Group #20287719 — subgroups

Subgroups — #20287719

Order 6784.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16811348	C₂	1	2	yes	maximal
2	#16811001	C₂	1	2	yes	maximal
3	#16810831	C₂	1	4	yes	maximal
4	#16810830	C₂	1	2	yes	maximal
5	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₅₃	C₂	1	2	yes	maximal
8	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×#11719925	C₂²	1	4	yes	—
11	#13133215	C₂²	1	4	yes	—
12	#13133208	—	2	2	no	—
13	#13133207	—	2	4	no	—
14	Q₈×Dic₅₃	C₂²	1	4	yes	—
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₄×Dic₁₀₆	C₂²	1	2	yes	—
17	#13133182	—	2	2	no	—
18	#13133151	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	C₂²	1	2	yes	—
21	#13133142	C₂²	1	2	yes	—
22	#13133141	C₂²	1	4	yes	—
23	C₈×Dic₅₃	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
25	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
29	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
30	C₂×Dic₁₀₆	C₂³	1	2	yes	—
31	Q₁₆×C₅₃	—	2	4	no	—
32	Q₁₆×C₅₃	—	4	2	no	—
33	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#11719925	—	4	4	no	—
39	C₄⋊Dic₅₃	—	2	4	no	—
40	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
41	C₄⋊Dic₅₃	C₂³	1	2	yes	—
42	#11719919	—	2	2	no	—
43	#11719919	—	2	4	no	—
44	#11719919	—	2	4	no	—
45	#11719919	D₄	1	2	yes	—
46	C₄×Dic₅₃	—	2	2	no	—
47	C₄×Dic₅₃	—	2	4	no	—
48	C₄×Dic₅₃	C₂³	1	2	yes	—
49	C₂×#88322	C₂³	1	2	yes	—
50	Dic₂₁₂	—	2	2	no	—
51	Q₈×C₅₃	—	2	4	no	—
52	Q₈×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	2	no	—
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
60	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
61	Dic₁₀₆	—	2	2	no	—
62	Dic₁₀₆	—	2	2	no	—
63	Dic₁₀₆	Q₁₆	1	4	yes	complemented
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	#41	1	2	yes	—
66	#88322	—	2	2	no	—
67	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	4	4	no	—
70	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
71	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
72	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
73	Dic₅₃	—	2	2	no	—
74	Dic₅₃	—	4	2	no	—
75	Dic₅₃	—	4	2	no	—
76	Dic₅₃	C₂×Q₁₆	1	4	yes	—
77	#3395	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	#550	1	1	yes	characteristic
79	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
80	C₂×C₅₃	#578	1	1	yes	characteristic
81	#557	—	53	2	no	—
82	#494	—	53	2	no	—
83	#494	D₅₃	1	1	yes	characteristic
84	#462	—	53	4	no	—
85	#462	—	53	2	no	—
86	C₈×Q₈	—	53	1	no	—
87	C₄×Q₁₆	—	53	2	no	—
88	C₄×Q₁₆	—	53	1	no	—
89	C₅₃	#3395	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₁₆	—	53	4	no	—
91	C₂×Q₁₆	—	53	1	no	—
92	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
93	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
94	C₄⋊Q₈	—	53	4	no	—
95	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
96	#125	—	106	4	no	—
97	#125	—	106	2	no	—
98	C₄×Q₈	—	53	1	no	—
99	C₄×Q₈	—	53	4	no	—
100	C₄×Q₈	—	53	2	no	—
101	C₄×Q₈	—	106	2	no	—
102	#107	—	53	1	no	—
103	C₄⋊C₈	—	53	2	no	—
104	C₄⋊C₈	—	53	1	no	—
105	#103	—	53	4	no	—
106	#103	—	53	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₂×Q₈	—	53	1	no	—
110	C₂×Q₈	—	53	2	no	—
111	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
112	Q₁₆	—	2	4	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	106	2	no	—
115	Q₁₆	—	212	4	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	53	2	no	—
122	C₄⋊C₄	—	53	1	no	—
123	C₄⋊C₄	—	53	2	no	—
124	C₄⋊C₄	—	106	4	no	—
125	C₄⋊C₄	—	106	4	no	—
126	C₄⋊C₄	—	106	4	no	—
127	C₄⋊C₄	—	106	2	no	—
128	C₄²	—	53	2	no	—
129	C₄²	—	106	4	no	—
130	C₄²	—	106	2	no	—
131	C₄²	C₂²×D₅₃	1	1	yes	characteristic
132	Q₈	—	2	4	no	—
133	Q₈	—	2	4	no	—
134	Q₈	—	53	4	no	—
135	Q₈	—	106	2	no	—
136	Q₈	—	106	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	53	2	no	—
139	C₂×C₄	—	53	2	no	—
140	C₂×C₄	—	106	2	no	—
141	C₂×C₄	—	106	2	no	—
142	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	106	2	no	—
147	C₈	#11719947	1	2	yes	—
148	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	4	no	—
151	C₄	—	53	4	no	—
152	C₄	—	106	2	no	—
153	C₄	—	212	2	no	—
154	C₄	—	212	2	no	—
155	C₄	Q₁₆×D₅₃	1	2	yes	—
156	C₄	C₂×#11719947	1	1	yes	characteristic
157	C₄	#13133262	1	1	yes	characteristic
158	C₂	#16811293	1	1	yes	characteristic
159	C₂	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
160	C₂	#16811436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811348

