Group #20273414 — subgroups

Subgroups — #20273414

Order 6784.

208 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16811178	C₂	1	1	yes	maximal, characteristic
2	#16810679	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810370	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810369	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₅₃	—	2	1	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133208	C₄	1	2	yes	—
11	#13133197	—	2	1	no	—
12	#13133193	—	2	1	no	—
13	Q₈×Dic₅₃	—	2	1	no	—
14	#13133182	—	2	1	no	—
15	#13133175	C₄	1	2	yes	—
16	C₄²×D₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₄⋊#88322	C₂²	1	1	yes	characteristic
19	#13133146	—	2	1	no	—
20	#13133129	—	2	1	no	—
21	#13133127	—	2	1	no	—
22	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₅₃	—	2	1	no	—
26	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₆	—	2	1	no	—
30	C₂×C₈×C₅₃	—	2	1	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#11719921	—	2	2	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
39	#11719919	—	2	2	no	—
40	C₄×Dic₅₃	—	2	2	no	—
41	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	D₄	1	1	yes	characteristic
43	C₄×Dic₅₃	D₄	1	1	yes	characteristic
44	C₂×#88322	—	2	1	no	—
45	C₂×#88322	—	2	1	no	—
46	C₈×D₅₃	—	4	2	no	—
47	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	Q₈×C₅₃	—	4	1	no	—
49	Q₈×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₅₃	—	2	2	no	—
59	C₄×D₅₃	—	2	1	no	—
60	C₄×D₅₃	—	2	1	no	—
61	C₄×D₅₃	—	2	1	no	—
62	C₄×D₅₃	—	2	1	no	—
63	Dic₁₀₆	—	4	1	no	—
64	Dic₁₀₆	—	4	1	no	—
65	C₈×C₅₃	—	4	1	no	—
66	C₈×C₅₃	—	4	1	no	—
67	#88322	—	4	1	no	—
68	#88322	—	4	1	no	—
69	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	—	2	1	no	—
72	C₂×D₅₃	#101	1	2	yes	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	4	2	no	—
75	C₄×C₅₃	#135	1	1	yes	characteristic
76	C₄×C₅₃	#135	1	1	yes	characteristic
77	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₅₃	#137	1	1	yes	characteristic
79	Dic₅₃	—	2	1	no	—
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	—	2	1	no	—
82	Dic₅₃	—	4	2	no	—
83	#1552	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
85	C₂×C₅₃	#416	1	1	yes	characteristic
86	C₂×C₅₃	#417	1	1	yes	characteristic
87	D₅₃	—	2	2	no	—
88	#533	—	53	1	no	—
89	#432	—	53	1	no	—
90	C₂×C₄⋊C₈	—	53	1	no	—
91	#332	—	53	1	no	—
92	#332	—	53	1	no	—
93	#332	—	53	1	no	—
94	#332	D₅₃	1	1	yes	characteristic, complemented
95	C₅₃	#1552	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	106	1	no	—
97	C₄⋊Q₈	—	53	1	no	—
98	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
99	#125	—	53	2	no	—
100	C₂²⋊Q₈	—	106	1	no	—
101	C₂²⋊Q₈	—	106	1	no	—
102	C₄×Q₈	—	106	1	no	—
103	C₄×Q₈	—	106	1	no	—
104	#117	—	53	2	no	—
105	C₂×C₄²	—	53	1	no	—
106	C₄⋊C₈	—	53	1	no	—
107	C₄⋊C₈	—	53	1	no	—
108	C₄⋊C₈	—	106	1	no	—
109	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
111	C₂²⋊C₈	—	106	1	no	—
112	#97	—	106	1	no	—
113	C₂×Q₈	—	2	1	no	—
114	C₂×Q₈	—	106	1	no	—
115	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	53	1	no	—
117	C₂²×C₄	—	53	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	106	1	no	—
121	C₂×C₈	—	106	1	no	—
122	C₂×C₈	—	212	2	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	53	2	no	—
125	C₄⋊C₄	—	106	2	no	—
126	C₄⋊C₄	—	106	2	no	—
127	C₄⋊C₄	—	106	1	no	—
128	C₄⋊C₄	C₄×D₅₃	1	2	yes	—
129	C₂²⋊C₄	—	106	2	no	—
130	C₄²	—	53	1	no	—
131	C₄²	—	53	1	no	—
132	C₄²	—	53	1	no	—
133	C₄²	—	106	2	no	—
134	C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	C₂³	—	53	1	no	—
136	Q₈	—	4	1	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	212	1	no	—
139	Q₈	—	212	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	53	1	no	—
142	C₂×C₄	—	53	1	no	—
143	C₂×C₄	—	53	1	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	—	106	1	no	—
146	C₂×C₄	—	106	1	no	—
147	C₂×C₄	—	106	1	no	—
148	C₂×C₄	—	106	1	no	—
149	C₂×C₄	—	106	2	no	—
150	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
151	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
152	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
153	C₈	—	4	1	no	—
154	C₈	—	4	1	no	—
155	C₈	—	212	1	no	—
156	C₈	—	212	1	no	—
157	C₂²	—	53	2	no	—
158	C₂²	—	106	1	no	—
159	C₂²	—	106	1	no	—
160	C₂²	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	4	2	no	—
163	C₄	—	106	1	no	—
164	C₄	—	106	1	no	—
165	C₄	—	106	1	no	—
166	C₄	—	212	2	no	—
167	C₄	#13133238	1	1	yes	characteristic
168	C₄	#13133240	1	1	yes	characteristic
169	C₄	#13133241	1	1	yes	characteristic
170	C₄	#13133246	1	1	yes	characteristic
171	C₂	—	106	2	no	—
172	C₂	#101×D₅₃	1	1	yes	characteristic
173	C₂	#16810618	1	1	yes	characteristic
174	C₂	#16810621	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811178

