Group #20285062 — subgroups

Subgroups — #20285062

Order 6784.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16811179	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811174	C₂	1	1	yes	maximal, characteristic
3	#16810991	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810989	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810988	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810984	C₂	1	4	yes	maximal, complemented
7	#16810618	C₂	1	4	yes	maximal, complemented
8	#97×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719949	C₂²	1	2	yes	—
11	C₂×C₈×D₅₃	C₂²	1	2	yes	—
12	#13133216	C₂²	1	2	yes	—
13	#13133214	C₂²	1	1	yes	characteristic
14	#13133209	C₂²	1	1	yes	characteristic
15	#13133206	C₂²	1	2	yes	—
16	#13133195	—	2	2	no	—
17	#13133192	—	2	2	no	—
18	#13133182	—	2	2	no	—
19	#13133179	—	2	2	no	—
20	#13133175	C₂²	1	2	yes	—
21	C₄²×D₅₃	C₂²	1	1	yes	characteristic
22	#13133141	C₂²	1	4	yes	—
23	#13133139	C₂²	1	4	yes	—
24	#13133137	C₂²	1	4	yes	—
25	#13133128	C₂²	1	1	yes	characteristic
26	#13133127	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
28	#125×C₅₃	C₂²	1	1	yes	characteristic
29	#103×C₅₃	C₂²	1	4	yes	—
30	#97×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
32	#11719949	—	2	8	no	—
33	C₂×D₂₁₂	—	2	2	no	—
34	C₂×D₂₁₂	—	2	2	no	—
35	C₂×D₂₁₂	C₂³	1	2	yes	—
36	C₂×C₄×D₅₃	—	2	2	no	—
37	C₂×C₄×D₅₃	D₄	1	2	yes	—
38	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
39	C₂×Dic₁₀₆	—	2	2	no	—
40	C₂×C₈×C₅₃	C₂³	1	2	yes	—
41	C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
44	C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#11719921	—	2	2	no	—
46	#11719921	—	2	4	no	—
47	C₄⋊Dic₅₃	C₂³	1	2	yes	—
48	#11719919	—	2	2	no	—
49	#11719919	—	2	2	no	—
50	C₄×Dic₅₃	—	2	2	no	—
51	C₄×Dic₅₃	D₄	1	2	yes	—
52	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
53	C₂×#88322	C₂³	1	2	yes	—
54	C₈×D₅₃	—	2	4	no	—
55	C₂²×D₅₃	—	2	2	no	—
56	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₅₃	—	2	2	no	—
63	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
64	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
65	D₂₁₂	—	2	4	no	—
66	D₂₁₂	—	4	4	no	—
67	D₂₁₂	—	4	4	no	—
68	C₄×D₅₃	—	4	2	no	—
69	C₄×D₅₃	—	4	4	no	—
70	C₄×D₅₃	#41	1	4	yes	—
71	Dic₁₀₆	—	4	4	no	—
72	C₈×C₅₃	—	2	2	no	—
73	#88322	—	2	2	no	—
74	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×D₅₃	—	2	2	no	—
76	C₂×D₅₃	—	4	2	no	—
77	C₂×D₅₃	—	4	4	no	—
78	C₂×D₅₃	#124	1	2	yes	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	4	2	no	—
81	C₄×C₅₃	—	4	2	no	—
