Group #28328536 — subgroups

Subgroups — #28328536

Order 13568.

526 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 270 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	#20291129	C₂	1	8	yes	maximal, complemented
2	#20291073	C₂	1	4	yes	maximal
3	#20291072	C₂	1	2	yes	maximal
4	C₂×#16810747	C₂	1	1	yes	maximal, characteristic
5	#16811544	C₂²	1	8	yes	—
6	C₄⋊#11719910	C₂²	1	4	yes	—
7	C₂²⋊#11719910	C₂²	1	4	yes	—
8	#16811509	C₂²	1	8	yes	—
9	C₄×#11719910	C₂²	1	4	yes	—
10	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
11	#16810747	C₄	1	4	yes	—
12	#16810747	C₂²	1	4	yes	—
13	C₂×#13133146	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#11719910	C₂³	1	8	yes	—
18	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂³	1	1	yes	characteristic
21	C₄×Dic₁₀₆	C₈	1	4	yes	—
22	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
24	C₂×#11719919	C₈	1	2	yes	—
25	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
26	#13133146	C₂×C₄	1	8	yes	—
27	C₄×#88322	C₂×C₄	1	2	yes	—
28	C₄×#88322	C₂³	1	2	yes	—
29	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
30	C₈×Dic₅₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
38	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
39	C₂×Dic₁₀₆	C₁₆	1	8	yes	complemented
40	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
43	C₄²×C₅₃	C₂×C₈	1	2	yes	—
44	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
45	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
46	#11719919	C₂×C₈	1	8	yes	—
47	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
48	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
49	C₂×#88322	—	2	2	no	—
50	C₂×#88322	—	4	1	no	—
51	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
52	C₂×#88322	C₂²×C₄	1	2	yes	—
53	C₂×#88322	C₂⁴	1	1	yes	characteristic
54	#11719910	—	2	8	no	—
55	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
60	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
61	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
64	C₂×Dic₅₃	—	4	2	no	—
65	C₂×Dic₅₃	C₂×C₁₆	1	8	yes	—
66	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
67	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
68	Dic₁₀₆	C₂×C₁₆	1	16	yes	—
69	C₈×C₅₃	—	2	4	no	—
70	#88322	—	2	1	no	—
71	#88322	—	2	2	no	—
72	#88322	2₋¹⁺⁴	1	2	yes	—
73	#33496	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
77	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
78	C₄×C₅₃	—	2	2	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	C₂²×C₁₆	1	4	yes	—
81	C₄×C₅₃	#519	1	1	yes	characteristic
82	C₄×C₅₃	#568	1	1	yes	characteristic
83	Dic₅₃	—	2	4	no	—
84	Dic₅₃	C₂²×C₁₆	1	8	yes	—
85	C₂×C₈×Q₈	—	53	1	no	—
86	Q₈×C₁₆	—	53	8	no	—
87	#2162	—	53	2	no	—
88	#2162	—	53	4	no	—
89	C₂×C₅₃	—	2	2	no	—
90	C₂×C₅₃	#2982	1	1	yes	characteristic
91	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
92	C₂×C₅₃	#3407	1	1	yes	characteristic
93	C₂×C₄×Q₈	—	53	1	no	—
94	C₈×Q₈	—	53	4	no	—
95	C₈×Q₈	—	53	4	no	—
96	C₂×C₄⋊C₈	—	53	2	no	—
97	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
98	C₂×C₄×C₈	—	53	2	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	C₄⋊C₁₆	—	53	4	no	—
101	C₄⋊C₁₆	—	53	8	no	—
102	C₂²⋊C₁₆	—	53	4	no	—
103	C₄×C₁₆	—	53	4	no	—
104	C₄×C₁₆	—	53	8	no	—
105	C₅₃	#33496	1	1	yes	characteristic, complemented, Sylow
106	C₂²×Q₈	—	53	1	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂²×C₈	C₂×#6157	1	2	yes	—
110	C₄×Q₈	—	53	4	no	—
111	C₄×Q₈	—	53	4	no	—
112	C₂×C₄⋊C₄	—	53	2	no	—
113	C₂×C₄⋊C₄	—	53	1	no	—
114	C₂×C₄²	—	53	2	no	—
115	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
116	C₂×C₁₆	—	53	8	no	—
117	C₄⋊C₈	—	53	8	no	—
118	C₄⋊C₈	C₂×#6157	1	4	yes	—
119	C₄×C₈	—	53	4	no	—
120	C₄×C₈	—	53	4	no	—
121	C₄×C₈	—	53	2	no	—
122	C₄×C₈	—	53	2	no	—
123	C₂×Q₈	—	53	8	no	—
124	C₂×Q₈	—	53	4	no	—
125	C₂²×C₄	—	53	2	no	—
126	C₂²×C₄	—	53	2	no	—
127	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
128	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
129	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	53	1	no	—
132	C₂×C₈	—	53	1	no	—
133	C₂×C₈	—	53	2	no	—
134	C₂×C₈	—	106	2	no	—
135	C₂×C₈	—	212	1	no	—
136	C₂×C₈	C₂²×#6157	1	4	yes	—
137	C₄⋊C₄	—	53	8	no	—
138	C₄⋊C₄	—	53	4	no	—
139	C₄²	—	53	4	no	—
140	C₄²	—	53	4	no	—
141	C₄²	D₅₃⋊C₈	1	2	yes	—
142	C₄²	C₂²×#6157	1	2	yes	—
143	C₁₆	—	106	8	no	—
144	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
145	Q₈	—	53	16	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	53	8	no	—
150	C₂×C₄	—	53	4	no	—
151	C₂×C₄	—	53	4	no	—
152	C₂×C₄	—	212	2	no	—
153	C₂×C₄	D₅₃⋊C₁₆	1	4	yes	—
154	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
155	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
156	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
157	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	53	2	no	—
160	C₈	—	106	1	no	—
161	C₈	—	106	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×D₅₃⋊C₁₆	1	2	yes	—
165	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	53	8	no	—
169	C₄	—	106	4	no	—
170	C₄	C₂×D₅₃⋊C₁₆	1	4	yes	—
171	C₄	#16811691	1	1	yes	characteristic
172	C₄	#16811701	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#20291610	1	1	yes	characteristic
175	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic
176	C₂	#20291684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291129

