Group #22601224 — subgroups

Subgroups — Q₁₆×Dic₁₄₂ #22601224

Order 9088.

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×Dic₁₄₂	C₂	1	2	yes	maximal
2	#17986404	C₂	1	1	yes	maximal, characteristic
3	#17986403	C₂	1	2	yes	maximal
4	#17986294	C₂	1	2	yes	maximal
5	#17986292	C₂	1	4	yes	maximal
6	C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₇₁	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#14556128	—	2	2	no	—
10	#14556128	C₂²	1	2	yes	—
11	#14556127	—	2	4	no	—
12	#14556127	C₂²	1	4	yes	—
13	Q₈×Dic₇₁	C₂²	1	4	yes	—
14	C₄×Dic₁₄₂	—	2	2	no	—
15	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
16	#14556062	C₂²	1	2	yes	—
17	#14556062	C₂²	1	2	yes	—
18	#14556061	C₂²	1	4	yes	—
19	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₇₁⋊C₈	C₂²	1	2	yes	—
21	#14556052	C₂²	1	4	yes	—
22	C₈×Dic₇₁	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₇₁	C₂²	1	2	yes	—
25	#107×C₇₁	C₂²	1	1	yes	characteristic
26	#103×C₇₁	C₂²	1	2	yes	—
27	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₄₂	—	2	2	no	—
29	C₂×Dic₁₄₂	—	2	4	no	—
30	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
31	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₇₁	—	2	2	no	—
33	C₄⋊Dic₇₁	—	2	4	no	—
34	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
35	C₄⋊Dic₇₁	C₂³	1	4	yes	—
36	Dic₇₁⋊C₄	—	2	4	no	—
37	Dic₇₁⋊C₄	D₄	1	2	yes	—
38	C₄×Dic₇₁	—	2	4	no	—
39	C₄×Dic₇₁	C₂³	1	2	yes	—
40	C₂×Q₈×C₇₁	C₂³	1	2	yes	—
41	Q₁₆×C₇₁	Q₈	1	4	yes	complemented
42	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₇₁	—	2	2	no	—
45	C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
46	C₄²×C₇₁	—	2	2	no	—
47	C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	Q₈×C₇₁	—	2	2	no	—
49	Q₈×C₇₁	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₁	—	2	4	no	—
56	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
57	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
58	Dic₁₄₂	—	8	4	no	—
59	Dic₁₄₂	—	8	2	no	—
60	Dic₁₄₂	Q₁₆	1	4	yes	direct product factor
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	C₂×Q₈	1	2	yes	—
63	C₇₁⋊C₈	—	2	2	no	—
64	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₇₁	—	2	4	no	—
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	—	4	2	no	—
68	C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
69	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
71	Dic₇₁	—	2	2	no	—
72	Dic₇₁	—	4	4	no	—
73	Dic₇₁	C₂×Q₁₆	1	4	yes	—
74	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
75	C₂×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₇₁	#576	1	1	yes	characteristic
77	Q₈×Q₁₆	—	71	1	no	maximal, Sylow
78	C₇₁	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
79	Q₈²	—	71	2	no	—
80	#500	—	71	2	no	—
81	#500	—	71	1	no	—
82	#477	—	71	2	no	—
83	#477	—	71	4	no	—
84	C₈×Q₈	—	71	1	no	—
85	C₄×Q₁₆	—	71	2	no	—
86	C₄×Q₁₆	D₇₁	1	1	yes	characteristic
87	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
88	C₄⋊Q₈	—	71	4	no	—
89	C₄⋊Q₈	—	71	2	no	—
90	C₄⋊Q₈	—	142	2	no	—
91	C₄⋊Q₈	—	142	4	no	—
92	C₄×Q₈	—	71	1	no	—
93	C₄×Q₈	—	71	4	no	—
94	C₄×Q₈	—	142	2	no	—
95	C₄×Q₈	C₂×D₇₁	1	2	yes	—
96	#107	—	71	2	no	—
97	#107	—	71	2	no	—
98	#107	—	71	4	no	—
99	#107	C₂×D₇₁	1	1	yes	characteristic
100	C₄⋊C₈	—	71	2	no	—
101	C₄⋊C₈	—	71	1	no	—
102	#103	—	71	4	no	—
103	#103	C₂×D₇₁	1	2	yes	—
104	C₄×C₈	—	71	2	no	—
105	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₂×Q₈	—	71	1	no	—
107	C₂×Q₈	—	142	4	no	—
108	C₂×Q₈	—	142	2	no	—
109	C₂×Q₈	C₂²×D₇₁	1	2	yes	—
110	Q₁₆	Dic₁₄₂	1	4	yes	direct product factor
111	C₂×C₈	—	71	2	no	—
112	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
113	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	71	2	no	—
116	C₄⋊C₄	—	71	1	no	—
117	C₄⋊C₄	—	71	4	no	—
118	C₄⋊C₄	—	142	4	no	—
119	C₄⋊C₄	—	142	4	no	—
120	C₄⋊C₄	—	142	2	no	—
121	C₄⋊C₄	C₂²×D₇₁	1	2	yes	—
122	C₄²	—	2	2	no	—
123	C₄²	—	71	2	no	—
124	C₄²	—	142	4	no	—
125	C₄²	C₂²×D₇₁	1	1	yes	characteristic
126	Q₈	—	2	2	no	—
127	Q₈	—	568	2	no	—
128	Q₈	—	568	4	no	—
129	Q₈	—	71	4	no	—
130	Q₈	C₂×Dic₁₄₂	1	4	yes	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	71	2	no	—
134	C₂×C₄	—	71	2	no	—
135	C₂×C₄	—	142	4	no	—
136	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
137	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	142	2	no	—
141	C₈	C₂×Dic₁₄₂	1	2	yes	—
142	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
143	C₄	—	2	4	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	2	no	—
146	C₄	—	284	4	no	—
147	C₄	—	71	4	no	—
148	C₄	—	142	2	no	—
149	C₄	Q₁₆×D₇₁	1	2	yes	—
150	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
151	C₄	#14556175	1	1	yes	characteristic
152	C₂	D₄×Dic₁₄₂	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₇₁	1	1	yes	characteristic
154	C₂	#17986801	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₄₂

