Group #20274760 — subgroups

Subgroups — #20274760

Order 6784.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#13133225	C₂	1	1	yes	maximal, characteristic
2	#16810659	C₂	1	2	yes	maximal, complemented
3	#16810596	C₂	1	2	yes	maximal
4	#16810380	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₂²	1	1	yes	characteristic
7	#13133225	—	2	2	no	—
8	#13133225	—	2	2	no	—
9	C₂×#11719912	C₄	1	2	yes	complemented
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₅₃	C₄	1	2	yes	complemented
12	#13133176	C₂²	1	2	yes	—
13	#13133150	—	2	2	no	—
14	#13133148	—	2	2	no	—
15	#13133131	—	2	4	no	—
16	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	#117×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₅₃	D₄	1	2	yes	—
20	#11719945	—	2	2	no	—
21	#11719945	—	2	2	no	—
22	C₂×D₂₁₂	Q₈	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
24	C₂×Dic₁₀₆	Q₈	1	1	yes	characteristic
25	M₁₆×C₅₃	—	2	2	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂²⋊C₄×C₅₃	—	2	2	no	—
29	C₄²×C₅₃	—	2	2	no	—
30	C₂²⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	2	no	—
32	C₄×Dic₅₃	—	2	2	no	—
33	C₂×#88322	C₂×C₄	1	1	yes	characteristic
34	C₂×#88322	C₂×C₄	1	1	yes	characteristic
35	C₂×#88322	D₄	1	2	yes	—
36	C₂×#88322	D₄	1	2	yes	—
37	#11719912	—	2	4	no	—
38	C₈×D₅₃	—	2	2	no	—
39	C₈×D₅₃	—	2	2	no	—
40	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂²⋊D₅₃	—	2	4	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
50	D₂₁₂	—	2	2	no	—
51	C₄×D₅₃	—	2	2	no	—
52	C₄×D₅₃	—	2	2	no	—
53	Dic₁₀₆	—	2	2	no	—
54	C₈×C₅₃	—	2	2	no	—
55	#88322	C₂²⋊C₄	1	1	yes	characteristic
56	#88322	C₂²⋊C₄	1	1	yes	characteristic
57	#88322	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₅₃	—	4	1	no	—
59	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₅₃	#121	1	2	yes	—
61	C₂×D₅₃	—	2	2	no	—
62	C₂×D₅₃	—	4	2	no	—
63	C₄×C₅₃	—	8	2	no	—
64	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₅₃	#117	1	1	yes	characteristic
66	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
67	Dic₅₃	—	2	2	no	—
68	Dic₅₃	—	8	2	no	—
69	#1812	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	2	no	—
71	C₂×C₅₃	—	2	1	no	—
72	C₂×C₅₃	#380	1	1	yes	characteristic
73	D₅₃	—	4	2	no	—
74	C₂×#131	—	53	1	no	—
75	#427	—	53	2	no	—
76	#410	—	53	2	no	—
77	H(3,ℤ/4)	—	53	1	no	—
78	H(3,ℤ/4)	D₅₃	1	1	yes	characteristic, complemented
79	C₅₃	#1812	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	53	1	no	—
81	#131	—	106	2	no	—
82	#131	—	106	2	no	—
83	C₂×M₁₆	—	53	2	no	—
84	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
85	C₂²×C₈	—	53	1	no	—
86	C₂²×C₈	—	53	2	no	—
87	#117	—	53	2	no	—
88	#117	C₂×D₅₃	1	2	yes	—
89	#107	—	106	2	no	—
90	#106	—	106	2	no	—
91	#99	—	106	4	no	—
92	#41	—	106	2	no	—
93	#41	—	106	2	no	—
94	C₂×Q₈	—	53	1	no	—
95	C₂×D₄	—	53	2	no	—
96	C₂×D₄	—	53	1	no	—
97	C₂²×C₄	—	53	2	no	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	106	4	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	53	2	no	—
104	C₂×C₈	—	53	2	no	—
105	C₂×C₈	—	106	2	no	—
106	C₂×C₈	—	106	2	no	—
107	C₂×C₈	C₄×D₅₃	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	106	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	106	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	106	2	no	—
114	C₂³	—	53	2	no	—
115	C₂³	D₄×D₅₃	1	1	yes	characteristic
116	Q₈	—	106	2	no	—
117	D₄	—	106	4	no	—
118	D₄	—	106	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	53	2	no	—
121	C₂×C₄	—	106	2	no	—
122	C₂×C₄	—	106	2	no	—
123	C₂×C₄	—	212	2	no	—
124	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
125	C₂×C₄	#11719945	1	2	yes	—
126	C₂×C₄	D₄×D₅₃	1	2	yes	—
127	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	53	1	no	—
130	C₈	—	53	1	no	—
131	C₈	—	53	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	106	2	no	—
134	C₂²	—	212	2	no	—
135	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
136	C₂²	#13133190	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	106	2	no	—
139	C₄	—	424	2	no	—
140	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
141	C₄	#13133174	1	1	yes	characteristic
142	C₄	#13133195	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	212	2	no	—
146	C₂	#16810482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133225

C₂

yes

maximal, characteristic

#16810659

C₂

yes

maximal, complemented

#16810596

C₂

yes

maximal

#16810380

C₂

yes

maximal, characteristic, complemented

C₅₃×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

complemented

#13133176

C₂²

yes

—

#13133150

—

#13133148

—

#13133131

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

D₄

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

Q₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

Q₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

characteristic

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

#1812

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#380

yes

characteristic

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1812

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₅₃

yes

—

#107

—

106

—

#106

—

106

—

#99

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

100

M₁₆

—

106

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

—

106

C₂×C₈

—

106

—

107

C₂×C₈

C₄×D₅₃

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

106

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

106

—

112

C₄²

—

113

C₄²

—

106

—

114

C₂³

—

115

C₂³

D₄×D₅₃

yes

characteristic

116

Q₈

—

106

—

117

D₄

—

106

—

118

D₄

—

106

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

212

—

124

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

125

C₂×C₄

#11719945

yes

—

126

C₂×C₄

D₄×D₅₃

yes

—

127

C₂×C₄

Q₈×D₅₃

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

106

—

134

C₂²

—

212

—

135

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

136

C₂²

#13133190

yes

—

137

C₄

—

138

C₄

—

106

—

139

C₄

—

424

—

140

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

141

C₄

#13133174

yes

characteristic

142

C₄

#13133195

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

212

—

146

C₂

#16810482

yes

characteristic

	subgroup	G/H
center	C₄	#13133174
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×H(3,ℤ/4)	C₂
socle	C₂×C₅₃	#380
hypercenter	H(3,ℤ/4)	D₅₃
hypocenter	C₅₃	#1812
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#13133174

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×H(3,ℤ/4)

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #16810659	C₂	semidirect
4. #16810380	C₂	semidirect
5. C₅₃×H(3,ℤ/4)	C₂	semidirect
9. C₂×#11719912	C₄	semidirect
11. C₂×C₈×D₅₃	C₄	semidirect
78. H(3,ℤ/4)	D₅₃	semidirect
79. C₅₃	#1812	semidirect

H ≅ G/N

type

2. #16810659

C₂

semidirect

4. #16810380

C₂

semidirect

5. C₅₃×H(3,ℤ/4)

C₂

semidirect

9. C₂×#11719912

C₄

semidirect

11. C₂×C₈×D₅₃

C₄

semidirect

78. H(3,ℤ/4)

D₅₃

semidirect

79. C₅₃

#1812

semidirect

Subgroups — #20274760

Conjugacy classes of subgroups

Complements