Group #20274863 — subgroups

Subgroups — #20274863

Order 6784.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810394	C₂	1	1	yes	maximal, characteristic
3	#16810384	C₂	1	1	yes	maximal, characteristic
4	#16810382	C₂	1	1	yes	maximal, characteristic
5	#16810381	C₂	1	1	yes	maximal, characteristic
6	#16810379	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	#13133176	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133127	C₄	1	2	yes	complemented
13	C₈×Dic₅₃	C₄	1	2	yes	complemented
14	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#117×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	1	no	—
19	M₁₆×C₅₃	—	2	1	no	—
20	M₁₆×C₅₃	—	2	1	no	—
21	M₁₆×C₅₃	—	2	1	no	—
22	M₁₆×C₅₃	—	2	1	no	—
23	C₂×C₈×C₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₂²⋊C₄×C₅₃	—	2	1	no	—
27	C₄²×C₅₃	—	2	1	no	—
28	C₂²⋊Dic₅₃	—	2	1	no	—
29	C₄⋊Dic₅₃	—	2	1	no	—
30	C₄×Dic₅₃	—	2	1	no	—
31	C₄×Dic₅₃	—	2	1	no	—
32	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	#11719917	—	2	1	no	—
35	#11719917	—	2	1	no	—
36	#11719917	—	2	1	no	—
37	#11719917	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	C₂×C₄	1	2	yes	—
40	C₂³×C₅₃	#41	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	1	no	—
49	C₂×Dic₅₃	—	2	1	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	4	1	no	—
52	C₈×C₅₃	—	2	2	no	—
53	C₈×C₅₃	—	4	1	no	—
54	#88322	—	2	2	no	—
55	#88322	—	4	1	no	—
56	C₂²×C₅₃	—	4	1	no	—
57	C₂²×C₅₃	#117	1	1	yes	characteristic
58	C₂²×C₅₃	#126	1	1	yes	characteristic
59	C₂²×C₅₃	#126	1	1	yes	characteristic
60	C₄×C₅₃	—	8	1	no	—
61	C₄×C₅₃	#117	1	1	yes	characteristic
62	C₄×C₅₃	#117	1	1	yes	characteristic
63	C₄×C₅₃	#126	1	1	yes	characteristic
64	C₄×C₅₃	#126	1	1	yes	characteristic
65	Dic₅₃	—	2	2	no	—
66	Dic₅₃	—	4	1	no	—
67	Dic₅₃	—	8	1	no	—
68	#1841	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	—	2	1	no	—
72	C₂×C₅₃	#383	1	1	yes	characteristic
73	C₄×M₁₆	—	53	1	no	—
74	#344	—	53	1	no	—
75	#344	D₅₃	1	1	yes	characteristic
76	#339	—	53	1	no	—
77	#339	—	53	1	no	—
78	#338	—	53	1	no	—
79	H(3,ℤ/4)	—	53	1	no	—
80	C₅₃	#1841	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	53	1	no	—
82	C₂×M₁₆	—	53	1	no	—
83	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
84	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
85	#117	—	53	1	no	—
86	#117	C₂×D₅₃	1	1	yes	characteristic
87	C₂×C₄²	—	53	1	no	—
88	#97	—	53	2	no	—
89	C₄×C₈	—	53	2	no	—
90	C₂²×C₄	—	106	1	no	—
91	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
92	M₁₆	—	2	1	no	—
93	M₁₆	—	2	1	no	—
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	106	1	no	—
97	M₁₆	—	106	1	no	—
98	M₁₆	—	106	1	no	—
99	M₁₆	—	106	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	53	2	no	—
102	C₂×C₈	—	106	1	no	—
103	C₂×C₈	C₄×D₅₃	1	2	yes	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	106	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₂²⋊C₄	—	106	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	53	1	no	—
110	C₄²	—	53	1	no	—
111	C₄²	—	106	1	no	—
112	C₄²	—	106	1	no	—
113	C₂³	#11719945	1	1	yes	characteristic
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	106	1	no	—
116	C₂×C₄	—	106	1	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	—	212	1	no	—
119	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
120	C₂×C₄	#11719945	1	1	yes	characteristic
121	C₂×C₄	#11719947	1	1	yes	characteristic
122	C₂×C₄	#11719947	1	1	yes	characteristic
123	C₂×C₄	#11719947	1	1	yes	characteristic
124	C₂×C₄	#11719949	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	4	1	no	—
127	C₈	—	106	2	no	—
128	C₈	—	212	1	no	—
129	C₂²	—	4	1	no	—
130	C₂²	#13133174	1	1	yes	characteristic
131	C₂²	#13133210	1	1	yes	characteristic
132	C₂²	#13133212	1	1	yes	characteristic
133	C₄	—	8	1	no	—
134	C₄	—	106	2	no	—
135	C₄	—	424	1	no	—
136	C₄	—	212	1	no	—
137	C₄	#13133173	1	1	yes	characteristic
138	C₄	#13133175	1	1	yes	characteristic
139	C₄	#13133210	1	1	yes	characteristic
140	C₄	#13133212	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#16810490	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#16810394

C₂

yes

maximal, characteristic

#16810384

C₂

yes

maximal, characteristic

#16810382

C₂

yes

maximal, characteristic

#16810381

C₂

yes

maximal, characteristic

#16810379

C₂

yes

maximal, characteristic

C₅₃×#344

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133127

C₄

yes

complemented

C₈×Dic₅₃

C₄

yes

complemented

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

C₂×C₄×C₅₃

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

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1841

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#383

yes

characteristic

C₄×M₁₆

—

#344

—

#344

D₅₃

yes

—

—

—

—

C₅₃

#1841

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂×M₁₆

C₂×D₅₃

yes

characteristic

#117

—

#117

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

106

—

103

C₂×C₈

C₄×D₅₃

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

106

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

106

—

112

C₄²

—

106

—

113

C₂³

#11719945

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

212

—

118

C₂×C₄

—

212

—

119

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

120

C₂×C₄

#11719945

yes

characteristic

121

C₂×C₄

#11719947

yes

characteristic

122

C₂×C₄

#11719947

yes

characteristic

123

C₂×C₄

#11719947

yes

characteristic

124

C₂×C₄

#11719949

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

106

—

128

C₈

—

212

—

129

C₂²

—

130

C₂²

#13133174

yes

characteristic

131

C₂²

#13133210

yes

characteristic

132

C₂²

#13133212

yes

characteristic

133

C₄

—

134

C₄

—

106

—

135

C₄

—

424

—

136

C₄

—

212

—

137

C₄

#13133173

yes

characteristic

138

C₄

#13133175

yes

characteristic

139

C₄

#13133210

yes

characteristic

140

C₄

#13133212

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

#16810490

yes

characteristic

	subgroup	G/H
center	C₄	#13133173
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#344	C₂
socle	C₂×C₅₃	#383
hypercenter	#344	D₅₃
hypocenter	C₅₃	#1841
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#13133173

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#344

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
12. #13133127	C₄	semidirect
13. C₈×Dic₅₃	C₄	semidirect
80. C₅₃	#1841	semidirect

H ≅ G/N

type

12. #13133127

C₄

semidirect

13. C₈×Dic₅₃

C₄

semidirect

80. C₅₃

#1841

semidirect

Subgroups — #20274863

Conjugacy classes of subgroups

Complements