Group #20276324 — subgroups

Subgroups — #20276324

Order 6784.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133156	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810402	C₂	1	2	yes	maximal, complemented
5	C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₄	1	1	yes	characteristic
7	C₂²×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×D₅₃	—	2	4	no	—
9	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#13133156	—	2	4	no	—
14	C₂×#11719908	C₂²	1	2	yes	complemented
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂²	1	2	yes	complemented
17	C₂³×D₅₃	C₈	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
20	C₂×C₄×D₅₃	—	2	2	no	—
21	C₂×C₄×D₅₃	—	2	2	no	—
22	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
24	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×#88322	C₂×C₄	1	1	yes	characteristic
32	C₈×D₅₃	—	2	2	no	—
33	C₈×D₅₃	—	2	2	no	—
34	C₈×D₅₃	—	2	2	no	—
35	C₈×D₅₃	D₄	1	4	yes	—
36	C₁₆×C₅₃	—	2	2	no	—
37	#11719908	—	2	2	no	—
38	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×D₅₃	—	2	2	no	—
40	C₂²×D₅₃	—	2	2	no	—
41	C₂²×D₅₃	—	2	2	no	—
42	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	2	no	—
49	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
51	C₄×D₅₃	—	2	2	no	—
52	C₄×D₅₃	—	2	1	no	—
53	C₄×D₅₃	—	2	1	no	—
54	C₄×D₅₃	—	2	2	no	—
55	C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
56	C₄×D₅₃	M₁₆	1	2	yes	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	C₂×D₄	1	2	yes	—
59	#88322	—	2	1	no	—
60	#88322	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₅₃	—	2	1	no	—
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	M₃₂	1	2	yes	complemented
64	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×D₅₃	—	2	8	no	—
66	C₂×D₅₃	—	2	1	no	—
67	C₂×D₅₃	—	2	1	no	—
68	C₂×D₅₃	—	2	1	no	—
69	C₂×D₅₃	—	2	1	no	—
70	C₂×D₅₃	C₂²⋊C₈	1	2	yes	—
71	C₂×D₅₃	M₃₂	1	2	yes	complemented
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	Dic₅₃	—	2	1	no	—
76	Dic₅₃	C₂²⋊C₈	1	2	yes	—
77	#2112	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	—	2	2	no	—
79	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
81	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
82	D₅₃	—	2	2	no	—
83	D₅₃	—	2	2	no	—
84	C₂³×C₈	—	53	1	no	—
85	C₂×M₃₂	—	53	2	no	—
86	C₂²⋊C₁₆	—	53	1	no	—
87	C₂²⋊C₁₆	—	53	2	no	—
88	C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
89	C₅₃	#2112	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	—	53	1	no	—
91	C₂²×C₈	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₂²×C₈	—	53	1	no	—
94	C₂²×C₈	—	53	1	no	—
95	C₂²×C₈	—	53	1	no	—
96	C₂²×C₈	—	106	4	no	—
97	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
98	M₃₂	—	106	4	no	—
99	C₂×C₁₆	—	53	2	no	—
100	C₂×C₁₆	C₂×D₅₃	1	2	yes	complemented
101	C₂⁴	—	53	1	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	1	no	—
104	C₂²×C₄	—	53	1	no	—
105	C₂²×C₄	—	53	1	no	—
106	C₂²×C₄	—	53	1	no	—
107	C₂²×C₄	—	106	2	no	—
108	C₂²×C₄	—	106	2	no	—
109	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	4	no	—
113	C₂×C₈	—	53	1	no	—
114	C₂×C₈	—	106	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	—	106	2	no	—
117	C₂×C₈	—	106	2	no	—
118	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	106	2	no	—
122	C₂³	—	53	1	no	—
123	C₂³	—	53	1	no	—
124	C₂³	—	106	2	no	—
125	C₂³	—	106	2	no	—
126	C₂³	—	106	2	no	—
127	C₂³	C₈×D₅₃	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	53	1	no	—
131	C₂×C₄	—	53	2	no	—
132	C₂×C₄	—	53	2	no	—
133	C₂×C₄	—	53	1	no	—
134	C₂×C₄	—	106	2	no	—
135	C₂×C₄	—	106	2	no	—
136	C₂×C₄	—	106	2	no	—
137	C₂×C₄	—	106	1	no	—
138	C₂×C₄	—	106	1	no	—
139	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	53	2	no	—
143	C₈	—	106	1	no	—
144	C₈	D₄×D₅₃	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	53	2	no	—
148	C₂²	—	53	2	no	—
149	C₂²	—	106	8	no	—
150	C₂²	—	106	1	no	—
151	C₂²	—	106	1	no	—
152	C₂²	—	106	1	no	—
153	C₂²	—	106	1	no	—
154	C₂²	#13133156	1	2	yes	complemented
155	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	53	2	no	—
158	C₄	—	106	1	no	—
159	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
160	C₄	M₁₆×D₅₃	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	—	106	2	no	—
163	C₂	—	106	2	no	—
164	C₂	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
165	C₂	M₃₂×D₅₃	1	1	yes	characteristic
166	C₂	C₂×#13133156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133156

