Group #20274825 — subgroups

Subgroups — #20274825

Order 6784.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16811097	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133133	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133133	C₂	1	1	yes	maximal, characteristic
4	#16810395	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810394	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810393	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#344	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	C₄×D₂₁₂	—	2	2	no	—
12	#13133179	—	2	2	no	—
13	#13133176	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×D₅₃	C₄	1	4	yes	—
15	#13133133	—	2	2	no	—
16	#13133133	—	2	2	no	—
17	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	#117×C₅₃	C₂²	1	1	yes	characteristic
20	C₂³×D₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×D₂₁₂	—	2	1	no	—
23	C₂×D₂₁₂	—	2	1	no	—
24	C₂×D₂₁₂	—	2	1	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	D₄	1	2	yes	—
27	C₂×D₂₁₂	D₄	1	2	yes	—
28	C₂×C₄×D₅₃	—	2	4	no	—
29	M₁₆×C₅₃	—	2	1	no	—
30	M₁₆×C₅₃	—	2	1	no	—
31	M₁₆×C₅₃	—	2	1	no	—
32	M₁₆×C₅₃	—	2	1	no	—
33	C₂×C₈×C₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	—	2	1	no	—
35	C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₄²×C₅₃	—	2	1	no	—
38	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
39	#11719921	—	2	4	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄×Dic₅₃	—	2	1	no	—
42	#11719917	—	2	1	no	—
43	#11719917	—	2	1	no	—
44	#11719917	—	2	1	no	—
45	#11719917	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₂×#88322	—	2	1	no	—
48	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×D₅₃	—	2	2	no	—
50	C₂²×D₅₃	—	2	2	no	—
51	C₂²×D₅₃	—	8	2	no	—
52	C₂²×D₅₃	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂×Dic₅₃	—	2	2	no	—
61	D₂₁₂	—	2	2	no	—
62	D₂₁₂	—	2	2	no	—
63	D₂₁₂	—	2	2	no	—
64	D₂₁₂	—	2	2	no	—
65	C₄×D₅₃	—	8	4	no	—
66	C₈×C₅₃	—	4	1	no	—
67	C₈×C₅₃	—	4	1	no	—
68	#88322	—	4	1	no	—
69	#88322	—	4	1	no	—
70	C₂²×C₅₃	—	4	1	no	—
71	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₅₃	#117	1	1	yes	characteristic
73	C₂²×C₅₃	#117	1	1	yes	characteristic
74	C₂×D₅₃	—	2	4	no	—
75	C₂×D₅₃	—	4	4	no	—
76	C₂×D₅₃	—	8	2	no	—
77	C₂×D₅₃	—	8	2	no	—
78	C₄×C₅₃	—	4	2	no	—
79	C₄×C₅₃	#123	1	1	yes	characteristic
80	C₄×C₅₃	#123	1	1	yes	characteristic
81	C₄×C₅₃	#123	1	1	yes	characteristic
82	C₄×C₅₃	#123	1	1	yes	characteristic
83	Dic₅₃	—	4	2	no	—
84	#1829	—	53	1	no	maximal, Sylow
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	#381	1	1	yes	characteristic
89	D₅₃	—	4	4	no	—
90	#518	—	53	1	no	—
91	C₂×#100	—	53	1	no	—
92	C₂×#100	—	53	1	no	—
93	#344	—	53	1	no	—
94	#344	—	53	1	no	—
95	#344	—	53	1	no	—
96	#344	D₅₃	1	1	yes	characteristic, complemented
97	C₅₃	#1829	1	1	yes	characteristic, complemented, Sylow
98	C₂²×D₄	—	53	1	no	—
99	C₂×M₁₆	—	53	1	no	—
100	C₂×M₁₆	—	53	1	no	—
101	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
102	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
103	C₄×D₄	—	106	2	no	—
104	C₄×D₄	—	106	2	no	—
105	#117	—	53	1	no	—
106	#117	C₂×D₅₃	1	1	yes	characteristic
107	C₂×C₂²⋊C₄	—	53	4	no	—
108	#100	—	106	2	no	—
109	#100	—	106	2	no	—
110	C₂⁴	—	53	2	no	—
111	C₂×D₄	—	53	2	no	—
112	C₂×D₄	—	53	2	no	—
113	C₂×D₄	—	106	1	no	—
114	C₂×D₄	—	106	1	no	—
115	C₂×D₄	—	106	1	no	—
116	C₂×D₄	—	106	1	no	—
117	C₂²×C₄	—	106	4	no	—
118	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
119	M₁₆	—	2	1	no	—
120	M₁₆	—	2	1	no	—
121	M₁₆	—	2	1	no	—
122	M₁₆	—	2	1	no	—
123	M₁₆	—	106	1	no	—
124	M₁₆	—	106	1	no	—
125	M₁₆	—	106	1	no	—
126	M₁₆	—	106	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	106	1	no	—
130	C₂×C₈	—	106	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	106	1	no	—
133	C₂²⋊C₄	—	53	2	no	—
134	C₂²⋊C₄	—	106	4	no	—
135	C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
136	C₄²	—	2	1	no	—
137	C₄²	—	106	1	no	—
138	C₂³	—	53	4	no	—
139	C₂³	—	106	2	no	—
140	C₂³	—	106	2	no	—
141	C₂³	—	424	2	no	—
142	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
143	D₄	—	106	2	no	—
144	D₄	—	106	2	no	—
145	D₄	—	106	2	no	—
146	D₄	—	106	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	—	424	4	no	—
150	C₂×C₄	#11719945	1	1	yes	characteristic
151	C₂×C₄	#11719945	1	1	yes	characteristic
152	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
153	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	#11719947	1	1	yes	characteristic
155	C₂×C₄	#11719949	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	4	1	no	—
158	C₈	—	212	1	no	—
159	C₈	—	212	1	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	106	4	no	—
162	C₂²	—	424	2	no	—
163	C₂²	—	424	2	no	—
164	C₂²	—	212	4	no	—
165	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
166	C₂²	#13133174	1	1	yes	characteristic
167	C₂²	#13133175	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	212	2	no	—
170	C₄	#13133199	1	1	yes	characteristic
171	C₄	#13133199	1	1	yes	characteristic
172	C₄	#13133202	1	1	yes	characteristic
173	C₄	#13133202	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	212	4	no	—
178	C₂	#16810485	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811097

