Group #20287137 — subgroups

Subgroups — #20287137

Order 6784.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16811339	C₂	1	2	yes	maximal, complemented
2	#16810950	C₂	1	4	yes	maximal, complemented
3	#16810910	C₂	1	2	yes	maximal, complemented
4	#16810860	C₂	1	2	yes	maximal, complemented
5	C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810660	C₂	1	2	yes	maximal, complemented
8	C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₅₃	C₂²	1	2	yes	—
11	#13133206	—	2	4	no	—
12	#13133192	—	2	2	no	—
13	#13133192	C₂²	1	4	yes	—
14	C₄×D₂₁₂	C₂²	1	2	yes	—
15	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
16	#13133175	—	2	2	no	—
17	#13133175	C₂²	1	4	yes	—
18	#13133151	C₂²	1	1	yes	characteristic
19	#13133148	C₂²	1	4	yes	—
20	C₄⋊#88322	C₂²	1	1	yes	characteristic
21	#13133139	C₂²	1	2	yes	—
22	#13133137	C₂²	1	4	yes	—
23	#13133129	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
25	#107×C₅₃	C₂²	1	2	yes	—
26	#107×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	—	2	2	no	—
29	C₂×D₂₁₂	—	2	4	no	—
30	C₂×D₂₁₂	D₄	1	1	yes	characteristic
31	C₂×D₂₁₂	C₂³	1	2	yes	—
32	C₂×C₄×D₅₃	—	2	2	no	—
33	C₂×C₄×D₅₃	C₂³	1	2	yes	—
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#11719921	—	2	4	no	—
40	#11719921	—	2	2	no	—
41	#11719921	—	2	4	no	—
42	#11719921	D₄	1	2	yes	—
43	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
44	C₄⋊Dic₅₃	C₂³	1	2	yes	—
45	C₄×Dic₅₃	—	2	4	no	—
46	C₂×#88322	C₂³	1	2	yes	—
47	D₄₂₄	—	2	2	no	—
48	C₈×D₅₃	—	2	4	no	—
49	C₂²×D₅₃	—	2	2	no	—
50	C₂²×D₅₃	C₂×D₄	1	2	yes	—
51	Q₈×C₅₃	—	8	2	no	—
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
58	D₂₁₂	—	2	2	no	—
59	D₂₁₂	—	2	2	no	—
60	D₂₁₂	—	2	2	no	—
61	D₂₁₂	—	8	4	no	—
62	C₄×D₅₃	—	2	2	no	—
63	C₄×D₅₃	—	2	2	no	—
64	C₄×D₅₃	—	4	4	no	—
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	#41	1	2	yes	—
67	#88322	—	2	2	no	—
68	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₅₃	—	2	2	no	—
70	C₂×D₅₃	—	2	2	no	—
71	C₂×D₅₃	—	2	2	no	—
72	C₂×D₅₃	—	4	2	no	—
73	C₂×D₅₃	—	4	2	no	—
74	C₂×D₅₃	—	4	2	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	4	4	no	—
77	C₄×C₅₃	#135	1	2	yes	—
78	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
79	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
80	Dic₅₃	—	2	2	no	—
81	Dic₅₃	—	4	2	no	—
82	#3341	—	53	1	no	maximal, Sylow
83	C₂×C₅₃	#547	1	1	yes	characteristic
84	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
85	C₂×C₅₃	#576	1	1	yes	characteristic
86	D₅₃	—	4	2	no	—
87	D₅₃	—	8	2	no	—
88	#555	—	53	2	no	—
89	#500	D₅₃	1	1	yes	characteristic, complemented
90	#482	—	53	4	no	—
91	#476	—	53	2	no	—
92	#466	—	53	2	no	—
93	C₄×D₈	—	53	1	no	—
94	C₈×D₄	—	53	1	no	—
95	#427	—	53	2	no	—
96	C₅₃	#3341	1	1	yes	characteristic, complemented, Sylow
97	C₂×D₈	—	53	1	no	—
98	C₂²×C₈	—	53	2	no	—
99	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
100	#124	—	106	4	no	—
101	#121	—	53	4	no	—
102	#121	—	106	2	no	—
103	C₄×D₄	—	53	1	no	—
104	C₄×D₄	—	53	2	no	—
105	#117	—	53	4	no	—
106	#117	—	106	2	no	—
107	#107	—	53	1	no	—
108	#107	C₂×D₅₃	1	2	yes	—
109	#107	C₂×D₅₃	1	2	yes	—
110	#106	—	53	4	no	—
111	C₄⋊C₈	—	53	1	no	—
112	#102	—	53	2	no	—
113	#102	—	53	4	no	—
114	C₂²⋊C₈	—	53	2	no	—
115	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	C₂×Q₈	—	2	2	no	—
117	C₂×D₄	—	53	1	no	—
118	C₂×D₄	—	53	2	no	—
119	C₂×D₄	—	106	4	no	—
120	C₂²×C₄	—	53	2	no	—
121	C₂²×C₄	—	106	2	no	—
122	D₈	—	106	2	no	—
123	C₂×C₈	—	53	2	no	—
124	C₂×C₈	—	106	4	no	—
125	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	53	1	no	—
129	C₄⋊C₄	—	53	2	no	—
130	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
131	C₂²⋊C₄	—	53	2	no	—
132	C₂²⋊C₄	—	106	4	no	—
133	C₂²⋊C₄	—	106	4	no	—
134	C₂²⋊C₄	—	106	2	no	—
135	C₄²	—	106	4	no	—
136	C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₂³	—	53	2	no	—
138	C₂³	—	106	2	no	—
139	Q₈	—	8	2	no	—
140	D₄	—	106	2	no	—
141	D₄	—	106	2	no	—
142	D₄	—	106	2	no	—
143	D₄	—	424	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	53	2	no	—
146	C₂×C₄	—	106	2	no	—
147	C₂×C₄	—	106	2	no	—
148	C₂×C₄	—	106	2	no	—
149	C₂×C₄	—	212	4	no	—
150	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
151	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
152	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	106	2	no	—
155	C₈	#11719949	1	2	yes	—
156	C₂²	—	106	2	no	—
157	C₂²	—	106	2	no	—
158	C₂²	—	106	2	no	—
159	C₂²	—	212	2	no	—
160	C₂²	—	212	2	no	—
161	C₂²	—	212	2	no	—
162	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	4	4	no	—
165	C₄	—	106	2	no	—
166	C₄	—	212	2	no	—
167	C₄	#13133240	1	2	yes	—
168	C₄	C₂×#11719949	1	1	yes	characteristic
169	C₄	#13133265	1	1	yes	characteristic
170	C₂	—	424	2	no	—
171	C₂	—	212	2	no	—
172	C₂	#16811276	1	1	yes	characteristic
173	C₂	C₂×#13133240	1	1	yes	characteristic
174	C₂	#16811424	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811339

