Group #20276764 — subgroups

Subgroups — #20276764

Order 6784.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133163	C₂	1	1	yes	maximal, characteristic
3	#16811001	C₂	1	1	yes	maximal, characteristic
4	#16810717	C₂	1	1	yes	maximal, characteristic
5	#16810441	C₂	1	1	yes	maximal, characteristic
6	#16810426	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#358	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×#11719925	—	2	1	no	—
11	#13133215	—	2	1	no	—
12	#13133182	—	2	1	no	—
13	Dic₄₂₄	—	2	2	no	—
14	#13133163	—	2	2	no	—
15	C₂×#11719908	C₂²	1	1	yes	characteristic
16	#13133140	—	2	1	no	—
17	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	#107×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×Dic₁₀₆	—	2	1	no	—
23	C₂×Dic₁₀₆	—	2	1	no	—
24	Q₁₆×C₅₃	—	2	1	no	—
25	Q₁₆×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₅₃	—	2	1	no	—
28	#11719925	—	4	2	no	—
29	#11719919	—	4	1	no	—
30	#11719919	—	4	1	no	—
31	#11719919	—	4	1	no	—
32	C₄×Dic₅₃	—	4	1	no	—
33	C₄×Dic₅₃	D₄	1	1	yes	characteristic
34	C₂×#88322	D₄	1	1	yes	characteristic
35	Dic₂₁₂	—	2	1	no	—
36	Dic₂₁₂	—	2	1	no	—
37	Dic₂₁₂	—	2	1	no	—
38	Dic₂₁₂	D₄	1	2	yes	—
39	C₁₆×C₅₃	—	2	1	no	—
40	#11719908	—	2	1	no	—
41	Q₈×C₅₃	—	4	1	no	—
42	Q₈×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₅₃	—	4	1	no	—
47	C₂×Dic₅₃	—	4	1	no	—
48	C₂×Dic₅₃	D₈	1	1	yes	characteristic
49	C₂×Dic₅₃	D₈	1	1	yes	characteristic
50	Dic₁₀₆	—	2	2	no	—
51	Dic₁₀₆	—	4	1	no	—
52	Dic₁₀₆	—	4	1	no	—
53	Dic₁₀₆	—	4	1	no	—
54	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₈×C₅₃	#41	1	1	yes	characteristic
56	#88322	—	2	1	no	—
57	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₄×C₅₃	—	8	1	no	—
59	C₄×C₅₃	—	8	1	no	—
60	C₄×C₅₃	#121	1	1	yes	characteristic
61	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₅₃	—	2	1	no	—
63	Dic₅₃	—	4	2	no	—
64	Dic₅₃	—	8	1	no	—
65	Dic₅₃	Q₃₂	1	2	yes	complemented
66	#2209	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#459	1	1	yes	characteristic
68	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
69	C₂×C₅₃	#510	1	1	yes	characteristic
70	C₂×Q₃₂	—	53	1	no	—
71	C₂×Q₃₂	—	53	1	no	—
72	#494	—	53	1	no	—
73	C₄×Q₁₆	—	53	1	no	—
74	C₄⋊C₁₆	—	53	1	no	—
75	#358	—	53	1	no	—
76	#358	D₅₃	1	1	yes	characteristic
77	C₅₃	#2209	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	53	1	no	—
79	C₂×Q₁₆	—	53	1	no	—
80	C₂×Q₁₆	—	106	1	no	—
81	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
82	C₄⋊Q₈	—	106	1	no	—
83	C₄×Q₈	—	106	1	no	—
84	Q₃₂	—	106	2	no	—
85	Q₃₂	—	106	2	no	—
86	C₂×C₁₆	—	53	1	no	—
87	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
88	#107	C₂×D₅₃	1	1	yes	characteristic
89	#103	—	106	1	no	—
90	C₄×C₈	—	53	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	106	1	no	—
93	C₂×Q₈	—	106	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	53	2	no	—
97	Q₁₆	—	106	1	no	—
98	Q₁₆	—	106	1	no	—
99	Q₁₆	—	106	1	no	—
100	Q₁₆	—	212	2	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	212	1	no	—
105	C₄⋊C₄	—	212	1	no	—
106	C₄⋊C₄	—	212	1	no	—
107	C₄²	—	53	1	no	—
108	C₄²	—	212	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	106	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	106	2	no	—
114	Q₈	—	212	1	no	—
115	Q₈	—	212	1	no	—
116	Q₈	—	212	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	212	1	no	—
122	C₂×C₄	—	212	1	no	—
123	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
124	C₈	—	106	1	no	—
125	C₈	#11719945	1	1	yes	characteristic
126	C₈	D₄×D₅₃	1	1	yes	characteristic
127	C₂²	D₈×D₅₃	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	53	2	no	—
131	C₄	—	106	1	no	—
132	C₄	—	424	1	no	—
133	C₄	—	212	2	no	—
134	C₄	#13133190	1	1	yes	characteristic
135	C₄	#13133244	1	1	yes	characteristic
136	C₂	#16810817	1	1	yes	characteristic
137	C₂	Q₃₂×D₅₃	1	1	yes	characteristic
138	C₂	#16811062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₂₄

C₂

yes

maximal, characteristic

C₂×#13133163

C₂

yes

maximal, characteristic

#16811001

C₂

yes

maximal, characteristic

#16810717

C₂

yes

maximal, characteristic

#16810441

C₂

yes

maximal, characteristic

#16810426

C₂

yes

maximal, characteristic

C₅₃×#358

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×Dic₂₁₂

C₂²

yes

—

—

—

Dic₄₂₄

—

#13133163

—

C₂×#11719908

C₂²

yes

characteristic

#13133140

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

—

—

—

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

D₄

yes

—

C₁₆×C₅₃

—

#11719908

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

Q₃₂

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×Q₃₂

yes

characteristic

C₂×C₅₃

#510

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

D₅₃

yes

characteristic

C₅₃

#2209

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

106

—

C₄×Q₈

—

106

—

Q₃₂

—

106

—

Q₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

C₂×D₅₃

yes

characteristic

#103

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

106

—

Q₁₆

—

106

—

Q₁₆

—

106

—

100

Q₁₆

—

212

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₅₃

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

212

—

105

C₄⋊C₄

—

212

—

106

C₄⋊C₄

—

212

—

107

C₄²

—

108

C₄²

—

212

—

109

C₁₆

—

110

C₁₆

—

106

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

106

—

114

Q₈

—

212

—

115

Q₈

—

212

—

116

Q₈

—

212

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

212

—

122

C₂×C₄

—

212

—

123

C₂×C₄

D₄×D₅₃

yes

characteristic

124

C₈

—

106

—

125

C₈

#11719945

yes

characteristic

126

C₈

D₄×D₅₃

yes

characteristic

127

C₂²

D₈×D₅₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

106

—

132

C₄

—

424

—

133

C₄

—

212

—

134

C₄

#13133190

yes

characteristic

135

C₄

#13133244

yes

characteristic

136

C₂

#16810817

yes

characteristic

137

C₂

Q₃₂×D₅₃

yes

characteristic

138

C₂

#16811062

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#358	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#358	D₅₃
hypocenter	C₅₃	#2209
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#358

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
65. Dic₅₃	Q₃₂	semidirect
77. C₅₃	#2209	semidirect

H ≅ G/N

type

65. Dic₅₃

Q₃₂

semidirect

77. C₅₃

#2209

semidirect

Subgroups — #20276764

Conjugacy classes of subgroups

Complements