Group #20274465 — subgroups

Subgroups — #20274465

Order 6784.

240 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#13133127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810469	C₂	1	1	yes	maximal, characteristic
4	#16810378	C₂	1	1	yes	maximal, characteristic
5	#16810377	C₂	1	1	yes	maximal, characteristic
6	#16810377	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133127	C₄	1	4	yes	complemented
13	C₈×Dic₅₃	C₄	1	4	yes	complemented
14	#13133124	C₄	1	2	yes	—
15	#13133124	C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
22	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄²×C₅₃	—	2	1	no	—
31	C₄²×C₅₃	—	2	1	no	—
32	C₄×Dic₅₃	—	2	1	no	—
33	C₄×Dic₅₃	—	2	1	no	—
34	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	C₂×C₄	1	2	yes	—
41	C₂×#88322	C₂×C₄	1	2	yes	—
42	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
52	C₂×Dic₅₃	—	2	2	no	—
53	C₂×Dic₅₃	—	2	2	no	—
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	2	2	no	—
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	C₄²	1	2	yes	—
58	C₂×Dic₅₃	C₄²	1	2	yes	—
59	C₈×C₅₃	—	2	2	no	—
60	C₈×C₅₃	—	2	2	no	—
61	#88322	—	2	2	no	—
62	#88322	—	2	2	no	—
63	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₅₃	#117	1	1	yes	characteristic
65	C₂²×C₅₃	#117	1	1	yes	characteristic
66	C₂²×C₅₃	#131	1	2	yes	—
67	C₂²×C₅₃	#131	1	2	yes	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	#117	1	1	yes	characteristic
70	C₄×C₅₃	#117	1	1	yes	characteristic
71	C₄×C₅₃	#117	1	1	yes	characteristic
72	C₄×C₅₃	#117	1	1	yes	characteristic
73	Dic₅₃	—	2	2	no	—
74	Dic₅₃	—	2	2	no	—
75	Dic₅₃	—	4	2	no	—
76	#1746	—	53	1	no	maximal, Sylow
77	C₂×C₅₃	#376	1	1	yes	characteristic
78	C₂×C₅₃	#405	1	1	yes	characteristic
79	C₂×C₅₃	#405	1	1	yes	characteristic
80	C₂×C₅₃	#433	1	2	yes	—
81	C₂×C₅₃	#433	1	2	yes	—
82	C₂×#97	—	53	1	no	—
83	C₂×C₄×C₈	—	53	1	no	—
84	#376	—	53	1	no	—
85	#336	—	53	1	no	—
86	#336	—	53	1	no	—
87	#336	—	53	1	no	—
88	#336	D₅₃	1	1	yes	characteristic
89	C₅₃	#1746	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	53	1	no	—
91	C₂²×C₈	—	53	1	no	—
92	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
94	C₂×C₄²	—	53	1	no	—
95	C₂×C₄²	—	53	1	no	—
96	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
97	#97	—	53	4	no	—
98	C₄×C₈	—	53	4	no	—
99	#95	—	53	2	no	—
100	#95	—	53	2	no	—
101	C₂²×C₄	—	53	1	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	2	no	—
104	C₂²×C₄	C₄×D₅₃	1	2	yes	—
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	53	2	no	—
111	C₂×C₈	—	53	2	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	—	106	1	no	—
115	C₂×C₈	—	106	1	no	—
116	C₂×C₈	C₄×D₅₃	1	2	yes	—
117	C₂×C₈	C₄×D₅₃	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	53	2	no	—
121	C₄²	—	53	2	no	—
122	C₄²	—	106	1	no	—
123	C₄²	—	106	1	no	—
124	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	—	53	2	no	—
130	C₂×C₄	—	106	2	no	—
131	C₂×C₄	—	106	2	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	106	2	no	—
134	C₂×C₄	—	106	2	no	—
135	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₄	#11719947	1	1	yes	characteristic
138	C₂×C₄	#11719947	1	1	yes	characteristic
139	C₂×C₄	#11719947	1	1	yes	characteristic
140	C₂×C₄	#11719949	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	2	no	—
143	C₈	—	106	2	no	—
144	C₈	—	106	2	no	—
145	C₂²	C₄²×D₅₃	1	1	yes	characteristic
146	C₂²	#13133173	1	1	yes	characteristic
147	C₂²	#13133175	1	1	yes	characteristic
148	C₂²	#13133223	1	2	yes	—
149	C₂²	#13133225	1	2	yes	—
150	C₄	—	4	2	no	—
151	C₄	—	106	2	no	—
152	C₄	—	106	2	no	—
153	C₄	—	212	2	no	—
154	C₄	#13133173	1	1	yes	characteristic
155	C₄	#13133173	1	1	yes	characteristic
156	C₄	#13133175	1	1	yes	characteristic
157	C₄	#13133175	1	1	yes	characteristic
158	C₂	#16810468	1	1	yes	characteristic
159	C₂	#16810577	1	1	yes	characteristic
160	C₂	#16810580	1	1	yes	characteristic
161	C₂	#16810681	1	2	yes	—
162	C₂	#16810683	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133127

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810469

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal, characteristic

C₅₃×#336

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133127

C₄

yes

complemented

C₈×Dic₅₃

C₄

yes

complemented

#13133124

C₄

yes

—

#13133124

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

C₄²

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#433

yes

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#1746

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#97

—

C₄×C₈

—

#95

—

100

#95

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₄×D₅₃

yes

—

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

C₄×D₅₃

yes

—

117

C₂×C₈

C₄×D₅₃

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

106

—

123

C₄²

—

106

—

124

C₂³

C₂×C₄×D₅₃

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

106

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

106

—

135

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

136

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

137

C₂×C₄

#11719947

yes

characteristic

138

C₂×C₄

#11719947

yes

characteristic

139

C₂×C₄

#11719947

yes

characteristic

140

C₂×C₄

#11719949

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

106

—

144

C₈

—

106

—

145

C₂²

C₄²×D₅₃

yes

characteristic

146

C₂²

#13133173

yes

characteristic

147

C₂²

#13133175

yes

characteristic

148

C₂²

#13133223

yes

—

149

C₂²

#13133225

yes

—

150

C₄

—

151

C₄

—

106

—

152

C₄

—

106

—

153

C₄

—

212

—

154

C₄

#13133173

yes

characteristic

155

C₄

#13133173

yes

characteristic

156

C₄

#13133175

yes

characteristic

157

C₄

#13133175

yes

characteristic

158

C₂

#16810468

yes

characteristic

159

C₂

#16810577

yes

characteristic

160

C₂

#16810580

yes

characteristic

161

C₂

#16810681

yes

—

162

C₂

#16810683

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₅₃×#336	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1746
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Fitting

C₅₃×#336

C₂

socle

C₂³×C₅₃

C₂²×C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
12. #13133127	C₄	semidirect
13. C₈×Dic₅₃	C₄	semidirect
89. C₅₃	#1746	semidirect

H ≅ G/N

type

12. #13133127

C₄

semidirect

13. C₈×Dic₅₃

C₄

semidirect

89. C₅₃

#1746

semidirect

Subgroups — #20274465

Conjugacy classes of subgroups

Complements