Group #20274754 — subgroups

Subgroups — #20274754

Order 6784.

272 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810586	C₂	1	2	yes	maximal
4	#16810387	C₂	1	2	yes	maximal, complemented
5	C₂×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	C₂×#11719912	—	2	2	no	—
9	C₂×#11719912	—	2	2	no	—
10	C₂×#11719912	C₄	1	4	yes	—
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	#13133153	—	2	2	no	—
13	#13133129	—	2	4	no	—
14	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	#108×C₅₃	—	2	2	no	—
17	C₂³×D₅₃	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₅₃	—	2	1	no	—
21	C₂×C₄×D₅₃	—	2	1	no	—
22	C₂×C₄×D₅₃	—	2	1	no	—
23	C₂×C₄×D₅₃	—	2	1	no	—
24	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₅₃	D₄	1	2	yes	—
26	M₁₆×C₅₃	—	2	2	no	—
27	M₁₆×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₅₃	D₄	1	2	yes	—
31	C₂×C₈×C₅₃	D₄	1	2	yes	—
32	#11719917	—	2	2	no	—
33	#11719917	—	2	2	no	—
34	C₂×#88322	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	C₂×C₄	1	2	yes	—
38	#11719912	—	2	8	no	—
39	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×D₅₃	—	2	1	no	—
41	C₂²×D₅₃	—	2	1	no	—
42	C₂²×D₅₃	—	4	2	no	—
43	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	1	no	—
51	C₂×Dic₅₃	—	2	1	no	—
52	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
53	C₄×D₅₃	—	2	2	no	—
54	C₄×D₅₃	—	2	2	no	—
55	C₄×D₅₃	—	2	2	no	—
56	C₄×D₅₃	—	2	2	no	—
57	C₈×C₅₃	—	4	2	no	—
58	C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
59	#88322	—	2	2	no	—
60	#88322	—	4	2	no	—
61	C₂²×C₅₃	—	2	2	no	—
62	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×D₅₃	—	2	2	no	—
66	C₂×D₅₃	—	2	4	no	—
67	C₂×D₅₃	—	4	2	no	—
68	C₂×D₅₃	—	4	2	no	—
69	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₄×C₅₃	#117	1	1	yes	characteristic
71	C₄×C₅₃	#121	1	1	yes	characteristic
72	C₄×C₅₃	#121	1	1	yes	characteristic
73	Dic₅₃	—	2	2	no	—
74	#1811	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	#380	1	1	yes	characteristic
78	C₂×C₅₃	#430	1	2	yes	—
79	D₅₃	—	4	2	no	—
80	C₂²×M₁₆	—	53	1	no	—
81	C₂×#108	—	53	1	no	—
82	C₂×#108	D₅₃	1	1	yes	characteristic, complemented
83	#407	—	53	2	no	—
84	#341	—	53	2	no	—
85	C₅₃	#1811	1	1	yes	characteristic, complemented, Sylow
86	C₂³×C₄	—	53	1	no	—
87	C₂×M₁₆	—	53	4	no	—
88	C₂×M₁₆	—	53	2	no	—
89	C₂×M₁₆	—	106	2	no	—
90	C₂×M₁₆	—	106	2	no	—
91	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
92	C₂²×C₈	—	53	1	no	—
93	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
94	#108	—	2	2	no	—
95	#108	—	106	2	no	—
96	C₂²⋊C₈	—	106	4	no	—
97	C₂⁴	—	53	1	no	—
98	C₂²×C₄	—	53	1	no	—
99	C₂²×C₄	—	53	2	no	—
100	C₂²×C₄	—	53	2	no	—
101	C₂²×C₄	—	106	1	no	—
102	C₂²×C₄	—	106	1	no	—
103	C₂²×C₄	—	106	1	no	—
104	C₂²×C₄	—	106	1	no	—
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	106	8	no	—
109	M₁₆	—	106	2	no	—
110	M₁₆	—	106	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	—	106	1	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	C₄×D₅₃	1	2	yes	—
117	C₂×C₈	D₂₁₂	1	2	yes	—
118	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
119	C₂³	—	53	2	no	—
120	C₂³	—	106	1	no	—
121	C₂³	—	106	1	no	—
122	C₂³	—	212	2	no	—
123	C₂³	Q₈×D₅₃	1	1	yes	characteristic
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	106	1	no	—
126	C₂×C₄	—	106	1	no	—
127	C₂×C₄	—	106	2	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	—	106	2	no	—
130	C₂×C₄	—	106	2	no	—
131	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
133	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄	#11719947	1	1	yes	characteristic
135	C₂×C₄	#11719949	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₈	—	106	2	no	—
139	C₈	—	212	2	no	—
140	C₈	#11719921	1	4	yes	—
141	C₂²	—	2	2	no	—
142	C₂²	—	106	4	no	—
143	C₂²	—	106	2	no	—
144	C₂²	—	212	2	no	—
145	C₂²	—	212	2	no	—
146	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
147	C₂²	#13133193	1	1	yes	characteristic
148	C₂²	#13133197	1	1	yes	characteristic
149	C₄	—	106	2	no	—
150	C₄	C₂×#11719921	1	1	yes	characteristic
151	C₄	#13133175	1	1	yes	characteristic
152	C₄	#13133188	1	1	yes	characteristic
153	C₄	#13133192	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	—	212	2	no	—
157	C₂	#16810480	1	1	yes	characteristic
158	C₂	#16810670	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719912

