Group #20275243 — subgroups

Subgroups — #20275243

Order 6784.

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

normal subgroups only

This group has 344 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	#16810628	C₂	1	2	yes	maximal, complemented
3	#16810597	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810379	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	#13133176	C₂²	1	2	yes	—
10	#13133138	—	2	4	no	—
11	#13133132	—	2	2	no	—
12	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	#127×C₅₃	C₄	1	2	yes	complemented
15	#124×C₅₃	C₄	1	2	yes	complemented
16	#121×C₅₃	—	2	2	no	—
17	#120×C₅₃	—	2	2	no	—
18	#117×C₅₃	C₂²	1	1	yes	characteristic
19	C₂⁴×C₅₃	—	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	—	2	4	no	—
24	C₂×D₄×C₅₃	—	2	2	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	2	no	—
27	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₅₃	D₄	1	2	yes	—
30	C₂²×C₄×C₅₃	D₄	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂²⋊C₄×C₅₃	—	2	1	no	—
34	C₂²⋊C₄×C₅₃	—	2	4	no	—
35	C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²⋊Dic₅₃	—	2	2	no	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄×Dic₅₃	—	2	2	no	—
39	#11719917	—	2	4	no	—
40	#11719917	—	2	2	no	—
41	#11719917	—	2	2	no	—
42	C₂×#88322	—	2	4	no	—
43	C₂×#88322	D₄	1	4	yes	complemented
44	C₂³×C₅₃	—	2	2	no	—
45	C₂³×C₅₃	—	4	2	no	—
46	C₂³×C₅₃	—	8	2	no	—
47	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₅₃	—	2	2	no	—
50	D₄×C₅₃	—	2	4	no	—
51	D₄×C₅₃	—	2	2	no	—
52	D₄×C₅₃	—	2	4	no	—
53	D₄×C₅₃	—	4	1	no	—
54	D₄×C₅₃	—	4	1	no	—
55	D₄×C₅₃	—	8	4	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#41	1	2	yes	—
64	C₂×Dic₅₃	—	4	2	no	—
65	#88322	—	2	4	no	—
66	C₂²×C₅₃	—	2	2	no	—
67	C₂²×C₅₃	—	4	1	no	—
68	C₂²×C₅₃	—	4	2	no	—
69	C₂²×C₅₃	—	4	4	no	—
70	C₂²×C₅₃	—	4	2	no	—
71	C₂²×C₅₃	—	8	4	no	—
72	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₅₃	#121	1	2	yes	—
74	C₄×C₅₃	—	2	2	no	—
75	C₄×C₅₃	—	4	2	no	—
76	C₄×C₅₃	C₄×D₄	1	2	yes	—
77	C₄×C₅₃	#120	1	1	yes	characteristic
78	C₄×C₅₃	#123	1	1	yes	characteristic
79	Dic₅₃	—	8	2	no	—
80	#1910	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	—	2	1	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	#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	C₂×D₅₃	1	1	yes	characteristic
94	#131	—	106	4	no	—
95	C₂×M₁₆	—	53	2	no	—
96	C₂×M₁₆	—	106	2	no	—
97	#127	Dic₅₃	1	2	yes	complemented
98	#124	Dic₅₃	1	2	yes	complemented
99	#121	—	2	2	no	—
100	#120	—	2	2	no	—
101	#117	—	53	2	no	—
102	#117	C₂×D₅₃	1	1	yes	characteristic
103	#102	—	106	4	no	—
104	#99	—	106	2	no	—
105	C₂⁴	—	2	1	no	—
106	#41	—	2	2	no	—
107	#41	—	2	2	no	—
108	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
109	C₂×D₄	—	2	4	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
114	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
115	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
116	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
118	M₁₆	—	106	4	no	—
119	M₁₆	—	106	2	no	—
120	M₁₆	—	106	2	no	—
121	C₂×C₈	—	53	4	no	—
122	C₂×C₈	—	106	4	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	106	2	no	—
125	C₂²⋊C₄	—	2	1	no	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	106	2	no	—
128	C₄²	—	106	2	no	—
129	C₄²	C₂×Dic₅₃	1	2	yes	—
130	C₂³	—	2	2	no	—
131	C₂³	—	4	2	no	—
132	C₂³	—	8	2	no	—
133	C₂³	C₂²⋊Dic₅₃	1	2	yes	—
134	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
135	Q₈	—	2	2	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	2	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	8	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	212	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	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
151	C₈	—	106	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	8	4	no	—
158	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
159	C₂²	#13133189	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	—	424	2	no	—
163	C₄	D₄×Dic₅₃	1	2	yes	—
164	C₄	#13133185	1	1	yes	characteristic
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

