Group #20275263 — subgroups

Subgroups — #20275263

Order 6784.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16810607	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133134	C₂	1	2	yes	maximal, complemented
4	#16810393	C₂	1	2	yes	maximal, complemented
5	C₅₃×#535	C₂	1	1	yes	maximal, characteristic
6	#13133224	—	2	4	no	—
7	C₂×#11719917	—	2	2	no	—
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	#13133136	—	2	1	no	—
11	#13133134	—	2	4	no	—
12	#13133134	—	2	1	no	—
13	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	#127×C₅₃	—	2	1	no	—
16	#124×C₅₃	—	2	1	no	—
17	#121×C₅₃	—	2	2	no	—
18	#120×C₅₃	C₄	1	4	yes	—
19	#117×C₅₃	C₂²	1	1	yes	characteristic
20	C₂⁴×C₅₃	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	—	2	4	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₄×C₅₃	—	2	2	no	—
28	C₂×D₄×C₅₃	—	2	2	no	—
29	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	D₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂²⋊C₄×C₅₃	—	2	4	no	—
35	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₄²×C₅₃	—	2	1	no	—
37	#11719917	—	2	4	no	—
38	#11719917	—	2	2	no	—
39	#11719917	—	2	2	no	—
40	#11719917	—	2	2	no	—
41	#11719917	D₄	1	4	yes	complemented
42	C₂×#88322	—	2	4	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	2	2	no	—
45	C₂³×C₅₃	—	2	2	no	—
46	C₂³×C₅₃	—	2	4	no	—
47	C₂³×C₅₃	—	8	2	no	—
48	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₅₃	—	2	2	no	—
51	D₄×C₅₃	—	2	4	no	—
52	D₄×C₅₃	—	2	2	no	—
53	D₄×C₅₃	—	4	1	no	—
54	D₄×C₅₃	—	4	2	no	—
55	D₄×C₅₃	—	4	1	no	—
56	D₄×C₅₃	—	8	4	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#41	1	2	yes	—
65	#88322	—	2	4	no	—
66	#88322	—	4	2	no	—
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	—	4	1	no	—
69	C₂²×C₅₃	—	4	2	no	—
70	C₂²×C₅₃	—	4	4	no	—
71	C₂²×C₅₃	—	4	2	no	—
72	C₂²×C₅₃	—	8	4	no	—
73	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₅₃	C₄×D₄	1	2	yes	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	4	2	no	—
77	C₄×C₅₃	#120	1	1	yes	characteristic
78	C₄×C₅₃	#121	1	2	yes	—
79	C₄×C₅₃	#123	1	1	yes	characteristic
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	—	8	2	no	—
85	C₂×C₅₃	#386	1	1	yes	characteristic
86	#568	—	53	1	no	—
87	#535	D₅₃	1	1	yes	characteristic
88	#413	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
94	#131	—	106	4	no	—
95	C₂×M₁₆	—	53	2	no	—
96	C₂×M₁₆	—	53	2	no	—
97	C₂×M₁₆	—	106	2	no	—
98	#127	—	2	1	no	—
99	#124	—	2	1	no	—
100	#121	—	2	2	no	—
101	#120	Dic₅₃	1	4	yes	—
102	#117	C₂×D₅₃	1	1	yes	characteristic
103	#101	—	106	1	no	—
104	#100	—	106	4	no	—
105	#100	—	106	1	no	—
106	C₂⁴	C₂×Dic₅₃	1	2	yes	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂×D₄	—	2	4	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	2	1	no	—
115	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
116	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
118	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
119	M₁₆	—	53	4	no	—
120	M₁₆	—	106	4	no	—
121	M₁₆	—	106	2	no	—
122	M₁₆	—	106	2	no	—
123	M₁₆	—	106	2	no	—
124	C₂×C₈	—	106	4	no	—
125	C₂×C₈	—	106	2	no	—
126	C₂×C₈	—	106	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	2	4	no	—
129	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
130	C₄²	—	2	1	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	8	2	no	—
134	C₂³	C₂²⋊Dic₅₃	1	2	yes	—
135	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	2	no	—
141	D₄	—	4	1	no	—
142	D₄	—	8	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
147	C₂×C₄	D₄×D₅₃	1	2	yes	—
148	C₂×C₄	#11719947	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	C₈	—	106	4	no	—
152	C₈	—	212	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	4	no	—
159	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
160	C₂²	D₄×Dic₅₃	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	4	2	no	—
163	C₄	#13133185	1	1	yes	characteristic
164	C₄	#13133189	1	2	yes	—
165	C₄	#13133198	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	8	2	no	—
170	C₂	#16810503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811390