82	C₄×C₅₃	C₂×#41	1	2	yes	—
83	Dic₅₃	—	2	2	no	—
84	Dic₅₃	—	4	2	no	—
85	Dic₅₃	#124	1	2	yes	—
86	#3135	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
88	C₂×C₅₃	#575	1	2	yes	—
89	D₅₃	—	2	2	no	—
90	D₅₃	—	8	2	no	—
91	#533	—	53	1	no	—
92	#532	—	53	1	no	—
93	#491	—	53	1	no	—
94	#491	D₅₃	1	1	yes	characteristic, complemented
95	#490	—	53	1	no	—
96	#490	—	53	1	no	—
97	#489	—	53	4	no	—
98	#416	—	53	4	no	—
99	C₂×#97	—	53	1	no	—
100	C₅₃	#3135	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	—	53	2	no	—
102	C₂²×C₈	—	53	2	no	—
103	C₄⋊Q₈	—	53	2	no	—
104	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
105	#127	—	53	1	no	—
106	#125	—	53	1	no	—
107	#125	C₂×D₅₃	1	1	yes	characteristic
108	#124	—	53	2	no	—
109	C₂²⋊Q₈	—	106	2	no	—
110	#121	—	106	2	no	—
111	C₄×Q₈	—	106	2	no	—
112	C₄×D₄	—	106	2	no	—
113	#117	—	53	2	no	—
114	C₂×C₄²	—	53	1	no	—
115	#103	—	53	4	no	—
116	#103	C₂×D₅₃	1	4	yes	—
117	#102	—	53	4	no	—
118	#102	—	53	4	no	—
119	#97	—	53	2	no	—
120	#97	—	53	1	no	—
121	#97	C₂×D₅₃	1	1	yes	characteristic
122	#41	—	106	8	no	—
123	C₂×Q₈	—	106	2	no	—
124	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
125	C₂×D₄	—	53	2	no	—
126	C₂×D₄	—	106	2	no	—
127	C₂×D₄	—	106	2	no	—
128	C₂²×C₄	—	53	1	no	—
129	C₂²×C₄	—	53	2	no	—
130	C₂²×C₄	—	106	2	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	106	4	no	—
133	C₂×C₈	C₂²×D₅₃	1	2	yes	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	53	2	no	—
137	C₄⋊C₄	—	106	2	no	—
138	C₄⋊C₄	—	106	2	no	—
139	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
140	C₂²⋊C₄	—	106	4	no	—
141	C₂²⋊C₄	—	106	2	no	—
142	C₄²	—	53	2	no	—
143	C₄²	—	53	1	no	—
144	C₄²	—	106	2	no	—
145	C₄²	C₂²×D₅₃	1	1	yes	characteristic
146	C₂³	—	53	1	no	—
147	C₂³	—	106	2	no	—
148	Q₈	—	2	4	no	—
149	Q₈	—	212	4	no	—
150	D₄	—	106	4	no	—
151	D₄	—	212	4	no	—
152	D₄	—	212	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	53	1	no	—
156	C₂×C₄	—	53	4	no	—
157	C₂×C₄	—	53	2	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	—	212	2	no	—
160	C₂×C₄	—	212	4	no	—
161	C₂×C₄	D₄×D₅₃	1	2	yes	—
162	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	106	2	no	—
165	C₂²	—	53	2	no	—
166	C₂²	—	106	2	no	—
167	C₂²	—	212	2	no	—
168	C₂²	—	212	4	no	—
169	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	53	2	no	—
174	C₄	—	106	2	no	—
175	C₄	—	212	2	no	—
176	C₄	#41×D₅₃	1	2	yes	—
177	C₂	—	106	2	no	—
178	C₂	—	424	2	no	—
179	C₂	#124×D₅₃	1	1	yes	characteristic
180	C₂	#16811422	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811179