#16810628

C₂

yes

maximal, complemented

#16810597

C₂

yes

maximal, characteristic, complemented

#16810379

C₂

yes

maximal, complemented

C₅₃×#535

C₂

yes

maximal, characteristic

#13133224

—

C₂×#11719917

—

C₂×#11719917

C₂²

yes

—

#13133176

C₂²

yes

—

#13133138

—

#13133132

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂²×D₄×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₄

yes

complemented

#124×C₅₃

C₄

yes

complemented

#121×C₅₃

—

#120×C₅₃

—

#117×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

D₄

yes

complemented

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂×D₄

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1910

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

D₅₃

yes

—

—

—

C₅₃

#1910

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

C₂²×D₄

C₂×D₅₃

yes

characteristic

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

#127

Dic₅₃

yes

complemented

#124

Dic₅₃

yes

complemented

#121

—

100

#120

—

101

#117

—

102

#117

C₂×D₅₃

yes

characteristic

103

#102

—

106

—

104

#99

—

106

—

105

C₂⁴

—

106

#41

—

107

#41

—

108

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

C₂×Dic₅₃

yes

—

114

C₂×D₄

C₂×Dic₅₃

yes

characteristic

115

C₂×D₄

C₂²⋊D₅₃

yes

—

116

C₂²×C₄

C₂²⋊D₅₃

yes

—

117

C₂²×C₄

C₂²×D₅₃

yes

characteristic

118

M₁₆

—

106

—

119

M₁₆

—

106

—

120

M₁₆

—

106

—

121

C₂×C₈

—

122

C₂×C₈

—

106

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

106

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

106

—

128

C₄²

—

106

—

129

C₄²

C₂×Dic₅₃

yes

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

C₂²⋊Dic₅₃

yes

—

134

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

135

Q₈

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

212

—

146

C₂×C₄

C₂²⋊Dic₅₃

yes

—

147

C₂×C₄

D₄×D₅₃

yes

—

148

C₂×C₄

#11719947

yes

—

149

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

151

C₈

—

106

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

159

C₂²

#13133189

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

424

—

163

C₄

D₄×Dic₅₃

yes

—

164

C₄

#13133185

yes

characteristic

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	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#535	C₂
socle	C₂×C₅₃	#386
hypercenter	#535	D₅₃
hypocenter	C₅₃	#1910
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

subgroup

G/H

center

C₂

#16810503

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#535

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

N	H ≅ G/N	type
1. #16811390	C₂	semidirect
2. #16810628	C₂	semidirect
3. #16810597	C₂	semidirect
4. #16810379	C₂	semidirect
14. #127×C₅₃	C₄	semidirect
15. #124×C₅₃	C₄	semidirect
43. C₂×#88322	D₄	semidirect
91. C₅₃	#1910	semidirect
97. #127	Dic₅₃	semidirect
98. #124	Dic₅₃	semidirect

H ≅ G/N

type

1. #16811390

C₂

semidirect

2. #16810628

C₂

semidirect

3. #16810597

C₂

semidirect

4. #16810379

C₂

semidirect

14. #127×C₅₃

C₄

semidirect

15. #124×C₅₃

C₄

semidirect

43. C₂×#88322

D₄

semidirect

91. C₅₃

#1910

semidirect

97. #127

Dic₅₃

semidirect

98. #124

Dic₅₃

semidirect

Subgroups — #20275243

Conjugacy classes of subgroups

Complements