C₂

yes

maximal, complemented

C₂²⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810402

C₂

yes

maximal, complemented

C₅₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₄

yes

characteristic

C₂²×#88322

C₄

yes

characteristic

C₂×C₈×D₅₃

—

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133156

—

C₂×#11719908

C₂²

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

complemented

C₂³×D₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

D₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

—

#88322

—

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

M₃₂

yes

complemented

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊C₈

yes

—

C₂×D₅₃

M₃₂

yes

complemented

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

C₂²⋊C₈

yes

—

#2112

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

D₅₃

—

D₅₃

—

C₂³×C₈

—

C₂×M₃₂

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₅₃

yes

characteristic, complemented

C₅₃

#2112

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

M₃₂

—

106

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂×D₅₃

yes

complemented

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

106

—

108

C₂²×C₄

—

106

—

109

C₂²×C₄

C₄×D₅₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

C₄×D₅₃

yes

characteristic

119

C₂×C₈

C₂²×D₅₃

yes

characteristic

120

C₁₆

—

121

C₁₆

—

106

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

106

—

125

C₂³

—

106

—

126

C₂³

—

106

—

127

C₂³

C₈×D₅₃

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

C₈×D₅₃

yes

characteristic

140

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

106

—

144

C₈

D₄×D₅₃

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

106

—

150

C₂²

—

106

—

151

C₂²

—

106

—

152

C₂²

—

106

—

153

C₂²

—

106

—

154

C₂²

#13133156

yes

complemented

155

C₂²

C₂×C₈×D₅₃

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

106

—

159

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

160

C₄

M₁₆×D₅₃

yes

characteristic

161

C₂

—

162

C₂

—

106

—

163

C₂

—

106

—

164

C₂

C₂²⋊C₈×D₅₃

yes

characteristic

165

C₂

M₃₂×D₅₃

yes

characteristic

166

C₂

C₂×#13133156

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₅₃×C₂²⋊C₁₆	C₂
hypercenter	C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#2112
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₅₃×C₂²⋊C₁₆

C₂

hypercenter

C₂²⋊C₁₆

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. C₂×#13133156	C₂	semidirect
3. C₂²⋊#11719908	C₂	semidirect
4. #16810402	C₂	semidirect
5. C₅₃×C₂²⋊C₁₆	C₂	semidirect
14. C₂×#11719908	C₂²	semidirect
16. C₂×C₁₆×C₅₃	C₂²	semidirect
63. C₂²×C₅₃	M₃₂	semidirect
71. C₂×D₅₃	M₃₂	semidirect
88. C₂²⋊C₁₆	D₅₃	semidirect
89. C₅₃	#2112	semidirect
100. C₂×C₁₆	C₂×D₅₃	semidirect
154. C₂²	#13133156	semidirect

H ≅ G/N

type

2. C₂×#13133156

C₂

semidirect

3. C₂²⋊#11719908

C₂

semidirect

4. #16810402

C₂

semidirect

5. C₅₃×C₂²⋊C₁₆

C₂

semidirect

14. C₂×#11719908

C₂²

semidirect

16. C₂×C₁₆×C₅₃

C₂²

semidirect

63. C₂²×C₅₃

M₃₂

semidirect

71. C₂×D₅₃

M₃₂

semidirect

88. C₂²⋊C₁₆

D₅₃

semidirect

89. C₅₃

#2112

semidirect

100. C₂×C₁₆

C₂×D₅₃

semidirect

154. C₂²

#13133156

semidirect

Subgroups — #20276324

Conjugacy classes of subgroups

Complements