Group #20273621 — subgroups

Subgroups — #20273621

Order 6784.

148 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810754	C₂	1	1	yes	maximal, characteristic
2	#16810750	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719925	C₂	1	1	yes	maximal, characteristic
4	#16810373	C₂	1	1	yes	maximal, characteristic
5	#16810355	C₂	1	1	yes	maximal, characteristic
6	#16810345	C₂	1	1	yes	maximal, characteristic
7	C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719925	C₄	1	1	yes	characteristic
9	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#13133150	C₄	1	1	yes	characteristic
11	C₄⋊#88322	—	2	1	no	—
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133146	—	2	1	no	—
14	#13133141	C₄	1	1	yes	characteristic
15	#13133140	C₄	1	1	yes	characteristic
16	#13133128	—	2	1	no	—
17	#13133127	—	2	1	no	—
18	C₄×#88322	C₂²	1	1	yes	characteristic
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#11719925	—	2	2	no	—
32	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	#11719919	—	2	1	no	—
34	C₄×Dic₅₃	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	C₂×#88322	D₄	1	1	yes	characteristic
39	C₂×#88322	D₄	1	1	yes	characteristic
40	Q₈×C₅₃	—	2	1	no	—
41	Q₈×C₅₃	—	2	1	no	—
42	C₂×C₄×C₅₃	—	2	1	no	—
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	1	no	—
48	C₂×Dic₅₃	—	2	1	no	—
49	Dic₁₀₆	—	2	1	no	—
50	Dic₁₀₆	—	2	1	no	—
51	C₈×C₅₃	—	4	1	no	—
52	C₈×C₅₃	—	4	1	no	—
53	#88322	—	2	1	no	—
54	#88322	—	2	1	no	—
55	#88322	—	2	1	no	—
56	#88322	—	4	1	no	—
57	#88322	M₁₆	1	2	yes	—
58	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	4	1	no	—
61	C₄×C₅₃	—	4	1	no	—
62	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
63	C₄×C₅₃	#131	1	1	yes	characteristic
64	C₄×C₅₃	#137	1	1	yes	characteristic
65	C₄×C₅₃	#137	1	1	yes	characteristic
66	Dic₅₃	—	4	1	no	—
67	Dic₅₃	—	4	1	no	—
68	#1593	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	#436	1	1	yes	characteristic
70	C₂×C₅₃	#441	1	1	yes	characteristic
71	C₂×C₅₃	#444	1	1	yes	characteristic
72	#446	—	53	1	no	—
73	#446	—	53	1	no	—
74	C₄×Q₁₆	—	53	1	no	—
75	#334	—	53	1	no	—
76	Q₈⋊C₈	—	53	1	no	—
77	Q₈⋊C₈	D₅₃	1	1	yes	characteristic
78	#322	—	53	1	no	—
79	C₅₃	#1593	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	53	1	no	—
81	C₄×Q₈	—	53	1	no	—
82	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
83	#107	—	53	1	no	—
84	C₄⋊C₈	—	53	1	no	—
85	C₄⋊C₈	—	106	1	no	—
86	C₄⋊C₈	—	106	1	no	—
87	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
88	#103	—	53	1	no	—
89	#103	—	53	1	no	—
90	#97	—	106	1	no	—
91	#97	—	106	1	no	—
92	C₄×C₈	—	53	1	no	—
93	C₄×C₈	—	53	1	no	—
94	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₂×Q₈	—	53	1	no	—
96	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
97	Q₁₆	—	106	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	53	1	no	—
107	C₄⋊C₄	—	106	1	no	—
108	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	106	1	no	—
111	C₄²	C₂²×D₅₃	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	2	1	no	—
114	Q₈	—	106	1	no	—
115	Q₈	—	106	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	106	1	no	—
119	C₂×C₄	—	106	1	no	—
120	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
121	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
122	C₂×C₄	#11719947	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	53	2	no	—
126	C₈	—	106	1	no	—
127	C₈	—	106	1	no	—
128	C₈	—	106	1	no	—
129	C₈	—	212	1	no	—
130	C₂²	#13133177	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	212	1	no	—
135	C₄	—	212	1	no	—
136	C₄	M₁₆×D₅₃	1	1	yes	characteristic
137	C₄	#13133223	1	1	yes	characteristic
138	C₄	#13133246	1	1	yes	characteristic
139	C₄	#13133249	1	1	yes	characteristic
140	C₂	#16810698	1	1	yes	characteristic
141	C₂	#16810727	1	1	yes	characteristic
142	C₂	#16810742	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810754

C₂

yes

maximal, characteristic

#16810750

C₂

yes

maximal, characteristic

C₄×#11719925

C₂

yes

maximal, characteristic

#16810373

C₂

yes

maximal, characteristic

#16810355

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, characteristic

C₅₃×Q₈⋊C₈

C₂

yes

maximal, characteristic

C₂×#11719925

C₄

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133150

C₄

yes

characteristic

C₄⋊#88322

—

C₄⋊#88322

C₂²

yes

characteristic

#13133146

—

#13133141

C₄

yes

characteristic

#13133140

C₄

yes

—

—

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

C₂×C₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

—

M₁₆

yes

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

D₅₃

yes

characteristic

#322

—

C₅₃

#1593

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

—

—

—

106

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₄×D₅₃

yes

characteristic

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

106

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

C₄×D₅₃

yes

characteristic

109

C₄²

—

110

C₄²

—

106

—

111

C₄²

C₂²×D₅₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

106

—

115

Q₈

—

106

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

121

C₂×C₄

D₄×D₅₃

yes

characteristic

122

C₂×C₄

#11719947

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

106

—

127

C₈

—

106

—

128

C₈

—

106

—

129

C₈

—

212

—

130

C₂²

#13133177

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

212

—

135

C₄

—

212

—

136

C₄

M₁₆×D₅₃

yes

characteristic

137

C₄

#13133223

yes

characteristic

138

C₄

#13133246

yes

characteristic

139

C₄

#13133249

yes

characteristic

140

C₂

#16810698

yes

characteristic

141

C₂

#16810727

yes

characteristic

142

C₂

#16810742

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×Q₈⋊C₈	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	Q₈⋊C₈	D₅₃
hypocenter	C₅₃	#1593
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×Q₈⋊C₈

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

Q₈⋊C₈

D₅₃

hypocenter

C₅₃

#1593

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
79. C₅₃	#1593	semidirect

H ≅ G/N

type

79. C₅₃

#1593

semidirect

Subgroups — #20273621

Conjugacy classes of subgroups

Complements