Group #20287599 — subgroups

Subgroups — Q₁₆×Dic₁₀₆ #20287599

Order 6784.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	Q₈×Dic₁₀₆	C₂	1	2	yes	maximal
2	#16811028	C₂	1	1	yes	maximal, characteristic
3	#16811027	C₂	1	2	yes	maximal
4	#16810918	C₂	1	2	yes	maximal
5	#16810916	C₂	1	4	yes	maximal
6	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₅₃	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#13133217	—	2	2	no	—
10	#13133217	C₂²	1	2	yes	—
11	#13133216	—	2	4	no	—
12	#13133216	C₂²	1	4	yes	—
13	Q₈×Dic₅₃	C₂²	1	4	yes	—
14	C₄×Dic₁₀₆	—	2	2	no	—
15	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	#13133151	C₂²	1	2	yes	—
17	#13133151	C₂²	1	2	yes	—
18	#13133150	C₂²	1	4	yes	—
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	#13133146	C₂²	1	2	yes	—
21	#13133141	C₂²	1	4	yes	—
22	C₈×Dic₅₃	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
25	#107×C₅₃	C₂²	1	1	yes	characteristic
26	#103×C₅₃	C₂²	1	2	yes	—
27	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
29	C₂×Dic₁₀₆	—	2	2	no	—
30	C₂×Dic₁₀₆	—	2	4	no	—
31	C₂×Dic₁₀₆	D₄	1	1	yes	characteristic
32	Q₁₆×C₅₃	Q₈	1	4	yes	complemented
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	C₂³	1	2	yes	—
37	C₄²×C₅₃	—	2	2	no	—
38	C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₅₃	—	2	2	no	—
40	C₄⋊Dic₅₃	—	2	4	no	—
41	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₃	C₂³	1	4	yes	—
43	#11719919	—	2	4	no	—
44	#11719919	D₄	1	2	yes	—
45	C₄×Dic₅₃	—	2	4	no	—
46	C₄×Dic₅₃	C₂³	1	2	yes	—
47	C₂×#88322	C₂³	1	2	yes	—
48	Q₈×C₅₃	—	2	2	no	—
49	Q₈×C₅₃	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
57	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
58	Dic₁₀₆	—	8	4	no	—
59	Dic₁₀₆	—	8	2	no	—
60	Dic₁₀₆	Q₁₆	1	4	yes	direct product factor
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	C₂×Q₈	1	2	yes	—
63	#88322	—	2	2	no	—
64	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₅₃	—	2	4	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	4	2	no	—
68	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
69	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
71	Dic₅₃	—	2	2	no	—
72	Dic₅₃	—	4	4	no	—
73	Dic₅₃	C₂×Q₁₆	1	4	yes	—
74	Q₈×Q₁₆	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	D₄×Q₈	1	1	yes	characteristic
76	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
77	C₂×C₅₃	#576	1	1	yes	characteristic
78	Q₈²	—	53	2	no	—
79	#500	—	53	2	no	—
80	#500	—	53	1	no	—
81	#477	—	53	2	no	—
82	#477	—	53	4	no	—
83	C₈×Q₈	—	53	1	no	—
84	C₄×Q₁₆	—	53	2	no	—
85	C₄×Q₁₆	D₅₃	1	1	yes	characteristic
86	C₅₃	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
88	C₄⋊Q₈	—	53	4	no	—
89	C₄⋊Q₈	—	53	2	no	—
90	C₄⋊Q₈	—	106	2	no	—
91	C₄⋊Q₈	—	106	4	no	—
92	C₄×Q₈	—	53	1	no	—
93	C₄×Q₈	—	53	4	no	—
94	C₄×Q₈	—	106	2	no	—
95	C₄×Q₈	C₂×D₅₃	1	2	yes	—
96	#107	—	53	2	no	—
97	#107	—	53	2	no	—
98	#107	—	53	4	no	—
99	#107	C₂×D₅₃	1	1	yes	characteristic
100	C₄⋊C₈	—	53	2	no	—
101	C₄⋊C₈	—	53	1	no	—
102	#103	—	53	4	no	—
103	#103	C₂×D₅₃	1	2	yes	—
104	C₄×C₈	—	53	2	no	—
105	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
106	C₂×Q₈	—	53	1	no	—
107	C₂×Q₈	—	106	4	no	—
108	C₂×Q₈	—	106	2	no	—
109	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
110	Q₁₆	Dic₁₀₆	1	4	yes	direct product factor
111	C₂×C₈	—	53	2	no	—
112	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	53	2	no	—
116	C₄⋊C₄	—	53	1	no	—
117	C₄⋊C₄	—	53	4	no	—
118	C₄⋊C₄	—	106	4	no	—
119	C₄⋊C₄	—	106	4	no	—
120	C₄⋊C₄	—	106	2	no	—
121	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
122	C₄²	—	2	2	no	—
123	C₄²	—	53	2	no	—
124	C₄²	—	106	4	no	—
125	C₄²	C₂²×D₅₃	1	1	yes	characteristic
126	Q₈	—	2	2	no	—
127	Q₈	—	53	4	no	—
128	Q₈	—	424	2	no	—
129	Q₈	—	424	4	no	—
130	Q₈	C₂×Dic₁₀₆	1	4	yes	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	53	2	no	—
134	C₂×C₄	—	53	2	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
137	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	106	2	no	—
141	C₈	C₂×Dic₁₀₆	1	2	yes	—
142	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
143	C₄	—	2	4	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	2	no	—
146	C₄	—	53	4	no	—
147	C₄	—	106	2	no	—
148	C₄	—	212	4	no	—
149	C₄	Q₁₆×D₅₃	1	2	yes	—
150	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
151	C₄	#13133264	1	1	yes	characteristic
152	C₂	D₄×Dic₁₀₆	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
154	C₂	#16811425	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal

