Group #20274232 — subgroups

Subgroups — #20274232

Order 6784.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16811026	C₂	1	1	yes	maximal, characteristic
2	#16810834	C₂	1	1	yes	maximal, characteristic
3	#16810830	C₂	1	1	yes	maximal, characteristic
4	#16810750	C₂	1	1	yes	maximal, characteristic
5	#16810373	C₂	1	1	yes	maximal, characteristic
6	#16810368	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#332	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₂	—	2	1	no	—
9	C₂×#11719925	—	2	1	no	—
10	#13133217	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
12	#13133150	—	2	1	no	—
13	#13133150	—	2	1	no	—
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	#13133146	—	2	1	no	—
16	#13133142	—	2	1	no	—
17	#13133141	—	2	1	no	—
18	#13133127	—	2	1	no	—
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	—	2	1	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#11719925	—	4	2	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
35	#11719919	—	2	1	no	—
36	C₄×Dic₅₃	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	D₄	1	1	yes	characteristic
39	C₂×#88322	D₄	1	1	yes	characteristic
40	Dic₂₁₂	—	4	2	no	—
41	Q₈×C₅₃	—	4	1	no	—
42	Q₈×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	C₂×D₄	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	C₂×Dic₅₃	—	4	1	no	—
50	Dic₁₀₆	—	2	1	no	—
51	Dic₁₀₆	—	2	1	no	—
52	Dic₁₀₆	—	4	1	no	—
53	Dic₁₀₆	—	4	1	no	—
54	C₈×C₅₃	—	4	1	no	—
55	C₈×C₅₃	—	4	1	no	—
56	#88322	—	2	1	no	—
57	#88322	—	4	1	no	—
58	#88322	Q₁₆	1	2	yes	complemented
59	C₂²×C₅₃	#121	1	1	yes	characteristic
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	8	1	no	—
62	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄×C₅₃	#137	1	1	yes	characteristic
66	Dic₅₃	—	4	1	no	—
67	Dic₅₃	—	4	1	no	—
68	Dic₅₃	—	8	1	no	—
69	#1694	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	#462	1	1	yes	characteristic
71	C₂×C₅₃	#469	1	1	yes	characteristic
72	C₂×C₅₃	#473	1	1	yes	characteristic
73	#500	—	53	1	no	—
74	#462	—	53	1	no	—
75	#462	—	53	1	no	—
76	#446	—	53	1	no	—
77	#334	—	53	1	no	—
78	#332	—	53	1	no	—
79	#332	D₅₃	1	1	yes	characteristic
80	C₅₃	#1694	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	106	1	no	—
82	C₂×Q₁₆	—	106	1	no	—
83	C₄⋊Q₈	—	53	1	no	—
84	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
85	C₄×Q₈	—	53	1	no	—
86	#107	—	106	1	no	—
87	#107	—	106	1	no	—
88	C₄⋊C₈	—	53	1	no	—
89	C₄⋊C₈	—	106	1	no	—
90	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
92	#103	—	106	1	no	—
93	#103	—	106	1	no	—
94	#97	—	106	1	no	—
95	C₄×C₈	—	53	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	53	1	no	—
98	C₂×Q₈	—	106	1	no	—
99	Q₁₆	—	212	2	no	—
100	Q₁₆	—	212	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	106	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	53	1	no	—
109	C₄⋊C₄	—	106	1	no	—
110	C₄⋊C₄	—	106	1	no	—
111	C₄⋊C₄	—	106	1	no	—
112	C₄²	—	106	1	no	—
113	C₄²	C₂²×D₅₃	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	106	1	no	—
117	Q₈	—	106	1	no	—
118	Q₈	—	212	1	no	—
119	Q₈	—	212	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	212	1	no	—
124	C₂×C₄	#11719945	1	1	yes	characteristic
125	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
126	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	53	2	no	—
130	C₈	—	106	1	no	—
131	C₈	—	212	1	no	—
132	C₂²	#13133190	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	424	1	no	—
136	C₄	—	212	1	no	—
137	C₄	—	212	1	no	—
138	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
139	C₄	#13133241	1	1	yes	characteristic
140	C₄	#13133244	1	1	yes	characteristic
141	C₄	#13133246	1	1	yes	characteristic
142	C₂	#16810832	1	1	yes	characteristic
143	C₂	#16810875	1	1	yes	characteristic
144	C₂	#16810897	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811026

C₂

yes

maximal, characteristic

#16810834

C₂

yes

maximal, characteristic

#16810830

C₂

yes

maximal, characteristic

#16810750

C₂

yes

maximal, characteristic

#16810373

C₂

yes

maximal, characteristic

#16810368

C₂

yes

maximal, characteristic

C₅₃×#332

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

—

C₂×#11719925

—

#13133217

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

—

—

C₂²

yes

—

—

—

—

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×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

Q₁₆

yes

complemented

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#1694

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

#107

—

106

—

#107

—

106

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#103

—

106

—

#103

—

106

—

#97

—

106

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

Q₁₆

—

212

—

100

Q₁₆

—

212

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

—

106

—

111

C₄⋊C₄

—

106

—

112

C₄²

—

106

—

113

C₄²

C₂²×D₅₃

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

106

—

117

Q₈

—

106

—

118

Q₈

—

212

—

119

Q₈

—

212

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

212

—

124

C₂×C₄

#11719945

yes

characteristic

125

C₂×C₄

D₄×D₅₃

yes

characteristic

126

C₂×C₄

D₄×D₅₃

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

106

—

131

C₈

—

212

—

132

C₂²

#13133190

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

424

—

136

C₄

—

212

—

137

C₄

—

212

—

138

C₄

Q₁₆×D₅₃

yes

characteristic

139

C₄

#13133241

yes

characteristic

140

C₄

#13133244

yes

characteristic

141

C₄

#13133246

yes

characteristic

142

C₂

#16810832

yes

characteristic

143

C₂

#16810875

yes

characteristic

144

C₂

#16810897

yes

characteristic

	subgroup	G/H
center	C₂²	#13133190
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#332	C₂
socle	C₂²×C₅₃	#121
hypercenter	#332	D₅₃
hypocenter	C₅₃	#1694
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#13133190

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#332

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
58. #88322	Q₁₆	semidirect
80. C₅₃	#1694	semidirect

H ≅ G/N

type

58. #88322

Q₁₆

semidirect

80. C₅₃

#1694

semidirect

Subgroups — #20274232

Conjugacy classes of subgroups

Complements