Group #20291024 — subgroups

Subgroups — #20291024

Order 6784.

260 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 64 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	D₅₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811633	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811535	C₂	1	4	yes	maximal, complemented
4	C₂×D₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#13133296	—	2	1	no	—
6	#13133292	—	2	1	no	—
7	#13133291	—	2	1	no	—
8	D₅₃⋊C₁₆	C₂²	1	2	yes	complemented
9	#13133286	C₂²	1	2	yes	complemented
10	#13133285	—	2	1	no	—
11	C₂×D₄×D₅₃	—	2	1	no	—
12	C₂×D₄₂₄	C₄	1	1	yes	characteristic, complemented
13	C₂×#11719922	—	2	1	no	—
14	D₈×D₅₃	—	2	1	no	—
15	D₈×D₅₃	—	2	1	no	—
16	D₈×D₅₃	C₄	1	2	yes	complemented
17	D₈×D₅₃	C₂²	1	2	yes	—
18	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₂³×D₅₃	—	4	1	no	—
21	C₂×D₄×C₅₃	—	2	1	no	—
22	C₂×C₂²⋊D₅₃	—	4	1	no	—
23	D₄×D₅₃	—	2	2	no	—
24	D₄×D₅₃	—	2	2	no	—
25	D₄×D₅₃	—	4	1	no	—
26	D₄×D₅₃	—	4	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
29	C₂²⋊#6157	—	4	1	no	—
30	C₄⋊#6157	—	2	2	no	—
31	C₄×#6157	—	4	1	no	—
32	D₈×C₅₃	—	2	1	no	—
33	D₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#11719922	—	2	2	no	—
36	#11719922	—	2	2	no	—
37	C₂×#88322	D₄	1	1	yes	characteristic
38	D₄₂₄	—	2	1	no	—
39	D₄₂₄	C₂×C₄	1	2	yes	—
40	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
41	C₈×D₅₃	D₄	1	1	yes	characteristic
42	C₈×D₅₃	D₄	1	1	yes	characteristic
43	C₈×D₅₃	C₂³	1	1	yes	characteristic
44	#11719910	—	2	2	no	—
45	C₂³×C₅₃	—	4	1	no	—
46	C₂²×D₅₃	—	4	1	no	—
47	C₂²×D₅₃	—	4	2	no	—
48	C₂²×D₅₃	—	4	2	no	—
49	C₂²×D₅₃	—	8	4	no	—
50	C₂²×D₅₃	D₈	1	1	yes	characteristic
51	C₂×#6157	—	4	2	no	—
52	D₄×C₅₃	—	2	2	no	—
53	D₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²⋊D₅₃	—	4	2	no	—
56	C₂²⋊D₅₃	—	4	2	no	—
57	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
58	D₂₁₂	—	2	2	no	—
59	D₂₁₂	—	4	1	no	—
60	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×D₅₃	D₈	1	1	yes	characteristic
62	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
63	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
64	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	#88322	C₂²⋊C₄	1	1	yes	characteristic
67	#88322	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₅₃	—	4	2	no	—
69	C₂²×C₅₃	—	8	2	no	—
70	C₂²×C₅₃	#102	1	1	yes	characteristic
71	C₂×D₅₃	—	2	2	no	—
72	C₂×D₅₃	—	4	2	no	—
73	C₂×D₅₃	—	8	4	no	—
74	C₂×D₅₃	—	8	2	no	—
75	C₂×D₅₃	—	8	4	no	—
76	C₂×D₅₃	#102	1	1	yes	characteristic
77	C₂×D₅₃	C₂×D₈	1	1	yes	characteristic
78	#6157	—	8	2	no	—
79	C₄×C₅₃	#102	1	1	yes	characteristic
80	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Dic₅₃	#102	1	1	yes	characteristic
82	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	#2140	—	53	1	no	maximal, Sylow
84	C₂×C₅₃	—	2	1	no	—
85	C₂×C₅₃	—	8	2	no	—
86	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
87	D₅₃	—	2	1	no	—
88	D₅₃	—	8	2	no	—
89	D₅₃	#509	1	2	yes	—
90	C₂²×D₈	—	53	1	no	—
91	C₂×M₃₂	—	53	1	no	—
92	#427	—	53	1	no	—
93	#357	—	53	4	no	—
94	C₅₃	#2140	1	1	yes	characteristic, complemented, Sylow
95	C₂²×D₄	—	106	1	no	—
96	C₂×D₈	—	53	2	no	—
97	C₂×D₈	—	53	2	no	—
98	C₂×D₈	—	53	1	no	—
99	C₂×D₈	—	106	1	no	—
100	C₂×D₈	—	106	1	no	—
101	C₂×D₈	—	106	1	no	—
102	C₂×D₈	#6157	1	1	yes	characteristic, complemented
103	C₂²×C₈	—	53	1	no	—
104	#117	—	106	1	no	—
105	M₃₂	—	106	1	no	—
106	M₃₂	—	106	1	no	—
107	C₂×C₁₆	—	53	2	no	—
108	#107	—	53	2	no	—
109	#106	—	106	1	no	—
110	C₂⁴	—	212	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	106	2	no	—
113	C₂×D₄	—	106	2	no	—
114	C₂×D₄	—	106	1	no	—
115	C₂×D₄	—	212	1	no	—
116	C₂×D₄	—	212	1	no	—
117	C₂×D₄	—	212	1	no	—
118	C₂²×C₄	—	53	1	no	—
119	D₈	—	2	1	no	—
120	D₈	—	53	2	no	—
121	D₈	—	106	2	no	—
122	D₈	—	106	2	no	—
123	D₈	—	106	1	no	—
124	D₈	C₂×#6157	1	2	yes	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	53	1	no	—
127	C₂×C₈	—	53	1	no	—
128	C₂×C₈	—	53	1	no	—
129	C₂×C₈	—	53	1	no	—
130	C₂×C₈	C₂×#6157	1	1	yes	characteristic
131	C₄⋊C₄	—	106	2	no	—
132	C₂²⋊C₄	—	212	1	no	—
133	C₄²	—	212	1	no	—
134	C₁₆	—	106	2	no	—
135	C₂³	—	4	1	no	—
136	C₂³	—	53	1	no	—
137	C₂³	—	424	4	no	—
138	C₂³	—	212	2	no	—
139	C₂³	—	212	2	no	—
140	C₂³	—	212	1	no	—
141	D₄	—	2	2	no	—
142	D₄	—	4	1	no	—
143	D₄	—	106	2	no	—
144	D₄	—	212	2	no	—
145	D₄	—	212	2	no	—
146	D₄	—	212	1	no	—
147	C₂×C₄	—	53	1	no	—
148	C₂×C₄	—	53	1	no	—
149	C₂×C₄	—	53	1	no	—
150	C₂×C₄	—	53	1	no	—
151	C₂×C₄	—	53	1	no	—
152	C₂×C₄	—	212	2	no	—
153	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
154	C₈	—	53	1	no	—
155	C₈	—	53	1	no	—
156	C₈	C₂²⋊#6157	1	1	yes	characteristic
157	C₈	C₂²×#6157	1	1	yes	characteristic
158	C₂²	—	4	2	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	53	1	no	—
161	C₂²	—	53	1	no	—
162	C₂²	—	106	2	no	—
163	C₂²	—	424	2	no	—
164	C₂²	—	424	4	no	—
165	C₂²	—	424	4	no	—
166	C₂²	—	212	2	no	—
167	C₂²	#13133279	1	1	yes	characteristic
168	C₄	—	53	1	no	—
169	C₄	—	53	1	no	—
170	C₄	—	424	2	no	—
171	C₄	#13133279	1	1	yes	characteristic
172	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	8	2	no	—
175	C₂	—	53	2	no	—
176	C₂	—	106	1	no	—
177	C₂	—	424	2	no	—
178	C₂	C₂×#13133279	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal, characteristic, complemented

