Group #28328544 — subgroups

Subgroups — #28328544

Order 13568.

494 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 232 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	#20291131	C₂	1	6	yes	maximal, complemented
2	Q₈×#11719910	C₂	1	2	yes	maximal, complemented
3	#20291079	C₂	1	3	yes	maximal
4	#20291074	C₂	1	3	yes	maximal
5	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811544	C₂²	1	6	yes	—
7	C₄⋊#11719910	C₂²	1	6	yes	—
8	#16811518	C₂²	1	1	yes	characteristic
9	#16811518	C₂²	1	3	yes	—
10	#16811512	C₂²	1	6	yes	—
11	C₄×#11719910	C₂²	1	6	yes	—
12	C₄×Q₈×D₅₃	C₄	1	1	yes	characteristic
13	Q₈×#88322	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₀₆	C₄	1	3	yes	—
15	#16810747	C₂²	1	3	yes	—
16	C₄⋊C₈×D₅₃	C₄	1	3	yes	—
17	C₄×C₈×D₅₃	C₂²	1	3	yes	—
18	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×#11719910	C₂³	1	6	yes	—
20	C₂×#11719910	C₂³	1	2	yes	—
21	C₂×Q₈×D₅₃	C₈	1	1	yes	characteristic
22	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
23	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
24	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
25	C₄×Dic₁₀₆	C₂×C₄	1	3	yes	—
26	#13133182	C₈	1	3	yes	—
27	C₄⋊C₄×D₅₃	C₈	1	3	yes	—
28	C₄²×D₅₃	C₂×C₄	1	3	yes	—
29	C₄⋊#88322	C₂×C₄	1	3	yes	—
30	#13133146	C₂×C₄	1	6	yes	—
31	C₄×#88322	C₂³	1	3	yes	—
32	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
33	C₈×Dic₅₃	C₂³	1	3	yes	—
34	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
37	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
38	Q₈×D₅₃	—	2	4	no	—
39	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
40	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
41	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₁₀₆	C₂×C₈	1	3	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
45	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
46	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
47	C₄⋊Dic₅₃	C₂×C₈	1	3	yes	—
48	#11719919	C₂×C₈	1	6	yes	—
49	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
50	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
51	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
52	C₂×#88322	C₂²×C₄	1	3	yes	—
53	C₂×#88322	C₂⁴	1	1	yes	characteristic
54	C₈×D₅₃	—	2	2	no	—
55	C₈×D₅₃	—	4	3	no	—
56	#11719910	—	2	6	no	—
57	#11719910	—	2	2	no	—
58	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
59	Q₈×C₅₃	M₃₂	1	4	yes	—
60	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
61	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
62	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
63	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
64	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
65	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₄×D₅₃	—	2	6	no	—
67	C₄×D₅₃	—	2	1	no	—
68	C₄×D₅₃	—	2	1	no	—
69	C₄×D₅₃	—	2	6	no	—
70	Dic₁₀₆	—	2	6	no	—
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	3	no	—
73	#88322	—	2	3	no	—
74	#88322	2₋¹⁺⁴	1	2	yes	—
