Group #19147479 — subgroups

Subgroups — Q₁₆×Dic₈₆ #19147479

Order 5504.

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×Dic₈₆	C₂	1	2	yes	maximal
2	#14583304	C₂	1	1	yes	maximal, characteristic
3	#14583303	C₂	1	2	yes	maximal
4	#14583194	C₂	1	2	yes	maximal
5	#14583192	C₂	1	4	yes	maximal
6	C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₄₃	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
9	#13063325	—	2	2	no	—
10	#13063325	C₂²	1	2	yes	—
11	#13063324	—	2	4	no	—
12	#13063324	C₂²	1	4	yes	—
13	Q₈×Dic₄₃	C₂²	1	4	yes	—
14	C₄×Dic₈₆	—	2	2	no	—
15	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
16	#13063259	C₂²	1	2	yes	—
17	#13063259	C₂²	1	2	yes	—
18	#13063258	C₂²	1	4	yes	—
19	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₄₃⋊C₈	C₂²	1	2	yes	—
21	#13063249	C₂²	1	4	yes	—
22	C₈×Dic₄₃	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
25	#107×C₄₃	C₂²	1	1	yes	characteristic
26	#103×C₄₃	C₂²	1	2	yes	—
27	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
29	C₂×Dic₈₆	—	2	2	no	—
30	C₂×Dic₈₆	—	2	4	no	—
31	C₂×Dic₈₆	D₄	1	1	yes	characteristic
32	Q₁₆×C₄₃	Q₈	1	4	yes	complemented
33	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₄₃	—	2	2	no	—
36	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
37	C₄²×C₄₃	—	2	2	no	—
38	C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₄₃	—	2	2	no	—
40	C₄⋊Dic₄₃	—	2	4	no	—
41	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₃	C₂³	1	4	yes	—
43	Dic₄₃⋊C₄	—	2	4	no	—
44	Dic₄₃⋊C₄	D₄	1	2	yes	—
45	C₄×Dic₄₃	—	2	4	no	—
46	C₄×Dic₄₃	C₂³	1	2	yes	—
47	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
48	Q₈×C₄₃	—	2	2	no	—
49	Q₈×C₄₃	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₃	—	2	4	no	—
56	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
57	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
58	Dic₈₆	—	8	4	no	—
59	Dic₈₆	—	8	2	no	—
60	Dic₈₆	Q₁₆	1	4	yes	direct product factor
61	C₈×C₄₃	—	2	1	no	—
62	C₈×C₄₃	C₂×Q₈	1	2	yes	—
63	C₄₃⋊C₈	—	2	2	no	—
64	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₃	—	2	4	no	—
66	C₄×C₄₃	—	2	1	no	—
67	C₄×C₄₃	—	4	2	no	—
68	C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
69	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
71	Dic₄₃	—	2	2	no	—
72	Dic₄₃	—	4	4	no	—
73	Dic₄₃	C₂×Q₁₆	1	4	yes	—
74	Q₈×Q₁₆	—	43	1	no	maximal, Sylow
75	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
76	C₂×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
77	C₂×C₄₃	#576	1	1	yes	characteristic
78	Q₈²	—	43	2	no	—
79	#500	—	43	2	no	—
80	#500	—	43	1	no	—
81	#477	—	43	2	no	—
82	#477	—	43	4	no	—
83	C₈×Q₈	—	43	1	no	—
84	C₄×Q₁₆	—	43	2	no	—
85	C₄×Q₁₆	D₄₃	1	1	yes	characteristic
86	C₄₃	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
88	C₄⋊Q₈	—	43	4	no	—
89	C₄⋊Q₈	—	43	2	no	—
90	C₄⋊Q₈	—	86	2	no	—
91	C₄⋊Q₈	—	86	4	no	—
92	C₄×Q₈	—	43	1	no	—
93	C₄×Q₈	—	43	4	no	—
94	C₄×Q₈	—	86	2	no	—
95	C₄×Q₈	C₂×D₄₃	1	2	yes	—
96	#107	—	43	2	no	—
97	#107	—	43	2	no	—
98	#107	—	43	4	no	—
99	#107	C₂×D₄₃	1	1	yes	characteristic
100	C₄⋊C₈	—	43	2	no	—
101	C₄⋊C₈	—	43	1	no	—
102	#103	—	43	4	no	—
103	#103	C₂×D₄₃	1	2	yes	—
104	C₄×C₈	—	43	2	no	—
105	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
106	C₂×Q₈	—	43	1	no	—
107	C₂×Q₈	—	86	4	no	—
108	C₂×Q₈	—	86	2	no	—
109	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
110	Q₁₆	Dic₈₆	1	4	yes	direct product factor
111	C₂×C₈	—	43	2	no	—
112	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
113	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	43	2	no	—
116	C₄⋊C₄	—	43	1	no	—
117	C₄⋊C₄	—	43	4	no	—
118	C₄⋊C₄	—	86	4	no	—
119	C₄⋊C₄	—	86	4	no	—
120	C₄⋊C₄	—	86	2	no	—
121	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
122	C₄²	—	2	2	no	—
123	C₄²	—	43	2	no	—
124	C₄²	—	86	4	no	—
125	C₄²	C₂²×D₄₃	1	1	yes	characteristic
126	Q₈	—	2	2	no	—
127	Q₈	—	43	4	no	—
128	Q₈	—	344	2	no	—
129	Q₈	—	344	4	no	—
130	Q₈	C₂×Dic₈₆	1	4	yes	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	43	2	no	—
134	C₂×C₄	—	43	2	no	—
135	C₂×C₄	—	86	4	no	—
136	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
137	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	86	2	no	—
141	C₈	C₂×Dic₈₆	1	2	yes	—
142	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
143	C₄	—	2	4	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	2	no	—
146	C₄	—	43	4	no	—
147	C₄	—	86	2	no	—
148	C₄	—	172	4	no	—
149	C₄	Q₁₆×D₄₃	1	2	yes	—
150	C₄	C₂²×Dic₈₆	1	1	yes	characteristic
151	C₄	#13063372	1	1	yes	characteristic
152	C₂	D₄×Dic₈₆	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
154	C₂	#14583701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₈₆