C₂

yes

maximal, characteristic

#16810679

C₂

yes

maximal, characteristic

C₄⋊C₈×D₅₃

C₂

yes

maximal, characteristic

#16810370

C₂

yes

maximal, characteristic, complemented

#16810369

C₂

yes

maximal, characteristic, complemented

#16810368

C₂

yes

maximal, characteristic, complemented

C₅₃×#332

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₅₃

—

#13133217

C₂²

yes

characteristic

#13133208

C₄

yes

—

#13133197

—

#13133193

—

Q₈×Dic₅₃

—

#13133182

—

#13133175

C₄

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

—

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

—

C₈×D₅₃

—

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×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×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#101

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#533

—

#432

—

C₂×C₄⋊C₈

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1552

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#125

—

100

C₂²⋊Q₈

—

106

—

101

C₂²⋊Q₈

—

106

—

102

C₄×Q₈

—

106

—

103

C₄×Q₈

—

106

—

104

#117

—

105

C₂×C₄²

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

106

—

109

C₄⋊C₈

C₂×D₅₃

yes

characteristic

110

C₄⋊C₈

C₂×D₅₃

yes

characteristic

111

C₂²⋊C₈

—

106

—

112

#97

—

106

—

113

C₂×Q₈

—

114

C₂×Q₈

—

106

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

106

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

212

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

106

—

128

C₄⋊C₄

C₄×D₅₃

yes

—

129

C₂²⋊C₄

—

106

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

106

—

134

C₄²

C₂²×D₅₃

yes

characteristic

135

C₂³

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

212

—

139

Q₈

—

212

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

106

—

145

C₂×C₄

—

106

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

106

—

150

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

151

C₂×C₄

D₄×D₅₃

yes

characteristic

152

C₂×C₄

D₄×D₅₃

yes

characteristic

153

C₈

—

154

C₈

—

155

C₈

—

212

—

156

C₈

—

212

—

157

C₂²

—

158

C₂²

—

106

—

159

C₂²

—

106

—

160

C₂²

C₂²⋊C₄×D₅₃

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

106

—

164

C₄

—

106

—

165

C₄

—

106

—

166

C₄

—

212

—

167

C₄

#13133238

yes

characteristic

168

C₄

#13133240

yes

characteristic

169

C₄

#13133241

yes

characteristic

170

C₄

#13133246

yes

characteristic

171

C₂

—

106

—

172

C₂

#101×D₅₃

yes

characteristic

173

C₂

#16810618

yes

characteristic

174

C₂

#16810621

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#332	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#332	D₅₃
hypocenter	C₅₃	#1552
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#332

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #16810370	C₂	semidirect
5. #16810369	C₂	semidirect
6. #16810368	C₂	semidirect
7. C₅₃×#332	C₂	semidirect
94. #332	D₅₃	semidirect
95. C₅₃	#1552	semidirect

H ≅ G/N

type

4. #16810370

C₂

semidirect

5. #16810369

C₂

semidirect

6. #16810368

C₂

semidirect

7. C₅₃×#332

C₂

semidirect

94. #332

D₅₃

semidirect

95. C₅₃

#1552

semidirect

Subgroups — #20273414

Conjugacy classes of subgroups

Complements