75	#33498	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
77	C₂×D₅₃	—	2	1	no	—
78	C₂×D₅₃	—	2	1	no	—
79	C₂×D₅₃	#504	1	2	yes	—
80	C₄×C₅₃	—	2	3	no	—
81	C₄×C₅₃	C₂×M₃₂	1	6	yes	—
82	C₄×C₅₃	#519	1	1	yes	characteristic
83	C₄×C₅₃	#568	1	1	yes	characteristic
84	Dic₅₃	—	2	3	no	—
85	Dic₅₃	—	2	3	no	—
86	Dic₅₃	—	2	1	no	—
87	C₂×C₈×Q₈	—	53	1	no	—
88	#2183	—	53	6	no	—
89	Q₈×C₁₆	—	53	2	no	—
90	#2163	—	53	3	no	—
91	#2162	—	53	3	no	—
92	C₂×C₅₃	#2982	1	1	yes	characteristic
93	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
94	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
95	D₅₃	—	2	2	no	—
96	C₂×C₄×Q₈	—	53	1	no	—
97	C₈×Q₈	—	53	1	no	—
98	C₈×Q₈	—	53	3	no	—
99	C₈×Q₈	—	53	3	no	—
100	C₈×Q₈	#6157	1	1	yes	characteristic
101	C₂×C₄⋊C₈	—	53	3	no	—
102	C₂×C₄×C₈	—	53	3	no	—
103	C₄⋊C₁₆	—	53	6	no	—
104	C₄⋊C₁₆	—	53	6	no	—
105	C₂²⋊C₁₆	—	53	1	no	—
106	C₂²⋊C₁₆	—	53	3	no	—
107	#346	—	53	6	no	—
108	C₄×C₁₆	—	53	6	no	—
109	C₅₃	#33498	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	53	1	no	—
111	C₂²×C₈	—	53	1	no	—
112	C₂²×C₈	—	53	3	no	—
113	C₄×Q₈	—	53	1	no	—
114	C₄×Q₈	—	53	3	no	—
115	C₄×Q₈	—	53	3	no	—
116	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	53	3	no	—
118	C₂×C₄²	—	53	3	no	—
119	C₂×C₁₆	—	53	2	no	—
120	C₂×C₁₆	—	53	6	no	—
121	C₄⋊C₈	—	53	6	no	—
122	C₄⋊C₈	—	53	3	no	—
123	C₄⋊C₈	C₂×#6157	1	3	yes	—
124	C₄×C₈	—	53	3	no	—
125	C₄×C₈	—	53	3	no	—
126	C₄×C₈	—	53	3	no	—
127	C₄×C₈	C₂×#6157	1	3	yes	—
128	C₂×Q₈	—	53	3	no	—
129	C₂×Q₈	—	106	4	no	—
130	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
131	C₂²×C₄	—	53	3	no	—
132	C₂²×C₄	—	53	3	no	—
133	C₂²×C₄	—	53	1	no	—
134	C₂×C₈	—	53	1	no	—
135	C₂×C₈	—	53	3	no	—
136	C₂×C₈	—	106	2	no	—
137	C₂×C₈	—	212	3	no	—
138	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
139	C₂×C₈	C₂²×#6157	1	3	yes	—
140	C₄⋊C₄	—	53	6	no	—
141	C₄⋊C₄	—	53	3	no	—
142	C₄⋊C₄	D₅₃⋊C₈	1	3	yes	—
143	C₄²	—	53	3	no	—
144	C₄²	—	53	3	no	—
145	C₄²	—	53	3	no	—
146	C₄²	C₂²×#6157	1	3	yes	—
147	C₁₆	—	106	2	no	—
148	C₁₆	—	106	6	no	—
149	C₂³	—	53	1	no	—
150	Q₈	—	106	6	no	—
151	Q₈	#13133292	1	4	yes	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	53	3	no	—
154	C₂×C₄	—	53	3	no	—
155	C₂×C₄	—	106	6	no	—
156	C₂×C₄	—	106	6	no	—
157	C₂×C₄	—	106	1	no	—
158	C₂×C₄	—	106	1	no	—
159	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
160	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
161	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	3	no	—
164	C₈	—	53	2	no	—
165	C₈	—	106	3	no	—
166	C₂²	—	53	2	no	—
167	C₂²	—	106	1	no	—
168	C₂²	—	106	1	no	—
169	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
170	C₄	—	2	3	no	—
171	C₄	—	106	1	no	—
172	C₄	—	106	3	no	—
173	C₄	—	106	3	no	—
174	C₄	C₂×#13133292	1	6	yes	—
175	C₄	#16811691	1	1	yes	characteristic
176	C₄	#16811702	1	1	yes	characteristic
177	C₂	—	106	2	no	—
178	C₂	#20291609	1	1	yes	characteristic
179	C₂	C₂²×#13133292	1	1	yes	characteristic
180	C₂	D₅₃⋊#504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291131