C₂

yes

maximal

#16811001

C₂

yes

maximal

#16810831

C₂

yes

maximal

#16810830

C₂

yes

maximal

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

Q₁₆×Dic₅₃

C₂

yes

maximal

C₅₃×#494

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719925

C₂²

yes

—

#13133215

C₂²

yes

—

#13133208

—

#13133207

—

Q₈×Dic₅₃

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

C₄×Dic₁₀₆

C₂²

yes

—

#13133182

—

#13133151

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133142

C₂²

yes

—

#13133141

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×Dic₁₀₆

D₄

yes

characteristic

C₂×Dic₁₀₆

C₂³

yes

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

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

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

Q₁₆

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

characteristic

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×Q₁₆

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×Q₁₆

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#462

—

#462

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₅₃

#3395

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂×Q₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

—

#125

—

106

—

#125

—

106

—

C₄×Q₈

—

C₄×Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

—

106

—

102

#107

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#103

—

106

#103

—

107

C₄×C₈

—

108

C₄×C₈

C₂×D₅₃

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

C₂²×D₅₃

yes

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

106

—

115

Q₁₆

—

212

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₅₃

yes

characteristic

118

C₂×C₈

C₂²×D₅₃

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

106

—

125

C₄⋊C₄

—

106

—

126

C₄⋊C₄

—

106

—

127

C₄⋊C₄

—

106

—

128

C₄²

—

129

C₄²

—

106

—

130

C₄²

—

106

—

131

C₄²

C₂²×D₅₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

106

—

136

Q₈

—

106

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

106

—

141

C₂×C₄

—

106

—

142

C₂×C₄

D₄×D₅₃

yes

characteristic

143

C₂×C₄

D₄×D₅₃

yes

characteristic

144

C₂×C₄

C₂³×D₅₃

yes

characteristic

145

C₈

—

146

C₈

—

106

—

147

C₈

#11719947

yes

—

148

C₂²

C₂×D₄×D₅₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

106

—

153

C₄

—

212

—

154

C₄

—

212

—

155

C₄

Q₁₆×D₅₃

yes

—

156

C₄

C₂×#11719947

yes

characteristic

157

C₄

#13133262

yes

characteristic

158

C₂

#16811293

yes

characteristic

159

C₂

C₂×Q₁₆×D₅₃

yes

characteristic

160

C₂

#16811436

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#494	D₅₃
hypocenter	C₅₃	#3395
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#494

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
63. Dic₁₀₆	Q₁₆	semidirect
89. C₅₃	#3395	semidirect

H ≅ G/N

type

63. Dic₁₀₆

Q₁₆

semidirect

89. C₅₃

#3395

semidirect

Subgroups — #20287719

Conjugacy classes of subgroups

Complements