#16811633

C₂

yes

maximal, characteristic, complemented

#16811535

C₂

yes

maximal, complemented

C₂×D₈×D₅₃

C₂

yes

maximal, characteristic

#13133296

—

#13133292

—

#13133291

—

D₅₃⋊C₁₆

C₂²

yes

complemented

#13133286

C₂²

yes

complemented

#13133285

—

C₂×D₄×D₅₃

—

C₂×D₄₂₄

C₄

yes

characteristic, complemented

C₂×#11719922

—

D₈×D₅₃

—

D₈×D₅₃

—

D₈×D₅₃

C₄

yes

complemented

D₈×D₅₃

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂×D₈×C₅₃

C₄

yes

characteristic, complemented

C₂³×D₅₃

—

C₂×D₄×C₅₃

—

C₂×C₂²⋊D₅₃

—

D₄×D₅₃

—

D₄×D₅₃

—

D₄×D₅₃

—

D₄×D₅₃

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂²⋊#6157

—

C₄⋊#6157

—

C₄×#6157

—

D₈×C₅₃

—

D₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

—

D₄

yes

characteristic

D₄₂₄

—

D₄₂₄

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

D₄

yes

characteristic

C₈×D₅₃

C₂³

yes

characteristic

#11719910

—

C₂³×C₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

D₈

yes

characteristic

C₂×#6157

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

SD₁₆

yes

characteristic

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

C₂²⋊C₄

yes

characteristic

C₄×D₅₃

D₈

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₂²×C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂×D₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