C₂

yes

maximal, complemented

Q₈×#11719910

C₂

yes

maximal, complemented

#20291079

C₂

yes

maximal

#20291074

C₂

yes

maximal

C₈×Q₈×D₅₃

C₂

yes

maximal, characteristic

#16811544

C₂²

yes

—

C₄⋊#11719910

C₂²

yes

—

#16811518

C₂²

yes

characteristic

#16811518

C₂²

yes

—

#16811512

C₂²

yes

—

C₄×#11719910

C₂²

yes

—

C₄×Q₈×D₅₃

C₄

yes

characteristic

Q₈×#88322

C₂²

yes

characteristic

C₈×Dic₁₀₆

C₄

yes

—

#16810747

C₂²

yes

—

C₄⋊C₈×D₅₃

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

—

C₈×Q₈×C₅₃

C₄

yes

characteristic

C₂×#11719910

C₂³

yes

—

C₂×#11719910

C₂³

yes

—

C₂×Q₈×D₅₃

C₈

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₂³

yes

characteristic

Q₈×Dic₅₃

C₈

yes

characteristic

C₄×Dic₁₀₆

C₂×C₄

yes

—

#13133182

C₈

yes

—

C₄⋊C₄×D₅₃

C₈

yes

—

C₄²×D₅₃

C₂×C₄

yes

—

C₄⋊#88322

C₂×C₄

yes

—

#13133146

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₈×Dic₅₃

C₂³

yes

—

C₄×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₈

yes

characteristic

Q₈×D₅₃

—

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₄×D₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₆

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

#11719919

C₂×C₈

yes

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

C₂²×C₄

yes

—

C₂×#88322

C₂²×C₄

yes

—

C₂×#88322

C₂⁴

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

#11719910

—

#11719910

—

C₂²×D₅₃

C₂²×C₈

yes

characteristic

Q₈×C₅₃

M₃₂

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂³×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₂²×C₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

2₋¹⁺⁴

yes

—

#33498

—

maximal, Sylow

C₂²×C₅₃

C₂³×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#504

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₃₂

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₈×Q₈

—

#2183

—

Q₈×C₁₆

—

#2163

—

#2162

—

C₂×C₅₃

#2982

yes

characteristic

C₂×C₅₃

C₂²×M₃₂

yes

characteristic

C₂×C₅₃

C₂×#504

yes

characteristic

D₅₃

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

100

C₈×Q₈

#6157

yes

characteristic

101

C₂×C₄⋊C₈

—

102

C₂×C₄×C₈

—

103

C₄⋊C₁₆

—

104

C₄⋊C₁₆

—

105

C₂²⋊C₁₆

—

106

C₂²⋊C₁₆

—

107

#346

—

108

C₄×C₁₆

—

109

C₅₃

#33498

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄×Q₈

—

114

C₄×Q₈

—

115

C₄×Q₈

—

116

C₄×Q₈

C₂×#6157

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

C₂×#6157

yes

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

C₂×#6157

yes

—

128

C₂×Q₈

—

129

C₂×Q₈

—

106

—

130

C₂×Q₈

D₅₃⋊C₈

yes

characteristic

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

106

—

137

C₂×C₈

—

212

—

138

C₂×C₈

C₂²×#6157

yes

characteristic

139

C₂×C₈

C₂²×#6157

yes

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

D₅₃⋊C₈

yes

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂²×#6157

yes

—

147

C₁₆

—

106

—

148

C₁₆

—

106

—

149

C₂³

—

150

Q₈

—

106

—

151

Q₈

#13133292

yes

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

106

—

156

C₂×C₄

—

106

—

157

C₂×C₄

—

106

—

158

C₂×C₄

—

106

—

159

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

160

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

161

C₂×C₄

C₂³×#6157

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

106

—

166

C₂²

—

167

C₂²

—

106

—

168

C₂²

—

106

—

169

C₂²

C₂²×D₅₃⋊C₈

yes

characteristic

170

C₄

—

171

C₄

—

106

—

172

C₄

—

106

—

173

C₄

—

106

—

174

C₄

C₂×#13133292

yes

—

175

C₄

#16811691

yes

characteristic

176

C₄

#16811702

yes

characteristic

177

C₂

—

106

—

178

C₂

#20291609

yes

characteristic

179

C₂

C₂²×#13133292

yes

characteristic

180

C₂

D₅₃⋊#504

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₈×Q₈×C₅₃	C₄
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	#33498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#6157

commutator, socle

C₂²×C₅₃

C₂³×C₈

Fitting

C₈×Q₈×C₅₃

C₄

prime radical

C₂²×D₅₃

C₂²×C₈

C₂⁴

C₈×Q₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #20291131	C₂	semidirect
2. Q₈×#11719910	C₂	semidirect
109. C₅₃	#33498	semidirect

H ≅ G/N

type

1. #20291131

C₂

semidirect

2. Q₈×#11719910

C₂

semidirect

109. C₅₃

#33498

semidirect

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

subgroup	index
#20291131	2
Q₈×#11719910	2
#20291079	2
#20291074	2
C₈×Q₈×D₅₃	2
#33498	53

subgroup

index

C₈×Q₈×D₅₃

Subgroups — #28328544

Conjugacy classes of subgroups

Complements

Maximal subgroups