1	no	—
7	#16811486	—	2	1	no	—
8	#16811484	—	2	1	no	—
9	#16811484	C₂²	1	2	yes	—
10	#16811484	C₂²	1	2	yes	—
11	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
12	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16810399	C₄	1	1	yes	characteristic
14	#16810398	C₂²	1	2	yes	—
15	C₅₃×#346	C₄	1	1	yes	characteristic
16	C₂×#11719936	—	4	1	no	—
17	C₄⋊#88324	—	2	2	no	—
18	C₄⋊#88324	—	2	2	no	—
19	C₄⋊#88324	—	4	1	no	—
20	#13133274	—	4	1	no	—
21	#13133272	—	4	1	no	—
22	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
24	C₄²×D₅₃	D₄	1	1	yes	characteristic
25	C₂×#11719908	C₂×C₄	1	2	yes	—
26	C₁₆×D₅₃	—	2	4	no	—
27	C₄×#88322	Q₈	1	1	yes	characteristic
28	C₈×Dic₅₃	D₄	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
35	C₂×#88324	—	4	2	no	—
36	C₂×#88324	—	4	2	no	—
37	#11719936	—	8	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
42	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
43	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
46	C₈×D₅₃	—	2	2	no	—
47	C₈×D₅₃	—	2	2	no	—
48	C₁₆×C₅₃	—	2	2	no	—
49	#11719908	M₁₆	1	4	yes	complemented
50	C₂²×D₅₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	#106	1	1	yes	characteristic
55	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₄×D₅₃	—	2	2	no	—
58	C₄×D₅₃	—	2	1	no	—
59	C₄×D₅₃	—	2	1	no	—
60	C₄×D₅₃	—	4	1	no	—
61	#88324	—	8	2	no	—
62	#88324	—	8	2	no	—
63	C₈×C₅₃	#108	1	2	yes	—
64	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
65	#88322	—	2	1	no	—
66	#88322	—	2	1	no	—
67	#12019	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
69	C₂×D₅₃	—	2	1	no	—
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	#365	1	2	yes	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	#423	1	1	yes	characteristic
75	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
76	Dic₅₃	—	2	1	no	—
77	Dic₅₃	—	2	1	no	—
78	Dic₅₃	—	2	1	no	—
79	C₂×#346	—	53	1	no	—
80	#1568	—	53	1	no	—
81	#1566	—	53	1	no	—
82	#1368	—	53	4	no	—
83	C₂×C₅₃	#1566	1	1	yes	characteristic
84	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
85	C₂×C₅₃	#2155	1	1	yes	characteristic
86	D₅₃	—	2	2	no	—
87	C₂²×C₁₆	—	53	2	no	—
88	#432	—	106	1	no	—
89	#423	—	106	1	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	#346	—	53	2	no	—
92	#346	—	53	1	no	—
93	#346	#6157	1	1	yes	characteristic
94	#335	—	106	1	no	—
95	#334	—	53	2	no	—
96	#334	—	53	2	no	—
97	#334	—	106	1	no	—
98	C₅₃	#12019	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	212	1	no	—
100	C₂²×C₈	—	53	1	no	—
101	C₂²×C₈	—	53	1	no	—
102	C₂×C₄²	—	53	1	no	—
103	C₂×C₁₆	—	53	2	no	—
104	C₂×C₁₆	—	106	4	no	—
105	C₂×C₁₆	C₂×#6157	1	2	yes	—
106	C₄⋊C₈	—	106	2	no	—
107	C₄⋊C₈	—	106	2	no	—
108	C₄⋊C₈	—	212	1	no	—
109	C₂²⋊C₈	—	212	1	no	—
110	#97	—	212	1	no	—
111	C₄×C₈	—	53	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	C₂×#6157	1	1	yes	characteristic
115	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	—	53	1	no	—
118	M₁₆	—	424	2	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	106	2	no	—
122	C₂×C₈	—	106	2	no	—
123	C₂×C₈	—	212	2	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
126	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
127	C₄²	—	53	1	no	—
128	C₄²	—	53	1	no	—
129	C₄²	—	53	1	no	—
130	C₄²	C₄⋊#6157	1	1	yes	characteristic
131	C₁₆	—	2	2	no	—
132	C₁₆	—	53	4	no	—
133	C₂³	—	53	1	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	—	106	1	no	—
139	C₂×C₄	—	106	1	no	—
140	C₂×C₄	—	212	1	no	—
141	C₂×C₄	#13133285	1	1	yes	characteristic
142	C₂×C₄	#13133285	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
144	C₈	—	106	1	no	—
145	C₈	—	106	1	no	—
146	C₈	—	424	2	no	—
147	C₈	—	424	2	no	—
148	C₈	#13133288	1	2	yes	—
149	C₈	D₅₃⋊M₁₆	1	2	yes	—
150	C₂²	—	53	2	no	—
151	C₂²	—	106	1	no	—
152	C₂²	—	106	1	no	—
153	C₂²	C₂×#13133285	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	106	1	no	—
157	C₄	—	106	1	no	—
158	C₄	—	106	1	no	—
159	C₄	#16811628	1	1	yes	characteristic
160	C₄	C₂×#13133288	1	1	yes	characteristic
161	C₂	—	106	2	no	—
162	C₂	#20290271	1	1	yes	characteristic
163	C₂	D₅₃⋊#365	1	1	yes	characteristic
164	C₂	#20291063	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290276

