Group #20275591 — subgroups

Subgroups — #20275591

Order 6784.

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

normal subgroups only

This group has 376 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×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133139	C₂	1	2	yes	maximal, complemented
4	#16810525	C₂	1	2	yes	maximal, complemented
5	C₂×#97×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	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	C₂×#11719921	—	2	4	no	—
12	#13133149	—	2	2	no	—
13	#13133139	—	2	4	no	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	#97×C₅₃	—	2	2	no	—
18	C₂³×D₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₃	—	2	2	no	—
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	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₃	D₄	1	2	yes	—
32	C₂×C₈×C₅₃	D₄	1	2	yes	—
33	C₂×C₈×C₅₃	D₄	1	2	yes	—
34	C₄²×C₅₃	—	2	1	no	—
35	C₄²×C₅₃	—	2	1	no	—
36	#11719921	—	4	8	no	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	—	2	2	no	—
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	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	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	—	4	2	no	—
55	C₂×Dic₅₃	—	4	4	no	—
56	D₂₁₂	—	2	4	no	—
57	D₂₁₂	—	2	4	no	—
58	D₂₁₂	—	4	2	no	—
59	D₂₁₂	—	4	2	no	—
60	C₈×C₅₃	—	2	2	no	—
61	C₈×C₅₃	C₂²⋊C₄	1	4	yes	complemented
62	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂²×C₅₃	#121	1	1	yes	characteristic
64	C₂²×C₅₃	#127	1	1	yes	characteristic
65	C₂²×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
66	C₂×D₅₃	—	4	4	no	—
67	C₂×D₅₃	—	4	8	no	—
68	C₂×D₅₃	—	8	4	no	—
69	C₂×D₅₃	—	8	2	no	—
70	C₂×D₅₃	—	8	2	no	—
71	C₄×C₅₃	—	4	2	no	—
72	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₄×C₅₃	#121	1	1	yes	characteristic
75	C₄×C₅₃	#124	1	1	yes	characteristic
76	Dic₅₃	—	8	2	no	—
77	#1972	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	#390	1	1	yes	characteristic
79	C₂×C₅₃	#442	1	2	yes	—
80	C₂×C₅₃	#469	1	2	yes	—
81	C₂×C₅₃	#496	1	2	yes	—
82	D₅₃	—	8	4	no	—
83	C₂²×D₈	—	53	1	no	—
84	C₂×#106	—	53	1	no	—
85	C₂×#102	—	53	2	no	—
86	C₂×#97	D₅₃	1	1	yes	characteristic, complemented
87	#390	—	53	2	no	—
88	C₅₃	#1972	1	1	yes	characteristic, complemented, Sylow
89	C₂²×D₄	—	53	2	no	—
90	C₂×D₈	—	53	4	no	—
91	C₂×D₈	—	106	2	no	—
92	C₂×D₈	—	106	2	no	—
93	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	53	2	no	—
96	C₂×C₂²⋊C₄	—	106	4	no	—
97	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
98	#106	—	106	2	no	—
99	#102	—	106	4	no	—
100	#97	—	2	2	no	—
101	C₂⁴	—	106	2	no	—
102	C₂×D₄	—	53	4	no	—
103	C₂×D₄	—	106	2	no	—
104	C₂×D₄	—	106	2	no	—
105	C₂×D₄	—	106	2	no	—
106	C₂×D₄	—	106	2	no	—
107	C₂×D₄	—	106	2	no	—
108	C₂²×C₄	—	106	2	no	—
109	C₂²×C₄	D₂₁₂	1	2	yes	—
110	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
111	D₈	—	106	8	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	C₄×D₅₃	1	2	yes	—
115	C₂×C₈	D₂₁₂	1	2	yes	—
116	C₂×C₈	D₂₁₂	1	2	yes	—
117	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
118	C₄⋊C₄	—	106	2	no	—
119	C₄⋊C₄	—	106	2	no	—
120	C₂²⋊C₄	—	212	8	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₂³	—	106	4	no	—
124	C₂³	—	212	2	no	—
125	C₂³	—	212	2	no	—
126	C₂³	—	212	4	no	—
127	C₂³	—	212	4	no	—
128	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
129	D₄	—	106	4	no	—
130	D₄	—	106	4	no	—
131	D₄	—	212	2	no	—
132	D₄	—	212	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	212	2	no	—
136	C₂×C₄	—	212	4	no	—
137	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
139	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
140	C₂×C₄	#11719945	1	1	yes	characteristic
141	C₂×C₄	#11719945	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	#11719921	1	4	yes	complemented
145	C₂²	—	424	2	no	—
146	C₂²	—	424	2	no	—
147	C₂²	—	424	4	no	—
148	C₂²	—	212	4	no	—
149	C₂²	—	212	8	no	—
150	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
151	C₂²	#13133191	1	1	yes	characteristic
152	C₂²	#13133214	1	1	yes	characteristic
153	C₂²	#13133245	1	4	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	424	2	no	—
156	C₄	C₂×#11719921	1	1	yes	characteristic
157	C₄	C₄×D₂₁₂	1	1	yes	characteristic
158	C₄	#13133191	1	1	yes	characteristic
159	C₄	#13133205	1	1	yes	characteristic
160	C₂	—	424	4	no	—
161	C₂	#16810525	1	1	yes	characteristic
162	C₂	#16810736	1	2	yes	—
163	C₂	#16810876	1	2	yes	—
164	C₂	#16811012	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄₂₄