#16811028

C₂

yes

maximal, characteristic

#16811027

C₂

yes

maximal

#16810918

C₂

yes

maximal

#16810916

C₂

yes

maximal

C₈×Dic₁₀₆

C₂

yes

maximal, characteristic

Q₁₆×Dic₅₃

C₂

yes

maximal

C₄×Q₁₆×C₅₃

C₂

yes

maximal, characteristic

#13133217

—

#13133217

C₂²

yes

—

#13133216

—

#13133216

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133151

C₂²

yes

—

#13133150

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133141

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

characteristic

#103×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

D₄

yes

characteristic

Q₁₆×C₅₃

Q₈

yes

complemented

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

#11719919

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

C₂×Q₈

yes

—

C₂×C₄×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×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

Q₁₆

yes

direct product factor

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

—

#88322

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×Q₁₆

yes

—

Q₈×Q₁₆

—

maximal, Sylow

C₂×C₅₃

D₄×Q₈

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

#576

yes

characteristic

Q₈²

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

D₅₃

yes

characteristic

C₅₃

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

106

—

C₄×Q₈

C₂×D₅₃

yes

—

—

—

—

C₂×D₅₃

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#103

—

103

#103

C₂×D₅₃

yes

—

104

C₄×C₈

—

105

C₄×C₈

C₂×D₅₃

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

106

—

108

C₂×Q₈

—

106

—

109

C₂×Q₈

C₂²×D₅₃

yes

—

110

Q₁₆

Dic₁₀₆

yes

direct product factor

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₅₃

yes

characteristic

113

C₂×C₈

C₂²×D₅₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

106

—

120

C₄⋊C₄

—

106

—

121

C₄⋊C₄

C₂²×D₅₃

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

106

—

125

C₄²

C₂²×D₅₃

yes

characteristic

126

Q₈

—

127

Q₈

—

128

Q₈

—

424

—

129

Q₈

—

424

—

130

Q₈

C₂×Dic₁₀₆

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

D₄×D₅₃

yes

characteristic

137

C₂×C₄

D₄×D₅₃

yes

characteristic

138

C₂×C₄

C₂³×D₅₃

yes

characteristic

139

C₈

—

140

C₈

—

106

—

141

C₈

C₂×Dic₁₀₆

yes

—

142

C₂²

C₂×D₄×D₅₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

106

—

148

C₄

—

212

—

149

C₄

Q₁₆×D₅₃

yes

—

150

C₄

C₂²×Dic₁₀₆

yes

characteristic

151

C₄

#13133264

yes

characteristic

152

C₂

D₄×Dic₁₀₆

yes

characteristic

153

C₂

C₂×Q₁₆×D₅₃

yes

characteristic

154

C₂

#16811425

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₄×Q₁₆	D₅₃
hypocenter	C₅₃	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₄×Q₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

hypercenter

C₄×Q₁₆

D₅₃

hypocenter

C₅₃

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
32. Q₁₆×C₅₃	Q₈	semidirect
60. Dic₁₀₆	Q₁₆	direct (N × H)
86. C₅₃	Q₈×Q₁₆	semidirect
110. Q₁₆	Dic₁₀₆	direct (N × H)

H ≅ G/N

type

32. Q₁₆×C₅₃

Q₈

semidirect

60. Dic₁₀₆

Q₁₆

direct (N × H)

86. C₅₃

Q₈×Q₁₆

semidirect

110. Q₁₆

Dic₁₀₆

direct (N × H)

Subgroups — Q₁₆×Dic₁₀₆ #20287599

Conjugacy classes of subgroups

Complements

Subgroups — Q16×Dic106 #20287599

Conjugacy classes of subgroups

Complements

Subgroups — Q₁₆×Dic₁₀₆ #20287599