C₂

yes

maximal, characteristic, complemented

#16810607

C₂

yes

maximal, characteristic, complemented

C₂×#13133134

C₂

yes

maximal, complemented

#16810393

C₂

yes

maximal, complemented

C₅₃×#535

C₂

yes

maximal, characteristic

#13133224

—

C₂×#11719917

—

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133136

—

#13133134

—

#13133134

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂²×D₄×C₅₃

C₂²

yes

characteristic

#127×C₅₃

—

#124×C₅₃

—

#121×C₅₃

—

#120×C₅₃

C₄

yes

—

#117×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂×C₄

yes

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

—

—

—

—

D₄

yes

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#121

yes

—

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

C₅₃

#1914

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂²×D₄

C₂×D₅₃

yes

characteristic

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

#127

—

#124

—

100

#121

—

101

#120

Dic₅₃

yes

—

102

#117

C₂×D₅₃

yes

characteristic

103

#101

—

106

—

104

#100

—

106

—

105

#100

—

106

—

106

C₂⁴

C₂×Dic₅₃

yes

—

107

#41

—

108

#41

—

109

C₂×Q₈

C₂²⋊D₅₃

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

C₂×Dic₅₃

yes

—

116

C₂×D₄

C₂²⋊D₅₃

yes

characteristic

117

C₂²×C₄

C₂²⋊D₅₃

yes

—

118

C₂²×C₄

C₂²×D₅₃

yes

characteristic

119

M₁₆

—

120

M₁₆

—

106

—

121

M₁₆

—

106

—

122

M₁₆

—

106

—

123

M₁₆

—

106

—

124

C₂×C₈

—

106

—

125

C₂×C₈

—

106

—

126

C₂×C₈

—

106

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

C₂×Dic₅₃

yes

—

130

C₄²

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂²⋊Dic₅₃

yes

—

135

C₂³

C₂²×Dic₅₃

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊Dic₅₃

yes

—

147

C₂×C₄

D₄×D₅₃

yes

—

148

C₂×C₄

#11719947

yes

—

149

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

C₈

—

106

—

152

C₈

—

212

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

160

C₂²

D₄×Dic₅₃

yes

—

161

C₄

—

162

C₄

—

163

C₄

#13133185

yes

characteristic

164

C₄

#13133189

yes

—

165

C₄

#13133198

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

#16810503

yes

characteristic

	subgroup	G/H
center	C₂	#16810503
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#535	C₂
socle	C₂×C₅₃	#386
hypercenter	#535	D₅₃
hypocenter	C₅₃	#1914
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

subgroup

G/H

center

C₂

#16810503

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#535

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

N	H ≅ G/N	type
1. #16811390	C₂	semidirect
2. #16810607	C₂	semidirect
3. C₂×#13133134	C₂	semidirect
4. #16810393	C₂	semidirect
41. #11719917	D₄	semidirect
91. C₅₃	#1914	semidirect

H ≅ G/N

type

1. #16811390

C₂

semidirect

2. #16810607

C₂

semidirect

3. C₂×#13133134

C₂

semidirect

4. #16810393

C₂

semidirect

41. #11719917

D₄

semidirect

91. C₅₃

#1914

semidirect

Subgroups — #20275263

Conjugacy classes of subgroups

Complements