C₂

yes

maximal, characteristic

C₂×#13133133

C₂

yes

maximal, characteristic

C₂×#13133133

C₂

yes

maximal, characteristic

#16810395

C₂

yes

maximal, characteristic, complemented

#16810394

C₂

yes

maximal, characteristic, complemented

#16810393

C₂

yes

maximal, characteristic, complemented

C₅₃×#344

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

C₄×D₂₁₂

—

#13133179

—

#13133176

C₂²

yes

characteristic

C₂²⋊C₄×D₅₃

C₄

yes

—

#13133133

—

#13133133

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

—

C₂×D₂₁₂

D₄

yes

—

C₂×C₄×D₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1829

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#381

yes

characteristic

D₅₃

—

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1829

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×M₁₆

—

100

C₂×M₁₆

—

101

C₂×M₁₆

C₂×D₅₃

yes

characteristic

102

C₂×M₁₆

C₂×D₅₃

yes

characteristic

103

C₄×D₄

—

106

—

104

C₄×D₄

—

106

—

105

#117

—

106

#117

C₂×D₅₃

yes

characteristic

107

C₂×C₂²⋊C₄

—

108

#100

—

106

—

109

#100

—

106

—

110

C₂⁴

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

106

—

114

C₂×D₄

—

106

—

115

C₂×D₄

—

106

—

116

C₂×D₄

—

106

—

117

C₂²×C₄

—

106

—

118

C₂²×C₄

C₂²×D₅₃

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

106

—

124

M₁₆

—

106

—

125

M₁₆

—

106

—

126

M₁₆

—

106

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

106

—

130

C₂×C₈

—

106

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

106

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

106

—

135

C₂²⋊C₄

C₄×D₅₃

yes

—

136

C₄²

—

137

C₄²

—

106

—

138

C₂³

—

139

C₂³

—

106

—

140

C₂³

—

106

—

141

C₂³

—

424

—

142

C₂³

C₂×C₄×D₅₃

yes

characteristic

143

D₄

—

106

—

144

D₄

—

106

—

145

D₄

—

106

—

146

D₄

—

106

—

147

C₂×C₄

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

424

—

150

C₂×C₄

#11719945

yes

characteristic

151

C₂×C₄

#11719945

yes

characteristic

152

C₂×C₄

D₄×D₅₃

yes

characteristic

153

C₂×C₄

D₄×D₅₃

yes

characteristic

154

C₂×C₄

#11719947

yes

characteristic

155

C₂×C₄

#11719949

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

212

—

159

C₈

—

212

—

160

C₂²

—

161

C₂²

—

106

—

162

C₂²

—

424

—

163

C₂²

—

424

—

164

C₂²

—

212

—

165

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

166

C₂²

#13133174

yes

characteristic

167

C₂²

#13133175

yes

characteristic

168

C₄

—

169

C₄

—

212

—

170

C₄

#13133199

yes

characteristic

171

C₄

#13133199

yes

characteristic

172

C₄

#13133202

yes

characteristic

173

C₄

#13133202

yes

characteristic

174

C₂

—

175

C₂

—

176

C₂

—

177

C₂

—

212

—

178

C₂

#16810485

yes

characteristic

	subgroup	G/H
center	C₂	#16810485
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#344	C₂
socle	C₂×C₅₃	#381
hypercenter	#344	D₅₃
hypocenter	C₅₃	#1829
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₂

#16810485

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#344

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
4. #16810395	C₂	semidirect
5. #16810394	C₂	semidirect
6. #16810393	C₂	semidirect
7. C₅₃×#344	C₂	semidirect
96. #344	D₅₃	semidirect
97. C₅₃	#1829	semidirect

H ≅ G/N

type

4. #16810395

C₂

semidirect

5. #16810394

C₂

semidirect

6. #16810393

C₂

semidirect

7. C₅₃×#344

C₂

semidirect

96. #344

D₅₃

semidirect

97. C₅₃

#1829

semidirect

Subgroups — #20274825

Conjugacy classes of subgroups

Complements