C₂

yes

maximal, characteristic

#20290270

C₂

yes

maximal, characteristic

#20289741

C₂

yes

maximal, complemented

#346×D₅₃

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#16811484

C₂²

yes

—

C₂×C₁₆×D₅₃

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810399

C₄

yes

characteristic

#16810398

C₂²

yes

—

C₅₃×#346

C₄

yes

—

—

—

—

—

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₁₆×D₅₃

—

C₄×#88322

Q₈

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

—

—

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

characteristic

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

M₁₆

yes

C₂²×D₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

#106

yes

characteristic

C₂×Dic₅₃

C₂×SD₁₆

yes

characteristic

C₂×Dic₅₃

C₂×SD₁₆

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

#88324

—

C₈×C₅₃

#108

yes

—

C₈×C₅₃

C₂×M₁₆

yes

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#365

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

—

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂²×C₁₆

—

#432

—

106

—

#423

—

106

—

C₂×C₄×C₈

—

—

—

yes

—

106

—

#334

—

#334

—

#334

—

106

—

C₅₃

#12019

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

106

—

105

C₂×C₁₆

C₂×#6157

yes

—

106

C₄⋊C₈

—

106

—

107

C₄⋊C₈

—

106

—

108

C₄⋊C₈

—

212

—

109

C₂²⋊C₈

—

212

—

110

#97

—

212

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#6157

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

M₁₆

—

424

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

212

—

124

C₂×C₈

—

212

—

125

C₂×C₈

C₄⋊#6157

yes

characteristic

126

C₂×C₈

C₂²×#6157

yes

characteristic

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₄⋊#6157

yes

characteristic

131

C₁₆

—

132

C₁₆

—

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

212

—

141

C₂×C₄

#13133285

yes

characteristic

142

C₂×C₄

#13133285

yes

characteristic

143

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

144

C₈

—

106

—

145

C₈

—

106

—

146

C₈

—

424

—

147

C₈

—

424

—

148

C₈

#13133288

yes

—

149

C₈

D₅₃⋊M₁₆

yes

—

150

C₂²

—

151

C₂²

—

106

—

152

C₂²

—

106

—

153

C₂²

C₂×#13133285

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

106

—

157

C₄

—

106

—

158

C₄

—

106

—

159

C₄

#16811628

yes

characteristic

160

C₄

C₂×#13133288

yes

characteristic

161

C₂

—

106

—

162

C₂

#20290271

yes

characteristic

163

C₂

D₅₃⋊#365

yes

characteristic

164

C₂

#20291063

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13133285
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#346	C₄
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂²×D₅₃	#106
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	#12019
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#13133285

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#346

C₄

C₂²×C₅₃

C₂²×D₅₃

C₈×Dic₅₃

C₂³

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
3. #20289741	C₂	semidirect
49. #11719908	M₁₆	semidirect
98. C₅₃	#12019	semidirect

H ≅ G/N

type

3. #20289741

C₂

semidirect

49. #11719908

M₁₆

semidirect

98. C₅₃

#12019

semidirect

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

subgroup	index
#20290276	2
#20290270	2
#20289741	2
#346×D₅₃	2
#12019	53

subgroup

index

#346×D₅₃

Subgroups — #28309901

Conjugacy classes of subgroups

Complements

Maximal subgroups