Group #28298777 — subgroups

Subgroups — #28298777

Order 13568.

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20291073	C₂	1	1	yes	maximal, characteristic
2	#20290272	C₂	1	1	yes	maximal, characteristic
3	#20274616	C₂	1	1	yes	maximal, characteristic
4	#16811626	—	2	1	no	—
5	#16811544	C₄	1	2	yes	—
6	C₂²⋊#11719910	—	2	1	no	—
7	#16811509	—	2	1	no	—
8	#16811487	—	2	1	no	—
9	#16811486	C₄	1	2	yes	—
10	C₂×#13133152	—	2	1	no	—
11	M₁₆×Dic₅₃	—	2	1	no	—
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810388	C₄	1	1	yes	characteristic
14	C₅₃×#341	C₄	1	1	yes	characteristic
15	C₂×#11719940	—	4	1	no	—
16	C₂×#11719910	—	2	2	no	—
17	#13133284	—	4	1	no	—
18	C₄⋊#88324	—	2	2	no	—
19	C₂×#11719917	—	2	1	no	—
20	C₂²×#88322	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
22	#13133152	—	2	4	no	—
23	#13133127	—	4	1	no	—
24	C₈×Dic₅₃	—	4	1	no	—
25	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₅₃	D₄	1	1	yes	characteristic
28	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
33	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
34	#11719940	—	8	2	no	—
35	C₂×#88324	—	4	2	no	—
36	M₁₆×C₅₃	—	2	2	no	—
37	M₁₆×C₅₃	—	4	1	no	—
38	C₂×C₈×C₅₃	—	4	1	no	—
39	C₂×C₈×C₅₃	—	4	1	no	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	D₈	1	1	yes	characteristic
45	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
46	#11719917	—	2	2	no	—
47	#11719917	—	4	1	no	—
48	C₂×#88322	—	2	2	no	—
49	C₂×#88322	—	4	1	no	—
50	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
52	#11719910	—	4	2	no	—
53	C₂³×C₅₃	#103	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	2	1	no	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	#102	1	1	yes	characteristic
62	C₂×Dic₅₃	#103	1	1	yes	characteristic
63	C₂×Dic₅₃	#106	1	1	yes	characteristic
64	C₂×Dic₅₃	#107	1	1	yes	characteristic
65	#88324	—	8	2	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	—	4	2	no	—
69	#88322	—	2	1	no	—
70	#88322	—	4	2	no	—
71	#88322	#104	1	2	yes	—
72	#7298	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	#340	1	1	yes	characteristic
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	#339	1	1	yes	characteristic
79	C₄×C₅₃	#344	1	1	yes	characteristic
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	—	2	1	no	—
82	Dic₅₃	#361	1	1	yes	characteristic
83	Dic₅₃	#362	1	1	yes	characteristic
84	Dic₅₃	#368	1	2	yes	—
85	#2162	—	53	1	no	—
86	C₄×#108	—	53	1	no	—
87	#1567	—	53	1	no	—
88	C₂×C₅₃	—	4	1	no	—
89	C₂×C₅₃	#1303	1	1	yes	characteristic
90	C₂×C₅₃	#1381	1	1	yes	characteristic
91	C₂×C₅₃	#1387	1	1	yes	characteristic
92	C₂×#108	—	106	1	no	—
93	#423	—	106	1	no	—
94	C₄×M₁₆	—	106	1	no	—
95	C₂×C₄×C₈	—	53	1	no	—
96	C₄⋊C₁₆	—	53	2	no	—
97	C₂²⋊C₁₆	—	106	1	no	—
98	C₄×C₁₆	—	106	1	no	—
99	#341	—	53	1	no	—
100	#341	#6157	1	1	yes	characteristic
101	#335	—	53	2	no	—
102	#335	—	106	1	no	—
103	C₅₃	#7298	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	—	106	1	no	—
106	C₂×M₁₆	—	212	1	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
109	C₂×C₄²	—	53	1	no	—
110	C₂×C₁₆	—	106	2	no	—
111	#108	—	106	4	no	—
112	C₄⋊C₈	—	106	2	no	—
113	C₄⋊C₈	—	212	1	no	—
114	#97	—	212	1	no	—
115	C₄×C₈	—	53	1	no	—
116	C₄×C₈	—	53	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	212	1	no	—
120	C₂²×C₄	—	53	1	no	—
121	C₂²×C₄	—	53	1	no	—
122	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
123	M₁₆	—	2	2	no	—
124	M₁₆	—	4	1	no	—
125	M₁₆	—	106	2	no	—
126	M₁₆	—	424	2	no	—
127	M₁₆	—	212	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	53	1	no	—
131	C₂×C₈	—	53	1	no	—
132	C₂×C₈	—	106	2	no	—
133	C₂×C₈	—	212	2	no	—
134	C₂×C₈	—	212	1	no	—
135	C₂×C₈	C₄×#6157	1	1	yes	characteristic
136	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
137	C₄²	—	53	1	no	—
138	C₄²	—	53	1	no	—
139	C₄²	—	53	1	no	—
140	C₄²	—	53	1	no	—
141	C₁₆	—	212	2	no	—
142	C₂³	#13133280	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	53	1	no	—
148	C₂×C₄	—	53	1	no	—
149	C₂×C₄	—	106	1	no	—
150	C₂×C₄	—	106	2	no	—
151	C₂×C₄	—	106	1	no	—
152	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
153	C₂×C₄	#13133279	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	4	2	no	—
157	C₈	—	53	2	no	—
158	C₈	—	106	1	no	—
159	C₈	—	424	2	no	—
160	C₈	—	212	2	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#16811496	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	53	2	no	—
167	C₄	—	53	1	no	—
168	C₄	—	53	1	no	—
169	C₄	—	106	1	no	—
170	C₄	—	106	1	no	—
171	C₄	#16811494	1	1	yes	characteristic
172	C₄	#16811504	1	1	yes	characteristic
173	C₂	—	4	1	no	—
174	C₂	#20289613	1	1	yes	characteristic
175	C₂	#20289774	1	1	yes	characteristic
176	C₂	#20289791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291073