C₂

yes

maximal

#14583304

C₂

yes

maximal, characteristic

#14583303

C₂

yes

maximal

#14583194

C₂

yes

maximal

#14583192

C₂

yes

maximal

C₈×Dic₈₆

C₂

yes

maximal, characteristic

Q₁₆×Dic₄₃

C₂

yes

maximal

C₄×Q₁₆×C₄₃

C₂

yes

maximal, characteristic

#13063325

—

#13063325

C₂²

yes

—

#13063324

—

#13063324

C₂²

yes

—

Q₈×Dic₄₃

C₂²

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₂²

yes

characteristic

#13063259

C₂²

yes

—

#13063259

C₂²

yes

—

#13063258

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

Dic₄₃⋊C₈

C₂²

yes

—

#13063249

C₂²

yes

—

C₈×Dic₄₃

C₂²

yes

—

C₂×Q₁₆×C₄₃

C₂²

yes

characteristic

C₄×Q₈×C₄₃

C₂²

yes

—

#107×C₄₃

C₂²

yes

characteristic

#103×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

characteristic

Q₁₆×C₄₃

Q₈

yes

complemented

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂³

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

D₄

yes

—

C₄×Dic₄₃

—

C₄×Dic₄₃

C₂³

yes

—

C₂×C₄₃⋊C₈

C₂³

yes

—

Q₈×C₄₃

—

Q₈×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×D₄

yes

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

Q₁₆

yes

direct product factor

C₈×C₄₃

—

C₈×C₄₃

C₂×Q₈

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×Q₁₆

yes

—

C₄×C₄₃

C₂²×Q₈

yes

characteristic

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

C₂×Q₁₆

yes

—

Q₈×Q₁₆

—

maximal, Sylow

C₂×C₄₃

D₄×Q₈

yes

characteristic

C₂×C₄₃

C₂²×Q₁₆

yes

characteristic

C₂×C₄₃

#576

yes

characteristic

Q₈²

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

D₄₃

yes

characteristic

C₄₃

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₄₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₃

yes

—

—

—

—

C₂×D₄₃

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#103

—

103

#103

C₂×D₄₃

yes

—

104

C₄×C₈

—

105

C₄×C₈

C₂×D₄₃

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

C₂²×D₄₃

yes

—

110

Q₁₆

Dic₈₆

yes

direct product factor

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₄₃

yes

characteristic

113

C₂×C₈

C₂²×D₄₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

C₂²×D₄₃

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂²×D₄₃

yes

characteristic

126

Q₈

—

127

Q₈

—

128

Q₈

—

344

—

129

Q₈

—

344

—

130

Q₈

C₂×Dic₈₆

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

D₄×D₄₃

yes

characteristic

137

C₂×C₄

D₄×D₄₃

yes

characteristic

138

C₂×C₄

C₂³×D₄₃

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

C₂×Dic₈₆

yes

—

142

C₂²

C₂×D₄×D₄₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

172

—

149

C₄

Q₁₆×D₄₃

yes

—

150

C₄

C₂²×Dic₈₆

yes

characteristic

151

C₄

#13063372

yes

characteristic

152

C₂

D₄×Dic₈₆

yes

characteristic

153

C₂

C₂×Q₁₆×D₄₃

yes

characteristic

154

C₂

#14583701

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄×Q₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×Q₁₆	D₄₃
hypocenter	C₄₃	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄×Q₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂²×D₄

hypercenter

C₄×Q₁₆

D₄₃

hypocenter

C₄₃

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
32. Q₁₆×C₄₃	Q₈	semidirect
60. Dic₈₆	Q₁₆	direct (N × H)
86. C₄₃	Q₈×Q₁₆	semidirect
110. Q₁₆	Dic₈₆	direct (N × H)

H ≅ G/N

type

32. Q₁₆×C₄₃

Q₈

semidirect

60. Dic₈₆

Q₁₆

direct (N × H)

86. C₄₃

Q₈×Q₁₆

semidirect

110. Q₁₆

Dic₈₆

direct (N × H)

Subgroups — Q₁₆×Dic₈₆ #19147479

Conjugacy classes of subgroups

Complements

Subgroups — Q16×Dic86 #19147479

Conjugacy classes of subgroups

Complements

Subgroups — Q₁₆×Dic₈₆ #19147479