Group #20275242 — subgroups

Subgroups — #20275242

Order 6784.

344 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16811187	C₂	1	1	yes	maximal, characteristic
2	#16810629	C₂	1	2	yes	maximal, complemented
3	#16810597	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810380	C₂	1	2	yes	maximal, complemented
5	C₅₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
7	#13133213	C₄	1	2	yes	complemented
8	#13133206	C₄	1	2	yes	complemented
9	#13133191	—	2	2	no	—
10	#13133185	—	2	2	no	—
11	#13133176	C₂²	1	2	yes	—
12	#13133176	C₂²	1	1	yes	characteristic
13	#13133139	—	2	4	no	—
14	#13133132	—	2	2	no	—
15	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
16	#131×C₅₃	—	2	4	no	—
17	C₂×M₁₆×C₅₃	—	2	2	no	—
18	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂³×D₅₃	—	2	1	no	—
20	#41×C₅₃	—	2	2	no	—
21	#41×C₅₃	—	2	2	no	—
22	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₅₃	D₄	1	2	yes	—
25	C₂²×C₄×C₅₃	D₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊D₅₃	—	2	4	no	—
28	C₂×D₂₁₂	—	2	2	no	—
29	C₂×D₂₁₂	—	2	2	no	—
30	C₂×D₂₁₂	—	2	1	no	—
31	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
32	M₁₆×C₅₃	—	2	4	no	—
33	M₁₆×C₅₃	—	2	2	no	—
34	M₁₆×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	4	no	—
36	C₂×C₈×C₅₃	D₄	1	4	yes	complemented
37	C₂²⋊Dic₅₃	—	2	1	no	—
38	C₂²⋊Dic₅₃	—	2	2	no	—
39	#11719921	—	2	4	no	—
40	C₄⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	2	no	—
42	C₄×Dic₅₃	—	2	2	no	—
43	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
44	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₅₃	—	2	2	no	—
47	C₂²×D₅₃	—	4	2	no	—
48	C₂²×D₅₃	—	8	2	no	—
49	Q₈×C₅₃	—	2	2	no	—
50	D₄×C₅₃	—	2	4	no	—
51	D₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	2	yes	—
59	C₂²⋊D₅₃	—	8	4	no	—
60	C₂×Dic₅₃	—	2	2	no	—
61	C₂×Dic₅₃	—	4	2	no	—
62	D₂₁₂	—	2	4	no	—
63	D₂₁₂	—	4	1	no	—
64	D₂₁₂	—	4	1	no	—
65	C₈×C₅₃	—	2	4	no	—
66	C₂²×C₅₃	—	2	2	no	—
67	C₂²×C₅₃	—	4	1	no	—
68	C₂²×C₅₃	—	4	2	no	—
69	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₅₃	#121	1	2	yes	—
71	C₂×D₅₃	—	4	4	no	—
72	C₂×D₅₃	—	4	2	no	—
73	C₂×D₅₃	—	8	4	no	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	C₄×D₄	1	2	yes	—
76	C₄×C₅₃	#120	1	1	yes	characteristic
77	C₄×C₅₃	#123	1	1	yes	characteristic
78	Dic₅₃	—	4	2	no	—
79	Dic₅₃	—	8	2	no	—
80	#1910	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	—	2	2	no	—
83	C₂×C₅₃	—	4	2	no	—
84	C₂×C₅₃	#386	1	1	yes	characteristic
85	D₅₃	—	8	2	no	—
86	#568	D₅₃	1	1	yes	characteristic, complemented
87	#535	—	53	1	no	—
88	#418	—	53	2	no	—
89	#410	—	53	1	no	—
90	H(3,ℤ/4)	—	53	2	no	—
91	C₅₃	#1910	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
93	C₂²×D₄	—	53	1	no	—
94	#131	—	2	4	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
97	#127	—	53	2	no	—
98	#124	—	53	2	no	—
99	#121	—	106	2	no	—
100	#120	—	106	2	no	—
101	#117	—	53	1	no	—
102	#117	—	53	2	no	—
103	#102	—	106	4	no	—
104	#99	—	106	2	no	—
105	C₂⁴	—	106	1	no	—
106	#41	—	2	2	no	—
107	#41	—	2	2	no	—
108	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
109	C₂×D₄	—	53	2	no	—
110	C₂×D₄	—	106	4	no	—
111	C₂×D₄	—	106	2	no	—
112	C₂×D₄	—	106	2	no	—
113	C₂×D₄	—	106	1	no	—
114	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
115	C₂×D₄	D₂₁₂	1	2	yes	—
116	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	2	no	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	C₂²⋊D₅₃	1	4	yes	complemented
123	C₄⋊C₄	—	106	1	no	—
124	C₄⋊C₄	—	106	2	no	—
125	C₂²⋊C₄	—	106	1	no	—
126	C₂²⋊C₄	—	106	4	no	—
127	C₂²⋊C₄	—	106	2	no	—
128	C₄²	—	53	2	no	—
129	C₄²	—	106	2	no	—
130	C₂³	—	106	2	no	—
131	C₂³	—	424	2	no	—
132	C₂³	—	212	2	no	—
133	C₂³	#11719921	1	2	yes	—
134	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
135	Q₈	—	2	2	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	2	no	—
138	D₄	—	106	4	no	—
139	D₄	—	424	4	no	—
140	D₄	—	212	1	no	—
141	D₄	—	212	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	—	212	2	no	—
146	C₂×C₄	#11719921	1	2	yes	—
147	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₄	#11719945	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	424	4	no	—
156	C₂²	—	212	4	no	—
157	C₂²	—	212	2	no	—
158	C₂²	C₂×#11719921	1	1	yes	characteristic
159	C₂²	#13133191	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	424	2	no	—
162	C₄	—	212	2	no	—
163	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
164	C₄	#13133186	1	1	yes	characteristic
165	C₄	#13133200	1	1	yes	characteristic
166	C₂	—	2	2	no	—
167	C₂	—	2	1	no	—
168	C₂	—	4	2	no	—
169	C₂	—	424	2	no	—
170	C₂	#16810502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811187

