Group #20275579 — subgroups

Subgroups — #20275579

Order 6784.

384 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 109 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×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133137	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810524	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	2	yes	—
7	C₂×D₄₂₄	—	2	2	no	—
8	C₂×D₄₂₄	—	2	2	no	—
9	C₂×D₄₂₄	C₄	1	4	yes	complemented
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂×#11719921	—	2	4	no	—
12	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	#13133137	—	2	4	no	—
14	C₈×Dic₅₃	—	2	2	no	—
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	#107×C₅₃	—	2	2	no	—
18	C₂³×D₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₃	D₄	1	2	yes	—
22	C₂×D₂₁₂	—	2	2	no	—
23	C₂×D₂₁₂	—	2	2	no	—
24	C₂×D₂₁₂	—	2	2	no	—
25	C₂×D₂₁₂	—	2	2	no	—
26	C₂×D₂₁₂	—	2	2	no	—
27	C₂×D₂₁₂	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₅₃	D₄	1	2	yes	—
30	C₂×C₈×C₅₃	D₄	1	2	yes	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊C₄×C₅₃	—	2	2	no	—
33	#11719921	—	4	8	no	—
34	C₄×Dic₅₃	—	2	1	no	—
35	C₄×Dic₅₃	—	2	1	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	D₄	1	2	yes	—
39	D₄₂₄	—	2	8	no	—
40	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₅₃	—	2	4	no	—
42	C₂²×D₅₃	—	4	2	no	—
43	C₂²×D₅₃	—	4	2	no	—
44	C₂²×D₅₃	—	4	4	no	—
45	C₂²×D₅₃	—	4	4	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	—	4	4	no	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	2	2	no	—
56	C₂×Dic₅₃	D₈	1	4	yes	complemented
57	D₂₁₂	—	2	4	no	—
58	D₂₁₂	—	2	4	no	—
59	D₂₁₂	—	4	2	no	—
60	D₂₁₂	—	4	2	no	—
61	C₈×C₅₃	C₂²⋊C₄	1	4	yes	complemented
62	#88322	—	2	2	no	—
63	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₅₃	#121	1	1	yes	characteristic
65	C₂²×C₅₃	#127	1	1	yes	characteristic
66	C₂²×C₅₃	C₂×D₈	1	2	yes	—
67	C₂²×C₅₃	#135	1	2	yes	—
68	C₂×D₅₃	—	4	4	no	—
69	C₂×D₅₃	—	4	8	no	—
70	C₂×D₅₃	—	8	4	no	—
71	C₂×D₅₃	—	8	2	no	—
72	C₂×D₅₃	—	8	2	no	—
73	C₄×C₅₃	—	8	2	no	—
74	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₄×C₅₃	#121	1	1	yes	characteristic
77	C₄×C₅₃	#124	1	1	yes	characteristic
78	Dic₅₃	—	2	4	no	—
79	#1970	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	#390	1	1	yes	characteristic
81	C₂×C₅₃	C₄×D₈	1	2	yes	—
82	C₂×C₅₃	#466	1	2	yes	—
83	C₂×C₅₃	#493	1	1	yes	characteristic
84	C₂×C₅₃	#495	1	1	yes	characteristic
85	D₅₃	—	8	4	no	—
86	C₂²×D₈	—	53	1	no	—
87	C₂×#107	D₅₃	1	1	yes	characteristic, complemented
88	C₂×#102	—	53	2	no	—
89	C₂×C₄×C₈	—	53	1	no	—
90	#390	—	53	2	no	—
91	C₅₃	#1970	1	1	yes	characteristic, complemented, Sylow
92	C₂²×D₄	—	53	2	no	—
93	C₂×D₈	—	53	4	no	—
94	C₂×D₈	—	106	2	no	—
95	C₂×D₈	—	106	2	no	—
96	C₂²×C₈	—	53	1	no	—
97	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
99	C₂×C₂²⋊C₄	—	106	4	no	—
100	C₂×C₄²	—	53	1	no	—
101	#107	—	2	2	no	—
102	#102	—	106	4	no	—
103	C₄×C₈	—	106	2	no	—
104	C₂⁴	—	106	2	no	—
105	C₂×D₄	—	53	4	no	—
106	C₂×D₄	—	106	2	no	—
107	C₂×D₄	—	106	2	no	—
108	C₂×D₄	—	106	2	no	—
109	C₂×D₄	—	106	2	no	—
110	C₂×D₄	—	106	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	53	2	no	—
113	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	D₈	—	106	8	no	—
115	C₂×C₈	—	53	2	no	—
116	C₂×C₈	—	106	1	no	—
117	C₂×C₈	—	106	1	no	—
118	C₂×C₈	C₄×D₅₃	1	2	yes	—
119	C₂×C₈	D₂₁₂	1	2	yes	—
120	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	212	8	no	—
124	C₄²	—	106	1	no	—
125	C₄²	—	106	1	no	—
126	C₂³	—	106	4	no	—
127	C₂³	—	212	2	no	—
128	C₂³	—	212	2	no	—
129	C₂³	—	212	4	no	—
130	C₂³	—	212	4	no	—
131	C₂³	D₄×D₅₃	1	1	yes	characteristic
132	D₄	—	106	4	no	—
133	D₄	—	106	4	no	—
134	D₄	—	212	2	no	—
135	D₄	—	212	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	53	4	no	—
139	C₂×C₄	—	106	2	no	—
140	C₂×C₄	—	106	2	no	—
141	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
142	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
143	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
144	C₂×C₄	#11719949	1	1	yes	characteristic
145	C₂×C₄	#11719949	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
147	C₈	—	106	2	no	—
148	C₈	#11719921	1	4	yes	complemented
149	C₂²	—	424	2	no	—
150	C₂²	—	424	4	no	—
151	C₂²	—	424	2	no	—
152	C₂²	—	212	4	no	—
153	C₂²	—	212	8	no	—
154	C₂²	#13133179	1	1	yes	characteristic
155	C₂²	#13133192	1	1	yes	characteristic
156	C₂²	#13133213	1	1	yes	characteristic
157	C₂²	D₈×D₅₃	1	2	yes	—
158	C₂²	#13133240	1	2	yes	—
159	C₄	—	8	2	no	—
160	C₄	—	106	4	no	—
161	C₄	C₂×#11719921	1	1	yes	characteristic
162	C₄	#13133179	1	1	yes	characteristic
163	C₄	#13133192	1	1	yes	characteristic
164	C₄	#13133206	1	1	yes	characteristic
165	C₂	—	424	4	no	—
166	C₂	#16810524	1	1	yes	characteristic
167	C₂	#16810705	1	2	yes	—
168	C₂	#16810860	1	2	yes	—
169	C₂	#16810998	1	1	yes	characteristic
170	C₂	#16811008	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₂₄

