Group #20275260 — subgroups

Subgroups — #20275260

Order 6784.

340 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16811390	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811187	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133133	C₂	1	2	yes	maximal, complemented
4	#16810394	C₂	1	2	yes	maximal, complemented
5	C₅₃×#413	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	C₂²	1	1	yes	characteristic
7	#13133224	—	2	4	no	—
8	C₂×#11719917	—	2	2	no	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	#13133213	—	2	1	no	—
11	#13133206	—	2	1	no	—
12	#13133191	—	2	2	no	—
13	#13133185	C₄	1	4	yes	—
14	#13133176	C₂²	1	1	yes	characteristic
15	#13133133	—	2	4	no	—
16	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
18	#101×C₅₃	—	2	1	no	—
19	#100×C₅₃	—	2	1	no	—
20	C₂³×D₅₃	C₂×C₄	1	2	yes	—
21	#41×C₅₃	—	2	2	no	—
22	#41×C₅₃	—	2	2	no	—
23	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	D₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₅₃	—	2	4	no	—
29	C₂×D₂₁₂	—	2	2	no	—
30	C₂×D₂₁₂	—	2	1	no	—
31	C₂×D₂₁₂	—	2	1	no	—
32	C₂×D₂₁₂	—	2	2	no	—
33	M₁₆×C₅₃	—	2	2	no	—
34	M₁₆×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂²⋊Dic₅₃	C₂×C₄	1	2	yes	—
37	#11719921	—	2	4	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₄×Dic₅₃	—	2	1	no	—
40	#11719917	—	2	4	no	—
41	#11719917	—	2	2	no	—
42	#11719917	D₄	1	4	yes	complemented
43	C₂×#88322	—	2	4	no	—
44	C₂×#88322	—	2	2	no	—
45	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×D₅₃	—	2	2	no	—
48	C₂²×D₅₃	—	2	4	no	—
49	C₂²×D₅₃	—	8	2	no	—
50	Q₈×C₅₃	—	2	2	no	—
51	D₄×C₅₃	—	2	4	no	—
52	D₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#41	1	2	yes	—
60	C₂²⋊D₅₃	—	8	4	no	—
61	C₂×Dic₅₃	—	2	2	no	—
62	D₂₁₂	—	4	1	no	—
63	D₂₁₂	—	4	2	no	—
64	D₂₁₂	—	4	1	no	—
65	C₈×C₅₃	—	4	2	no	—
66	#88322	—	2	4	no	—
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	—	4	2	no	—
70	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₅₃	C₄×D₄	1	2	yes	—
72	C₂×D₅₃	—	4	4	no	—
73	C₂×D₅₃	—	4	2	no	—
74	C₂×D₅₃	—	8	4	no	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	#120	1	1	yes	characteristic
77	C₄×C₅₃	#121	1	2	yes	—
78	C₄×C₅₃	#123	1	1	yes	characteristic
79	Dic₅₃	—	4	2	no	—
80	#1914	—	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	—	53	1	no	—
87	#535	—	53	1	no	—
88	#413	D₅₃	1	1	yes	characteristic, complemented
89	C₂×#100	—	53	2	no	—
90	#344	—	53	2	no	—
91	C₅₃	#1914	1	1	yes	characteristic, complemented, Sylow
92	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
93	C₂²×D₄	—	53	1	no	—
94	#131	—	106	4	no	—
95	C₂×M₁₆	—	53	2	no	—
96	C₂×M₁₆	—	106	2	no	—
97	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
98	#127	—	106	1	no	—
99	#124	—	106	1	no	—
100	#121	—	106	2	no	—
101	#120	—	53	4	no	—
102	#117	—	53	1	no	—
103	#101	—	2	1	no	—
104	#100	—	2	1	no	—
105	#100	—	106	4	no	—
106	C₂⁴	—	53	2	no	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂×D₄	—	106	4	no	—
111	C₂×D₄	—	106	1	no	—
112	C₂×D₄	—	106	2	no	—
113	C₂×D₄	—	106	2	no	—
114	C₂×D₄	—	106	1	no	—
115	C₂×D₄	C₄×D₅₃	1	2	yes	—
116	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂²×C₄	D₂₁₂	1	2	yes	—
118	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	2	no	—
121	M₁₆	—	53	4	no	—
122	M₁₆	—	106	4	no	—
123	M₁₆	—	106	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	106	4	no	—
126	C₂×C₈	—	106	2	no	—
127	C₄⋊C₄	—	106	1	no	—
128	C₂²⋊C₄	—	53	2	no	—
129	C₂²⋊C₄	—	106	4	no	—
130	C₄²	—	106	1	no	—
131	C₂³	—	106	4	no	—
132	C₂³	—	106	2	no	—
133	C₂³	—	424	2	no	—
134	C₂³	#11719921	1	2	yes	—
135	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	424	4	no	—
140	D₄	—	212	1	no	—
141	D₄	—	212	2	no	—
142	D₄	—	212	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	106	2	no	—
146	C₂×C₄	#11719921	1	2	yes	—
147	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
148	C₂×C₄	D₄×D₅₃	1	2	yes	—
149	C₂×C₄	#11719947	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₈	—	106	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	424	4	no	—
157	C₂²	—	212	4	no	—
158	C₂²	—	212	2	no	—
159	C₂²	C₂×#11719921	1	1	yes	characteristic
160	C₂²	#13133177	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	212	2	no	—
163	C₄	#13133187	1	1	yes	characteristic
164	C₄	#13133189	1	2	yes	—
165	C₄	#13133201	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	424	2	no	—
170	C₂	#16810500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811390