C₂

yes

maximal, characteristic

C₂×#13133149

C₂

yes

maximal, characteristic, complemented

C₂×#13133139

C₂

yes

maximal, complemented

#16810525

C₂

yes

maximal, complemented

C₂×#97×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

C₂×D₄₂₄

—

C₂×D₄₂₄

—

C₂×D₄₂₄

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719921

—

#13133149

—

#13133139

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

C₂³×D₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

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

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

complemented

C₂²×C₅₃

C₄×D₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

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

yes

—

C₂×C₅₃

#469

yes

—

C₂×C₅₃

#496

yes

—

D₅₃

—

C₂²×D₈

—

C₂×#106

—

C₂×#102

—

C₂×#97

D₅₃

yes

characteristic, complemented

#390

—

C₅₃

#1972

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#106

—

106

—

#102

—

106

—

100

#97

—

101

C₂⁴

—

106

—

102

C₂×D₄

—

103

C₂×D₄

—

106

—

104

C₂×D₄

—

106

—

105

C₂×D₄

—

106

—

106

C₂×D₄

—

106

—

107

C₂×D₄

—

106

—

108

C₂²×C₄

—

106

—

109

C₂²×C₄

D₂₁₂

yes

—

110

C₂²×C₄

C₂²×D₅₃

yes

characteristic

111

D₈

—

106

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄×D₅₃

yes

—

115

C₂×C₈

D₂₁₂

yes

—

116

C₂×C₈

D₂₁₂

yes

—

117

C₂×C₈

C₂²⋊D₅₃

yes

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₂²⋊C₄

—

212

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

106

—

124

C₂³

—

212

—

125

C₂³

—

212

—

126

C₂³

—

212

—

127

C₂³

—

212

—

128

C₂³

C₂×D₂₁₂

yes

characteristic

129

D₄

—

106

—

130

D₄

—

106

—

131

D₄

—

212

—

132

D₄

—

212

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

212

—

136

C₂×C₄

—

212

—

137

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

138

C₂×C₄

C₂×D₂₁₂

yes

characteristic

139

C₂×C₄

C₂×D₂₁₂

yes

characteristic

140

C₂×C₄

#11719945

yes

characteristic

141

C₂×C₄

#11719945

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

143

C₈

—

144

C₈

#11719921

yes

complemented

145

C₂²

—

424

—

146

C₂²

—

424

—

147

C₂²

—

424

—

148

C₂²

—

212

—

149

C₂²

—

212

—

150

C₂²

C₄×D₂₁₂

yes

characteristic

151

C₂²

#13133191

yes

characteristic

152

C₂²

#13133214

yes

characteristic

153

C₂²

#13133245

yes

—

154

C₄

—

155

C₄

—

424

—

156

C₄

C₂×#11719921

yes

characteristic

157

C₄

C₄×D₂₁₂

yes

characteristic

158

C₄

#13133191

yes

characteristic

159

C₄

#13133205

yes

characteristic

160

C₂

—

424

—

161

C₂

#16810525

yes

characteristic

162

C₂

#16810736

yes

—

163

C₂

#16810876

yes

—

164

C₂

#16811012

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#1972
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#97×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×#97

D₅₃

hypocenter

C₅₃

#1972

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#13133149	C₂	semidirect
3. C₂×#13133139	C₂	semidirect
4. #16810525	C₂	semidirect
5. C₂×#97×C₅₃	C₂	semidirect
9. C₂×D₄₂₄	C₄	semidirect
61. C₈×C₅₃	C₂²⋊C₄	semidirect
86. C₂×#97	D₅₃	semidirect
88. C₅₃	#1972	semidirect
144. C₈	#11719921	semidirect

H ≅ G/N

type

2. C₂×#13133149

C₂

semidirect

3. C₂×#13133139

C₂

semidirect

4. #16810525

C₂

semidirect

5. C₂×#97×C₅₃

C₂

semidirect

9. C₂×D₄₂₄

C₄

semidirect

61. C₈×C₅₃

C₂²⋊C₄

semidirect

86. C₂×#97

D₅₃

semidirect

88. C₅₃

#1972

semidirect

144. C₈

#11719921

semidirect

Subgroups — #20275591

Conjugacy classes of subgroups

Complements