C₂

yes

maximal, characteristic

#16810629

C₂

yes

maximal, complemented

#16810597

C₂

yes

maximal, characteristic, complemented

#16810380

C₂

yes

maximal, complemented

C₅₃×#568

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

characteristic

#13133213

C₄

yes

complemented

#13133206

C₄

yes

—

—

C₂²

yes

—

#13133176

C₂²

yes

characteristic

#13133139

—

#13133132

—

C₂×#41×C₅₃

C₂²

yes

characteristic

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂³×D₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂×C₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

complemented

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

#1910

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#386

yes

characteristic

D₅₃

—

#568

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#1910

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂²×D₄

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

#127

—

#124

—

#121

—

106

—

100

#120

—

106

—

101

#117

—

102

#117

—

103

#102

—

106

—

104

#99

—

106

—

105

C₂⁴

—

106

—

106

#41

—

107

#41

—

108

C₂×Q₈

C₄×D₅₃

yes

characteristic

109

C₂×D₄

—

110

C₂×D₄

—

106

—

111

C₂×D₄

—

106

—

112

C₂×D₄

—

106

—

113

C₂×D₄

—

106

—

114

C₂×D₄

C₄×D₅₃

yes

characteristic

115

C₂×D₄

D₂₁₂

yes

—

116

C₂²×C₄

C₂²⋊D₅₃

yes

—

117

C₂²×C₄

C₂²×D₅₃

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊D₅₃

yes

complemented

123

C₄⋊C₄

—

106

—

124

C₄⋊C₄

—

106

—

125

C₂²⋊C₄

—

106

—

126

C₂²⋊C₄

—

106

—

127

C₂²⋊C₄

—

106

—

128

C₄²

—

129

C₄²

—

106

—

130

C₂³

—

106

—

131

C₂³

—

424

—

132

C₂³

—

212

—

133

C₂³

#11719921

yes

—

134

C₂³

C₂×D₂₁₂

yes

characteristic

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

106

—

139

D₄

—

424

—

140

D₄

—

212

—

141

D₄

—

212

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

212

—

146

C₂×C₄

#11719921

yes

—

147

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

148

C₂×C₄

#11719945

yes

—

149

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

150

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

C₈

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

424

—

156

C₂²

—

212

—

157

C₂²

—

212

—

158

C₂²

C₂×#11719921

yes

characteristic

159

C₂²

#13133191

yes

—

160

C₄

—

161

C₄

—

424

—

162

C₄

—

212

—

163

C₄

C₄×C₂²⋊D₅₃

yes

—

164

C₄

#13133186

yes

characteristic

165

C₄

#13133200

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

424

—

170

C₂

#16810502

yes

characteristic

	subgroup	G/H
center	C₂	#16810502
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#568	C₂
socle	C₂×C₅₃	#386
hypercenter	#568	D₅₃
hypocenter	C₅₃	#1910
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#16810502

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#568

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
2. #16810629	C₂	semidirect
3. #16810597	C₂	semidirect
4. #16810380	C₂	semidirect
5. C₅₃×#568	C₂	semidirect
7. #13133213	C₄	semidirect
8. #13133206	C₄	semidirect
36. C₂×C₈×C₅₃	D₄	semidirect
86. #568	D₅₃	semidirect
91. C₅₃	#1910	semidirect
122. C₂×C₈	C₂²⋊D₅₃	semidirect

H ≅ G/N

type

2. #16810629

C₂

semidirect

3. #16810597

C₂

semidirect

4. #16810380

C₂

semidirect

5. C₅₃×#568

C₂

semidirect

7. #13133213

C₄

semidirect

8. #13133206

C₄

semidirect

36. C₂×C₈×C₅₃

D₄

semidirect

86. #568

D₅₃

semidirect

91. C₅₃

#1910

semidirect

122. C₂×C₈

C₂²⋊D₅₃

semidirect

Subgroups — #20275242

Conjugacy classes of subgroups

Complements