Group #20274571 — subgroups

Subgroups — #20274571

Order 6784.

260 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 121 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×#13133146	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133146	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133127	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810469	C₂	1	1	yes	maximal, characteristic
6	#16810378	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	#13133146	C₄	1	4	yes	complemented
13	#13133146	C₄	1	4	yes	complemented
14	#13133127	—	2	2	no	—
15	C₈×Dic₅₃	—	2	2	no	—
16	#13133124	C₄	1	2	yes	—
17	#13133124	C₄	1	2	yes	—
18	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄²×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	—	2	1	no	—
34	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₄×Dic₅₃	D₄	1	2	yes	—
37	C₄×Dic₅₃	Q₈	1	2	yes	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	2	1	no	—
41	C₂×#88322	—	2	1	no	—
42	C₂×#88322	C₂×C₄	1	2	yes	—
43	C₂×#88322	C₂×C₄	1	2	yes	—
44	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×Dic₅₃	—	2	2	no	—
55	C₂×Dic₅₃	—	2	2	no	—
56	C₂×Dic₅₃	—	2	2	no	—
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	C₄²	1	2	yes	—
59	C₂×Dic₅₃	C₄²	1	2	yes	—
60	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
61	C₈×C₅₃	—	2	2	no	—
62	C₈×C₅₃	—	2	2	no	—
63	#88322	—	2	2	no	—
64	#88322	—	2	2	no	—
65	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₅₃	#117	1	1	yes	characteristic
68	C₂²×C₅₃	#131	1	2	yes	—
69	C₂²×C₅₃	#131	1	2	yes	—
70	C₄×C₅₃	—	4	2	no	—
71	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	Dic₅₃	—	2	2	no	—
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	—	2	4	no	—
78	#1768	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₅₃	#405	1	1	yes	characteristic
81	C₂×C₅₃	#405	1	1	yes	characteristic
82	C₂×C₅₃	#424	1	2	yes	—
83	C₂×C₅₃	#433	1	2	yes	—
84	C₂×C₄⋊C₈	—	53	1	no	—
85	C₂×C₄⋊C₈	—	53	1	no	—
86	C₂×#97	—	53	1	no	—
87	C₂×C₄×C₈	—	53	1	no	—
88	#376	—	53	1	no	—
89	#336	—	53	1	no	—
90	#336	D₅₃	1	1	yes	characteristic
91	C₅₃	#1768	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	53	1	no	—
93	C₂²×C₈	—	53	1	no	—
94	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₂×C₄²	—	53	1	no	—
97	C₂×C₄²	—	53	1	no	—
98	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
99	C₄⋊C₈	—	53	4	no	—
100	C₄⋊C₈	—	53	4	no	—
101	#97	—	106	2	no	—
102	C₄×C₈	—	106	2	no	—
103	#95	—	53	2	no	—
104	#95	—	53	2	no	—
105	C₂²×C₄	—	53	1	no	—
106	C₂²×C₄	—	53	1	no	—
107	C₂²×C₄	—	53	2	no	—
108	C₂²×C₄	C₄×D₅₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	53	2	no	—
115	C₂×C₈	—	53	2	no	—
116	C₂×C₈	—	106	1	no	—
117	C₂×C₈	—	106	1	no	—
118	C₂×C₈	—	106	1	no	—
119	C₂×C₈	—	106	1	no	—
120	C₂×C₈	C₄×D₅₃	1	2	yes	—
121	C₂×C₈	C₄×D₅₃	1	2	yes	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	53	2	no	—
125	C₄²	—	53	2	no	—
126	C₄²	—	53	2	no	—
127	C₄²	—	53	2	no	—
128	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	53	2	no	—
133	C₂×C₄	—	53	2	no	—
134	C₂×C₄	—	53	4	no	—
135	C₂×C₄	—	106	2	no	—
136	C₂×C₄	—	106	2	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
141	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
142	C₂×C₄	#11719947	1	1	yes	characteristic
143	C₂×C₄	#11719947	1	1	yes	characteristic
144	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	106	2	no	—
148	C₈	—	106	2	no	—
149	C₂²	C₄²×D₅₃	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
151	C₂²	#13133173	1	1	yes	characteristic
152	C₂²	#13133223	1	2	yes	—
153	C₂²	#13133225	1	2	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	106	2	no	—
156	C₄	—	106	2	no	—
157	C₄	—	106	4	no	—
158	C₄	#13133177	1	1	yes	characteristic
159	C₄	#13133177	1	1	yes	characteristic
160	C₄	#13133182	1	1	yes	characteristic
161	C₄	#13133182	1	1	yes	characteristic
162	C₂	#16810474	1	1	yes	characteristic
163	C₂	#16810577	1	1	yes	characteristic
164	C₂	#16810580	1	1	yes	characteristic
165	C₂	#16810648	1	2	yes	—
166	C₂	#16810681	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133146

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal, characteristic

C₂×#13133127

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810469

C₂

yes

maximal, characteristic

#16810378

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

#13133146

C₄

yes

complemented

#13133146

C₄

yes

complemented

#13133127

—

C₈×Dic₅₃

—

#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

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

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

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

Q₈

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

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1768

—

maximal, Sylow

C₂×C₅₃

C₄×C₄⋊C₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#433

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

#336

D₅₃

yes

characteristic

C₅₃

#1768

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

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#97

—

106

—

102

C₄×C₈

—

106

—

103

#95

—

104

#95

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₅₃

yes

—

109

C₂²×C₄

C₂²×D₅₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

106

—

117

C₂×C₈

—

106

—

118

C₂×C₈

—

106

—

119

C₂×C₈

—

106

—

120

C₂×C₈

C₄×D₅₃

yes

—

121

C₂×C₈

C₄×D₅₃

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₂³

C₂×C₄×D₅₃

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

—

106

—

139

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

140

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

141

C₂×C₄

D₄×D₅₃

yes

characteristic

142

C₂×C₄

#11719947

yes

characteristic

143

C₂×C₄

#11719947

yes

characteristic

144

C₂×C₄

Q₈×D₅₃

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

106

—

148

C₈

—

106

—

149

C₂²

C₄²×D₅₃

yes

characteristic

150

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

151

C₂²

#13133173

yes

characteristic

152

C₂²

#13133223

yes

—

153

C₂²

#13133225

yes

—

154

C₄

—

155

C₄

—

106

—

156

C₄

—

106

—

157

C₄

—

106

—

158

C₄

#13133177

yes

characteristic

159

C₄

#13133177

yes

characteristic

160

C₄

#13133182

yes

characteristic

161

C₄

#13133182

yes

characteristic

162

C₂

#16810474

yes

characteristic

163

C₂

#16810577

yes

characteristic

164

C₂

#16810580

yes

characteristic

165

C₂

#16810648

yes

—

166

C₂

#16810681

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	#1768
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. #13133146	C₄	semidirect
13. #13133146	C₄	semidirect
91. C₅₃	#1768	semidirect

H ≅ G/N

type

12. #13133146

C₄

semidirect

13. #13133146

C₄

semidirect

91. C₅₃

#1768

semidirect

Subgroups — #20274571

Conjugacy classes of subgroups

Complements