C₂

yes

maximal, characteristic

#20290272

C₂

yes

maximal, characteristic

#20274616

C₂

yes

maximal, characteristic

#16811626

—

#16811544

C₄

yes

—

—

—

—

C₄

yes

—

C₂×#13133152

—

M₁₆×Dic₅₃

—

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810388

C₄

yes

characteristic

C₅₃×#341

C₄

yes

—

—

—

—

—

C₂²×#88322

D₄

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

#13133152

—

#13133127

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₅₃

SD₁₆

yes

characteristic

C₂²×Dic₅₃

Q₁₆

yes

characteristic

#11719940

—

C₂×#88324

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

D₈

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

#11719910

—

C₂³×C₅₃

#103

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

C₂×Dic₅₃

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

yes

—

#7298

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

Dic₅₃

yes

—

#2162

—

C₄×#108

—

#1567

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

106

—

#423

—

106

—

C₄×M₁₆

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

C₄×C₁₆

—

106

—

#341

—

100

yes

101

—

102

#335

—

106

—

103

C₅₃

#7298

yes

characteristic, complemented, Sylow

104

C₂×M₁₆

—

105

C₂×M₁₆

—

106

—

106

C₂×M₁₆

—

212

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂×#6157

yes

characteristic

109

C₂×C₄²

—

110

C₂×C₁₆

—

106

—

111

#108

—

106

—

112

C₄⋊C₈

—

106

—

113

C₄⋊C₈

—

212

—

114

#97

—

212

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

212

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂²⋊#6157

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

106

—

126

M₁₆

—

424

—

127

M₁₆

—

212

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

106

—

133

C₂×C₈

—

212

—

134

C₂×C₈

—

212

—

135

C₂×C₈

C₄×#6157

yes

characteristic

136

C₂×C₈

C₄⋊#6157

yes

characteristic

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₁₆

—

212

—

142

C₂³

#13133280

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

106

—

150

C₂×C₄

—

106

—

151

C₂×C₄

—

106

—

152

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

153

C₂×C₄

#13133279

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

106

—

159

C₈

—

424

—

160

C₈

—

212

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

#16811496

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

106

—

170

C₄

—

106

—

171

C₄

#16811494

yes

characteristic

172

C₄

#16811504

yes

characteristic

173

C₂

—

174

C₂

#20289613

yes

characteristic

175

C₂

#20289774

yes

characteristic

176

C₂

#20289791

yes

characteristic

	subgroup	G/H
center	C₂²	#16811496
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₅₃×#341	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#103
cosocle	C₂×C₈×Dic₅₃	C₂²
hypercenter	#341	#6157
hypocenter	C₅₃	#7298
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #341

subgroup

G/H

center

C₂²

#16811496

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂²×C₈

C₂×#6157

Fitting

C₅₃×#341

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂×C₈×Dic₅₃

C₂²

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #341

N	H ≅ G/N	type
103. C₅₃	#7298	semidirect

H ≅ G/N

type

103. C₅₃

#7298

semidirect

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

subgroup	index
#20291073	2
#20290272	2
#20274616	2
#7298	53

subgroup

index

Subgroups — #28298777

Conjugacy classes of subgroups

Complements

Maximal subgroups