C₂

yes

maximal, complemented

#16810950

C₂

yes

maximal, complemented

#16810910

C₂

yes

maximal, complemented

#16810860

C₂

yes

maximal, complemented

C₄×D₄₂₄

C₂

yes

maximal, characteristic, complemented

C₈×D₂₁₂

C₂

yes

maximal, characteristic, complemented

#16810660

C₂

yes

maximal, complemented

C₅₃×#500

C₂

yes

maximal, characteristic, complemented

C₂×D₄₂₄

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

—

#13133206

—

#13133192

—

#13133192

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

—

C₄×D₂₁₂

C₂²

yes

characteristic

#13133175

—

#13133175

C₂²

yes

—

#13133151

C₂²

yes

characteristic

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

#13133139

C₂²

yes

—

#13133137

C₂²

yes

—

#13133129

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

D₄

yes

characteristic

C₂×D₂₁₂

C₂³

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

—

—

D₄

yes

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

—

C₂×#88322

C₂³

yes

—

D₄₂₄

—

C₈×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×D₄

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

#555

—

#500

D₅₃

yes

characteristic, complemented

#482

—

#476

—

#466

—

C₄×D₈

—

C₈×D₄

—

#427

—

C₅₃

#3341

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

100

#124

—

106

—

101

#121

—

102

#121

—

106

—

103

C₄×D₄

—

104

C₄×D₄

—

105

#117

—

106

#117

—

106

—

107

#107

—

108

#107

C₂×D₅₃

yes

—

109

#107

C₂×D₅₃

yes

—

110

#106

—

111

C₄⋊C₈

—

112

#102

—

113

#102

—

114

C₂²⋊C₈

—

115

C₄×C₈

C₂×D₅₃

yes

characteristic

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

106

—

120

C₂²×C₄

—

121

C₂²×C₄

—

106

—

122

D₈

—

106

—

123

C₂×C₈

—

124

C₂×C₈

—

106

—

125

C₂×C₈

C₂²×D₅₃

yes

characteristic

126

C₂×C₈

C₂²×D₅₃

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₂²×D₅₃

yes

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

106

—

133

C₂²⋊C₄

—

106

—

134

C₂²⋊C₄

—

106

—

135

C₄²

—

106

—

136

C₄²

C₂²×D₅₃

yes

characteristic

137

C₂³

—

138

C₂³

—

106

—

139

Q₈

—

140

D₄

—

106

—

141

D₄

—

106

—

142

D₄

—

106

—

143

D₄

—

424

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

106

—

147

C₂×C₄

—

106

—

148

C₂×C₄

—

106

—

149

C₂×C₄

—

212

—

150

C₂×C₄

D₄×D₅₃

yes

characteristic

151

C₂×C₄

D₄×D₅₃

yes

characteristic

152

C₂×C₄

C₂³×D₅₃

yes

characteristic

153

C₈

—

154

C₈

—

106

—

155

C₈

#11719949

yes

—

156

C₂²

—

106

—

157

C₂²

—

106

—

158

C₂²

—

106

—

159

C₂²

—

212

—

160

C₂²

—

212

—

161

C₂²

—

212

—

162

C₂²

C₂×D₄×D₅₃

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

106

—

166

C₄

—

212

—

167

C₄

#13133240

yes

—

168

C₄

C₂×#11719949

yes

characteristic

169

C₄

#13133265

yes

characteristic

170

C₂

—

424

—

171

C₂

—

212

—

172

C₂

#16811276

yes

characteristic

173

C₂

C₂×#13133240

yes

characteristic

174

C₂

#16811424

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#500	D₅₃
hypocenter	C₅₃	#3341
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#500

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #16811339	C₂	semidirect
2. #16810950	C₂	semidirect
3. #16810910	C₂	semidirect
4. #16810860	C₂	semidirect
5. C₄×D₄₂₄	C₂	semidirect
6. C₈×D₂₁₂	C₂	semidirect
7. #16810660	C₂	semidirect
8. C₅₃×#500	C₂	semidirect
89. #500	D₅₃	semidirect
96. C₅₃	#3341	semidirect

H ≅ G/N

type

1. #16811339

C₂

semidirect

2. #16810950

C₂

semidirect

3. #16810910

C₂

semidirect

4. #16810860

C₂

semidirect

5. C₄×D₄₂₄

C₂

semidirect

6. C₈×D₂₁₂

C₂

semidirect

7. #16810660

C₂

semidirect

8. C₅₃×#500

C₂

semidirect

89. #500

D₅₃

semidirect

96. C₅₃

#3341

semidirect

Subgroups — #20287137

Conjugacy classes of subgroups

Complements