#102

yes

characteristic

C₂×D₅₃

C₂×D₈

yes

characteristic

#6157

—

C₄×C₅₃

#102

yes

characteristic

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Dic₅₃

#102

yes

characteristic

Dic₅₃

C₂×SD₁₆

yes

characteristic

#2140

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#102

yes

characteristic

D₅₃

—

D₅₃

—

D₅₃

#509

yes

—

C₂²×D₈

—

C₂×M₃₂

—

#427

—

#357

—

C₅₃

#2140

yes

characteristic, complemented, Sylow

C₂²×D₄

—

106

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

106

—

100

C₂×D₈

—

106

—

101

C₂×D₈

—

106

—

102

C₂×D₈

#6157

yes

characteristic, complemented

103

C₂²×C₈

—

104

#117

—

106

—

105

M₃₂

—

106

—

106

M₃₂

—

106

—

107

C₂×C₁₆

—

108

#107

—

109

#106

—

106

—

110

C₂⁴

—

212

—

111

C₂×D₄

—

112

C₂×D₄

—

106

—

113

C₂×D₄

—

106

—

114

C₂×D₄

—

106

—

115

C₂×D₄

—

212

—

116

C₂×D₄

—

212

—

117

C₂×D₄

—

212

—

118

C₂²×C₄

—

119

D₈

—

120

D₈

—

121

D₈

—

106

—

122

D₈

—

106

—

123

D₈

—

106

—

124

D₈

C₂×#6157

yes

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₂×#6157

yes

characteristic

131

C₄⋊C₄

—

106

—

132

C₂²⋊C₄

—

212

—

133

C₄²

—

212

—

134

C₁₆

—

106

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

424

—

138

C₂³

—

212

—

139

C₂³

—

212

—

140

C₂³

—

212

—

141

D₄

—

142

D₄

—

143

D₄

—

106

—

144

D₄

—

212

—

145

D₄

—

212

—

146

D₄

—

212

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

212

—

153

C₂×C₄

C₂²⋊#6157

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

C₂²⋊#6157

yes

characteristic

157

C₈

C₂²×#6157

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

106

—

163

C₂²

—

424

—

164

C₂²

—

424

—

165

C₂²

—

424

—

166

C₂²

—

212

—

167

C₂²

#13133279

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

424

—

171

C₄

#13133279

yes

characteristic

172

C₄

C₂×C₂²⋊#6157

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

—

176

C₂

—

106

—

177

C₂

—

424

—

178

C₂

C₂×#13133279

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13133279
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	C₂×D₈×C₅₃	C₄
socle	C₂×C₅₃	C₂×#102
hypercenter	C₂×D₈	#6157
hypocenter	C₅₃	#2140
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

subgroup

G/H

center

C₂

C₂×#13133279

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₈

C₂²×#6157

Fitting

C₂×D₈×C₅₃

C₄

C₂×C₅₃

C₂×D₈

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

N	H ≅ G/N	type
1. D₅₃⋊M₃₂	C₂	semidirect
2. #16811633	C₂	semidirect
3. #16811535	C₂	semidirect
8. D₅₃⋊C₁₆	C₂²	semidirect
9. #13133286	C₂²	semidirect
12. C₂×D₄₂₄	C₄	semidirect
16. D₈×D₅₃	C₄	semidirect
19. C₂×D₈×C₅₃	C₄	semidirect
94. C₅₃	#2140	semidirect
102. C₂×D₈	#6157	semidirect

H ≅ G/N

type

1. D₅₃⋊M₃₂

C₂

semidirect

2. #16811633

C₂

semidirect

3. #16811535

C₂

semidirect

8. D₅₃⋊C₁₆

C₂²

semidirect

9. #13133286

C₂²

semidirect

12. C₂×D₄₂₄

C₄

semidirect

16. D₈×D₅₃

C₄

semidirect

19. C₂×D₈×C₅₃

C₄

semidirect

94. C₅₃

#2140

semidirect

102. C₂×D₈

#6157

semidirect

Subgroups — #20291024

Conjugacy classes of subgroups

Complements