C₂

yes

maximal, characteristic, complemented

#16811174

C₂

yes

maximal, characteristic

#16810991

C₂

yes

maximal, characteristic, complemented

#16810989

C₂

yes

maximal, characteristic, complemented

#16810988

C₂

yes

maximal, characteristic, complemented

#16810984

C₂

yes

maximal, complemented

#16810618

C₂

yes

maximal, complemented

#97×D₅₃

C₂

yes

maximal, characteristic, complemented

C₅₃×#491

C₂

yes

maximal, characteristic, complemented

C₂×#11719949

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

—

#13133216

C₂²

yes

—

#13133214

C₂²

yes

characteristic

#13133209

C₂²

yes

characteristic

#13133206

C₂²

yes

—

—

—

—

—

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133141

C₂²

yes

—

#13133139

C₂²

yes

—

#13133137

C₂²

yes

—

#13133128

C₂²

yes

characteristic

#13133127

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#125×C₅₃

C₂²

yes

characteristic

#103×C₅₃

C₂²

yes

—

#97×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

#11719949

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

#11719921

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

C₈×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#124

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#124

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

D₅₃

—

—

—

—

D₅₃

yes

characteristic, complemented

—

—

—

—

C₂×#97

—

100

C₅₃

#3135

yes

characteristic, complemented, Sylow

101

C₂×#41

—

102

C₂²×C₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

105

#127

—

106

#125

—

107

#125

C₂×D₅₃

yes

characteristic

108

#124

—

109

C₂²⋊Q₈

—

106

—

110

#121

—

106

—

111

C₄×Q₈

—

106

—

112

C₄×D₄

—

106

—

113

#117

—

114

C₂×C₄²

—

115

#103

—

116

#103

C₂×D₅₃

yes

—

117

#102

—

118

#102

—

119

#97

—

120

#97

—

121

#97

C₂×D₅₃

yes

characteristic

122

#41

—

106

—

123

C₂×Q₈

—

106

—

124

C₂×Q₈

C₂²×D₅₃

yes

—

125

C₂×D₄

—

126

C₂×D₄

—

106

—

127

C₂×D₄

—

106

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

106

—

131

C₂×C₈

—

132

C₂×C₈

—

106

—

133

C₂×C₈

C₂²×D₅₃

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

106

—

138

C₄⋊C₄

—

106

—

139

C₄⋊C₄

C₂²×D₅₃

yes

—

140

C₂²⋊C₄

—

106

—

141

C₂²⋊C₄

—

106

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

106

—

145

C₄²

C₂²×D₅₃

yes

characteristic

146

C₂³

—

147

C₂³

—

106

—

148

Q₈

—

149

Q₈

—

212

—

150

D₄

—

106

—

151

D₄

—

212

—

152

D₄

—

212

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

106

—

159

C₂×C₄

—

212

—

160

C₂×C₄

—

212

—

161

C₂×C₄

D₄×D₅₃

yes

—

162

C₂×C₄

C₂³×D₅₃

yes

characteristic

163

C₈

—

164

C₈

—

106

—

165

C₂²

—

166

C₂²

—

106

—

167

C₂²

—

212

—

168

C₂²

—

212

—

169

C₂²

C₂×D₄×D₅₃

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

106

—

175

C₄

—

212

—

176

C₄

#41×D₅₃

yes

—

177

C₂

—

106

—

178

C₂

—

424

—

179

C₂

#124×D₅₃

yes

characteristic

180

C₂

#16811422

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#491	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#491	D₅₃
hypocenter	C₅₃	#3135
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#491

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
1. #16811179	C₂	semidirect
3. #16810991	C₂	semidirect
4. #16810989	C₂	semidirect
5. #16810988	C₂	semidirect
6. #16810984	C₂	semidirect
7. #16810618	C₂	semidirect
8. #97×D₅₃	C₂	semidirect
9. C₅₃×#491	C₂	semidirect
94. #491	D₅₃	semidirect
100. C₅₃	#3135	semidirect

H ≅ G/N

type

1. #16811179

C₂

semidirect

3. #16810991

C₂

semidirect

4. #16810989

C₂

semidirect

5. #16810988

C₂

semidirect

6. #16810984

C₂

semidirect

7. #16810618

C₂

semidirect

8. #97×D₅₃

C₂

semidirect

9. C₅₃×#491

C₂

semidirect

94. #491

D₅₃

semidirect

100. C₅₃

#3135

semidirect

Subgroups — #20285062

Conjugacy classes of subgroups

Complements