C₂

yes

maximal, characteristic, complemented

#16811187

C₂

yes

maximal, characteristic

C₂×#13133133

C₂

yes

maximal, complemented

#16810394

C₂

yes

maximal, complemented

C₅₃×#413

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

C₂²

yes

—

—

C₂²

yes

—

—

—

—

C₄

yes

—

#13133176

C₂²

yes

characteristic

#13133133

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

#101×C₅₃

—

#100×C₅₃

—

C₂³×D₅₃

C₂×C₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

#11719917

D₄

yes

complemented

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

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

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#121

yes

—

C₄×C₅₃

#123

yes

characteristic

Dic₅₃

—

#1914

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#386

yes

characteristic

D₅₃

—

#568

—

#535

—

#413

D₅₃

yes

characteristic, complemented

C₂×#100

—

#344

—

C₅₃

#1914

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂²×D₄

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

—

#127

—

106

—

#124

—

106

—

100

#121

—

106

—

101

#120

—

102

#117

—

103

#101

—

104

#100

—

105

#100

—

106

—

106

C₂⁴

—

107

#41

—

108

#41

—

109

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

110

C₂×D₄

—

106

—

111

C₂×D₄

—

106

—

112

C₂×D₄

—

106

—

113

C₂×D₄

—

106

—

114

C₂×D₄

—

106

—

115

C₂×D₄

C₄×D₅₃

yes

—

116

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

117

C₂²×C₄

D₂₁₂

yes

—

118

C₂²×C₄

C₂²×D₅₃

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

122

M₁₆

—

106

—

123

M₁₆

—

106

—

124

C₂×C₈

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

106

—

127

C₄⋊C₄

—

106

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

106

—

130

C₄²

—

106

—

131

C₂³

—

106

—

132

C₂³

—

106

—

133

C₂³

—

424

—

134

C₂³

#11719921

yes

—

135

C₂³

C₂×C₄×D₅₃

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

424

—

140

D₄

—

212

—

141

D₄

—

212

—

142

D₄

—

212

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

106

—

146

C₂×C₄

#11719921

yes

—

147

C₂×C₄

C₂×D₂₁₂

yes

characteristic

148

C₂×C₄

D₄×D₅₃

yes

—

149

C₂×C₄

#11719947

yes

—

150

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

C₈

—

152

C₈

—

106

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

424

—

157

C₂²

—

212

—

158

C₂²

—

212

—

159

C₂²

C₂×#11719921

yes

characteristic

160

C₂²

#13133177

yes

—

161

C₄

—

162

C₄

—

212

—

163

C₄

#13133187

yes

characteristic

164

C₄

#13133189

yes

—

165

C₄

#13133201

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

424

—

170

C₂

#16810500

yes

characteristic

	subgroup	G/H
center	C₂	#16810500
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#413	C₂
socle	C₂×C₅₃	#386
hypercenter	#413	D₅₃
hypocenter	C₅₃	#1914
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₂

#16810500

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#413

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
1. #16811390	C₂	semidirect
3. C₂×#13133133	C₂	semidirect
4. #16810394	C₂	semidirect
5. C₅₃×#413	C₂	semidirect
42. #11719917	D₄	semidirect
88. #413	D₅₃	semidirect
91. C₅₃	#1914	semidirect

H ≅ G/N

type

1. #16811390

C₂

semidirect

3. C₂×#13133133

C₂

semidirect

4. #16810394

C₂

semidirect

5. C₅₃×#413

C₂

semidirect

42. #11719917

D₄

semidirect

88. #413

D₅₃

semidirect

91. C₅₃

#1914

semidirect

Subgroups — #20275260

Conjugacy classes of subgroups

Complements