C₂

yes

maximal, complemented

#20291073

C₂

yes

maximal

#20291072

C₂

yes

maximal

C₂×#16810747

C₂

yes

maximal, characteristic

#16811544

C₂²

yes

—

C₄⋊#11719910

C₂²

yes

—

C₂²⋊#11719910

C₂²

yes

—

#16811509

C₂²

yes

—

C₄×#11719910

C₂²

yes

—

C₂×C₄×Dic₁₀₆

C₄

yes

characteristic

#16810747

C₄

yes

—

#16810747

C₂²

yes

—

C₂×#13133146

C₄

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×#11719910

C₂³

yes

—

C₂²×Dic₁₀₆

C₈

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂³

yes

characteristic

C₄×Dic₁₀₆

C₈

yes

—

C₄×Dic₁₀₆

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₃

C₈

yes

characteristic

C₂×#11719919

C₈

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

#13133146

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₈×Dic₅₃

C₂³

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₁₀₆

C₁₆

yes

complemented

C₂×Dic₁₀₆

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄²×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

C₂²×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂²×C₄

yes

characteristic

C₂×#88322

C₂²×C₄

yes

—

C₂×#88322

C₂⁴

yes

characteristic

#11719910

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂³×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₂×Dic₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₂²×C₈

yes

—

Dic₁₀₆

C₂×C₁₆

yes

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

2₋¹⁺⁴

yes

—

#33496

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₁₆

yes

—

C₂²×C₅₃

C₂³×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₁₆

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

C₂²×C₁₆

yes

—

C₂×C₈×Q₈

—

Q₈×C₁₆

—

#2162

—

#2162

—

C₂×C₅₃

—

C₂×C₅₃

#2982

yes

characteristic

C₂×C₅₃

C₂³×C₁₆

yes

characteristic

C₂×C₅₃

#3407

yes

characteristic

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#6157

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

100

C₄⋊C₁₆

—

101

C₄⋊C₁₆

—

102

C₂²⋊C₁₆

—

103

C₄×C₁₆

—

104

C₄×C₁₆

—

105

C₅₃

#33496

yes

characteristic, complemented, Sylow

106

C₂²×Q₈

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

C₂×#6157

yes

—

110

C₄×Q₈

—

111

C₄×Q₈

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂×#6157

yes

characteristic

116

C₂×C₁₆

—

117

C₄⋊C₈

—

118

C₄⋊C₈

C₂×#6157

yes

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

D₅₃⋊C₈

yes

characteristic

128

C₂²×C₄

D₅₃⋊C₈

yes

characteristic

129

C₂²×C₄

C₂²×#6157

yes

characteristic

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

106

—

135

C₂×C₈

—

212

—

136

C₂×C₈

C₂²×#6157

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

—

140

C₄²

—

141

C₄²

D₅₃⋊C₈

yes

—

142

C₄²

C₂²×#6157

yes

—

143

C₁₆

—

106

—

144

C₂³

C₂×D₅₃⋊C₈

yes

characteristic

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

212

—

153

C₂×C₄

D₅₃⋊C₁₆

yes

—

154

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

155

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

156

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

157

C₂×C₄

C₂³×#6157

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

—

106

—

161

C₈

—

106

—

162

C₂²

—

163

C₂²

—

164

C₂²

C₂×D₅₃⋊C₁₆

yes

—

165

C₂²

C₂²×D₅₃⋊C₈

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

106

—

170

C₄

C₂×D₅₃⋊C₁₆

yes

—

171

C₄

#16811691

yes

characteristic

172

C₄

#16811701

yes

characteristic

173

C₂

—

174

C₂

#20291610

yes

characteristic

175

C₂

C₂²×D₅₃⋊C₁₆

yes

characteristic

176

C₂

#20291684

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	#33496
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#6157

commutator

C₂×C₅₃

C₂³×C₁₆

Fitting

C₂×C₄⋊C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂³×C₈

prime radical

C₂³×C₅₃

C₂²×C₈

C₂⁴

C₂×C₄⋊C₈

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
1. #20291129	C₂	semidirect
39. C₂×Dic₁₀₆	C₁₆	semidirect
105. C₅₃	#33496	semidirect

H ≅ G/N

type

1. #20291129

C₂

semidirect

39. C₂×Dic₁₀₆

C₁₆

semidirect

105. C₅₃

#33496

semidirect

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

subgroup	index
#20291129	2
#20291073	2
#20291072	2
C₂×#16810747	2
#33496	53

subgroup

index

Subgroups — #28328536

Conjugacy classes of subgroups

Complements

Maximal subgroups