Group #20276038 — subgroups

Subgroups — #20276038

Order 6784.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133142	C₂	1	2	yes	maximal
3	#16810529	C₂	1	2	yes	maximal
4	#16810386	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	C₂×Dic₂₁₂	—	2	2	no	—
8	C₂×Dic₂₁₂	—	2	4	no	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#13133142	—	2	4	no	—
11	#13133124	—	2	2	no	—
12	#13133124	—	2	2	no	—
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	2	no	—
21	C₂²×Dic₅₃	—	2	2	no	—
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	—	2	2	no	—
24	C₂×Dic₁₀₆	—	2	2	no	—
25	C₂×Dic₁₀₆	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	D₄	1	4	yes	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	D₄	1	2	yes	—
32	C₄²×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	2	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	Dic₂₁₂	—	2	8	no	—
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	Q₁₆	1	2	yes	complemented
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	4	2	no	—
53	Dic₁₀₆	—	2	4	no	—
54	Dic₁₀₆	—	2	4	no	—
55	Dic₁₀₆	—	4	2	no	—
56	Dic₁₀₆	—	4	2	no	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	—	4	1	no	—
59	C₂²×C₅₃	#121	1	1	yes	characteristic
60	C₂²×C₅₃	#121	1	2	yes	—
61	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₅₃	#135	1	1	yes	characteristic
63	C₂²×C₅₃	#137	1	2	yes	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	—	4	1	no	—
66	C₄×C₅₃	#120	1	1	yes	characteristic
67	C₄×C₅₃	#123	1	2	yes	—
68	C₄×C₅₃	#124	1	1	yes	characteristic
69	Dic₅₃	—	4	4	no	—
70	Dic₅₃	—	8	2	no	—
71	#2056	—	53	1	no	maximal, Sylow
72	C₂×C₅₃	#394	1	1	yes	characteristic
73	C₂×C₅₃	#462	1	2	yes	—
74	C₂×C₅₃	#464	1	2	yes	—
75	C₂×C₅₃	#467	1	1	yes	characteristic
76	C₂×C₅₃	#470	1	1	yes	characteristic
77	C₂²×Q₁₆	—	53	1	no	—
78	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
79	C₂×#103	—	53	2	no	—
80	#391	—	53	2	no	—
81	#340	—	53	1	no	—
82	C₅₃	#2056	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	53	2	no	—
84	C₂×Q₁₆	—	106	4	no	—
85	C₂×Q₁₆	—	106	2	no	—
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	53	2	no	—
89	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	106	4	no	—
92	#95	—	106	2	no	—
93	#95	—	106	2	no	—
94	C₂×Q₈	—	53	4	no	—
95	C₂×Q₈	—	106	2	no	—
96	C₂×Q₈	—	106	2	no	—
97	C₂×Q₈	—	106	2	no	—
98	C₂×Q₈	—	106	2	no	—
99	C₂×Q₈	—	106	2	no	—
100	C₂²×C₄	—	106	2	no	—
101	C₂²×C₄	—	106	2	no	—
102	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
103	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
105	Q₁₆	—	106	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
110	C₄⋊C₄	—	106	2	no	—
111	C₄⋊C₄	—	106	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
115	Q₈	—	106	4	no	—
116	Q₈	—	106	4	no	—
117	Q₈	—	212	2	no	—
118	Q₈	—	212	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	106	4	no	—
124	C₂×C₄	—	212	2	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	—	212	2	no	—
127	C₂×C₄	—	212	2	no	—
128	C₂×C₄	—	212	2	no	—
129	C₂×C₄	Dic₂₁₂	1	2	yes	complemented
130	C₂×C₄	#11719945	1	1	yes	characteristic
131	C₂×C₄	D₄×D₅₃	1	2	yes	—
132	C₂×C₄	#11719949	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#13133191	1	1	yes	characteristic
137	C₂²	#13133192	1	2	yes	—
138	C₂²	C₂×Dic₂₁₂	1	1	yes	characteristic
139	C₂²	#13133239	1	1	yes	characteristic
140	C₂²	#13133250	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	424	2	no	—
144	C₄	—	212	4	no	—
145	C₄	#13133185	1	1	yes	characteristic
146	C₄	#13133202	1	2	yes	—
147	C₄	#13133206	1	1	yes	characteristic
148	C₂	#16810540	1	1	yes	characteristic
149	C₂	#16810834	1	2	yes	—
150	C₂	#16810846	1	2	yes	—
151	C₂	#16810864	1	1	yes	characteristic
152	C₂	#16810881	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂×#13133142

C₂

yes

maximal

#16810529

C₂

yes

maximal

#16810386

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133142

—

#13133124

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₂²×C₅₃

#121

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#120

yes

characteristic

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#124

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#464

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×Q₁₆

—

C₂×C₄⋊C₈

D₅₃

yes

—

—

—

C₅₃

#2056

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#103

—

106

—

#95

—

106

—

#95

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

100

C₂²×C₄

—

106

—

101

C₂²×C₄

—

106

—

102

C₂²×C₄

D₂₁₂

yes

characteristic

103

C₂²×C₄

D₂₁₂

yes

characteristic

104

C₂²×C₄

C₂²×D₅₃

yes

characteristic

105

Q₁₆

—

106

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₅₃

yes

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₂₁₂

yes

characteristic

115

Q₈

—

106

—

116

Q₈

—

106

—

117

Q₈

—

212

—

118

Q₈

—

212

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

212

—

125

C₂×C₄

—

212

—

126

C₂×C₄

—

212

—

127

C₂×C₄

—

212

—

128

C₂×C₄

—

212

—

129

C₂×C₄

Dic₂₁₂

yes

complemented

130

C₂×C₄

#11719945

yes

characteristic

131

C₂×C₄

D₄×D₅₃

yes

—

132

C₂×C₄

#11719949

yes

—

133

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

134

C₈

—

135

C₈

—

136

C₂²

#13133191

yes

characteristic

137

C₂²

#13133192

yes

—

138

C₂²

C₂×Dic₂₁₂

yes

characteristic

139

C₂²

#13133239

yes

characteristic

140

C₂²

#13133250

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

424

—

144

C₄

—

212

—

145

C₄

#13133185

yes

characteristic

146

C₄

#13133202

yes

—

147

C₄

#13133206

yes

characteristic

148

C₂

#16810540

yes

characteristic

149

C₂

#16810834

yes

—

150

C₂

#16810846

yes

—

151

C₂

#16810864

yes

characteristic

152

C₂

#16810881

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#2056
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₅₃

hypocenter

C₅₃

#2056

LCS

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₅₃	Q₁₆	semidirect
82. C₅₃	#2056	semidirect
129. C₂×C₄	Dic₂₁₂	semidirect

H ≅ G/N

type

42. C₂×C₄×C₅₃

Q₁₆

semidirect

82. C₅₃

#2056

semidirect

129. C₂×C₄

Dic₂₁₂

semidirect

Subgroups — #20276038

Conjugacy classes of subgroups

Complements