C₂

yes

maximal, characteristic

C₂×#13133137

C₂

yes

maximal, complemented

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic, complemented

#16810524

C₂

yes

maximal, complemented

C₂×#107×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

C₄

yes

complemented

C₂²×#88322

C₂²

yes

characteristic

C₂×#11719921

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133137

—

C₈×Dic₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

D₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

#11719921

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

D₄₂₄

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

complemented

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

C₂²⋊C₄

yes

complemented

#88322

—

C₂²×C₅₃

C₄×D₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×D₈

yes

—

C₂²×C₅₃

#135

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₄×D₈

yes

—

C₂×C₅₃

#466

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂²×D₈

—

C₂×#107

D₅₃

yes

characteristic, complemented

C₂×#102

—

C₂×C₄×C₈

—

#390

—

C₅₃

#1970

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

106

—

100

C₂×C₄²

—

101

#107

—

102

#102

—

106

—

103

C₄×C₈

—

106

—

104

C₂⁴

—

106

—

105

C₂×D₄

—

106

C₂×D₄

—

106

—

107

C₂×D₄

—

106

—

108

C₂×D₄

—

106

—

109

C₂×D₄

—

106

—

110

C₂×D₄

—

106

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₅₃

yes

characteristic

114

D₈

—

106

—

115

C₂×C₈

—

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

C₄×D₅₃

yes

—

119

C₂×C₈

D₂₁₂

yes

—

120

C₂×C₈

C₂²⋊D₅₃

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

212

—

124

C₄²

—

106

—

125

C₄²

—

106

—

126

C₂³

—

106

—

127

C₂³

—

212

—

128

C₂³

—

212

—

129

C₂³

—

212

—

130

C₂³

—

212

—

131

C₂³

D₄×D₅₃

yes

characteristic

132

D₄

—

106

—

133

D₄

—

106

—

134

D₄

—

212

—

135

D₄

—

212

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

106

—

140

C₂×C₄

—

106

—

141

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

142

C₂×C₄

C₂×D₂₁₂

yes

characteristic

143

C₂×C₄

D₄×D₅₃

yes

characteristic

144

C₂×C₄

#11719949

yes

characteristic

145

C₂×C₄

#11719949

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

147

C₈

—

106

—

148

C₈

#11719921

yes

complemented

149

C₂²

—

424

—

150

C₂²

—

424

—

151

C₂²

—

424

—

152

C₂²

—

212

—

153

C₂²

—

212

—

154

C₂²

#13133179

yes

characteristic

155

C₂²

#13133192

yes

characteristic

156

C₂²

#13133213

yes

characteristic

157

C₂²

D₈×D₅₃

yes

—

158

C₂²

#13133240

yes

—

159

C₄

—

160

C₄

—

106

—

161

C₄

C₂×#11719921

yes

characteristic

162

C₄

#13133179

yes

characteristic

163

C₄

#13133192

yes

characteristic

164

C₄

#13133206

yes

characteristic

165

C₂

—

424

—

166

C₂

#16810524

yes

characteristic

167

C₂

#16810705

yes

—

168

C₂

#16810860

yes

—

169

C₂

#16810998

yes

characteristic

170

C₂

#16811008

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#1970
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. C₂×#13133137	C₂	semidirect
3. C₂×C₈×Dic₅₃	C₂	semidirect
4. #16810524	C₂	semidirect
5. C₂×#107×C₅₃	C₂	semidirect
9. C₂×D₄₂₄	C₄	semidirect
56. C₂×Dic₅₃	D₈	semidirect
61. C₈×C₅₃	C₂²⋊C₄	semidirect
87. C₂×#107	D₅₃	semidirect
91. C₅₃	#1970	semidirect
148. C₈	#11719921	semidirect

H ≅ G/N

type

2. C₂×#13133137

C₂

semidirect

3. C₂×C₈×Dic₅₃

C₂

semidirect

4. #16810524

C₂

semidirect

5. C₂×#107×C₅₃

C₂

semidirect

9. C₂×D₄₂₄

C₄

semidirect

56. C₂×Dic₅₃

D₈

semidirect

61. C₈×C₅₃

C₂²⋊C₄

semidirect

87. C₂×#107

D₅₃

semidirect

91. C₅₃

#1970

semidirect

148. C₈

#11719921

semidirect

Subgroups — #20275579

Conjugacy classes of subgroups

Complements