C₂

yes

maximal, characteristic

C₂×#13133153

C₂

yes

maximal, characteristic, complemented

#16810586

C₂

yes

maximal

#16810387

C₂

yes

maximal, complemented

C₂×#108×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719912

—

C₂×#11719912

—

C₂×#11719912

C₄

yes

—

C₂²×#88322

C₂²

yes

characteristic

#13133153

—

#13133129

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#108×C₅₃

—

C₂³×D₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

Q₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

D₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

—

—

—

—

—

C₂×C₄

yes

—

#11719912

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1811

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#380

yes

characteristic

C₂×C₅₃

#430

yes

—

D₅₃

—

C₂²×M₁₆

—

C₂×#108

—

C₂×#108

D₅₃

yes

characteristic, complemented

#407

—

#341

—

C₅₃

#1811

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#108

—

#108

—

106

—

C₂²⋊C₈

—

106

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

106

—

102

C₂²×C₄

—

106

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

—

106

—

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

106

—

109

M₁₆

—

106

—

110

M₁₆

—

106

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

106

—

115

C₂×C₈

—

106

—

116

C₂×C₈

C₄×D₅₃

yes

—

117

C₂×C₈

D₂₁₂

yes

—

118

C₂×C₈

C₂²⋊D₅₃

yes

—

119

C₂³

—

120

C₂³

—

106

—

121

C₂³

—

106

—

122

C₂³

—

212

—

123

C₂³

Q₈×D₅₃

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

—

106

—

129

C₂×C₄

—

106

—

130

C₂×C₄

—

106

—

131

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

132

C₂×C₄

C₂×D₂₁₂

yes

characteristic

133

C₂×C₄

D₄×D₅₃

yes

characteristic

134

C₂×C₄

#11719947

yes

characteristic

135

C₂×C₄

#11719949

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

137

C₈

—

138

C₈

—

106

—

139

C₈

—

212

—

140

C₈

#11719921

yes

—

141

C₂²

—

142

C₂²

—

106

—

143

C₂²

—

106

—

144

C₂²

—

212

—

145

C₂²

—

212

—

146

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

147

C₂²

#13133193

yes

characteristic

148

C₂²

#13133197

yes

characteristic

149

C₄

—

106

—

150

C₄

C₂×#11719921

yes

characteristic

151

C₄

#13133175

yes

characteristic

152

C₄

#13133188

yes

characteristic

153

C₄

#13133192

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

—

212

—

157

C₂

#16810480

yes

characteristic

158

C₂

#16810670

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂×#108	D₅₃
hypocenter	C₅₃	#1811
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#108×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
2. C₂×#13133153	C₂	semidirect
4. #16810387	C₂	semidirect
5. C₂×#108×C₅₃	C₂	semidirect
82. C₂×#108	D₅₃	semidirect
85. C₅₃	#1811	semidirect

H ≅ G/N

type

2. C₂×#13133153

C₂

semidirect

4. #16810387

C₂

semidirect

5. C₂×#108×C₅₃

C₂

semidirect

82. C₂×#108

D₅₃

semidirect

85. C₅₃

#1811

semidirect

Subgroups — #20274754

Conjugacy classes of subgroups

Complements