C₂

yes

maximal

#17986404

C₂

yes

maximal, characteristic

#17986403

C₂

yes

maximal

#17986294

C₂

yes

maximal

#17986292

C₂

yes

maximal

C₈×Dic₁₄₂

C₂

yes

maximal, characteristic

Q₁₆×Dic₇₁

C₂

yes

maximal

C₄×Q₁₆×C₇₁

C₂

yes

maximal, characteristic

#14556128

—

#14556128

C₂²

yes

—

#14556127

—

#14556127

C₂²

yes

—

Q₈×Dic₇₁

C₂²

yes

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

#14556062

C₂²

yes

—

#14556062

C₂²

yes

—

#14556061

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

C₂²

yes

—

#14556052

C₂²

yes

—

C₈×Dic₇₁

C₂²

yes

—

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

—

#107×C₇₁

C₂²

yes

characteristic

#103×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

D₄

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

C₂³

yes

—

Q₁₆×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

C₂×D₄

yes

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

Q₁₆

yes

direct product factor

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₁₆

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

C₂×Q₁₆

yes

—

C₂×C₇₁

D₄×Q₈

yes

characteristic

C₂×C₇₁

C₂²×Q₁₆

yes

characteristic

C₂×C₇₁

#576

yes

characteristic

Q₈×Q₁₆

—

maximal, Sylow

C₇₁

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

Q₈²

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

D₇₁

yes

characteristic

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

142

—

C₄×Q₈

C₂×D₇₁

yes

—

—

—

—

C₂×D₇₁

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#103

—

103

#103

C₂×D₇₁

yes

—

104

C₄×C₈

—

105

C₄×C₈

C₂×D₇₁

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

142

—

108

C₂×Q₈

—

142

—

109

C₂×Q₈

C₂²×D₇₁

yes

—

110

Q₁₆

Dic₁₄₂

yes

direct product factor

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₇₁

yes

characteristic

113

C₂×C₈

C₂²×D₇₁

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

142

—

119

C₄⋊C₄

—

142

—

120

C₄⋊C₄

—

142

—

121

C₄⋊C₄

C₂²×D₇₁

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

142

—

125

C₄²

C₂²×D₇₁

yes

characteristic

126

Q₈

—

127

Q₈

—

568

—

128

Q₈

—

568

—

129

Q₈

—

130

Q₈

C₂×Dic₁₄₂

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

142

—

136

C₂×C₄

D₄×D₇₁

yes

characteristic

137

C₂×C₄

D₄×D₇₁

yes

characteristic

138

C₂×C₄

C₂³×D₇₁

yes

characteristic

139

C₈

—

140

C₈

—

142

—

141

C₈

C₂×Dic₁₄₂

yes

—

142

C₂²

C₂×D₄×D₇₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

284

—

147

C₄

—

148

C₄

—

142

—

149

C₄

Q₁₆×D₇₁

yes

—

150

C₄

C₂²×Dic₁₄₂

yes

characteristic

151

C₄

#14556175

yes

characteristic

152

C₂

D₄×Dic₁₄₂

yes

characteristic

153

C₂

C₂×Q₁₆×D₇₁

yes

characteristic

154

C₂

#17986801

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₄×Q₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	C₄×Q₁₆	D₇₁
hypocenter	C₇₁	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₄×Q₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

C₄×Q₁₆

D₇₁

hypocenter

C₇₁

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
41. Q₁₆×C₇₁	Q₈	semidirect
60. Dic₁₄₂	Q₁₆	direct (N × H)
78. C₇₁	Q₈×Q₁₆	semidirect
110. Q₁₆	Dic₁₄₂	direct (N × H)

H ≅ G/N

type

41. Q₁₆×C₇₁

Q₈

semidirect

60. Dic₁₄₂

Q₁₆

direct (N × H)

78. C₇₁

Q₈×Q₁₆

semidirect

110. Q₁₆

Dic₁₄₂

direct (N × H)

Subgroups — Q₁₆×Dic₁₄₂ #22601224

Conjugacy classes of subgroups

Complements

Subgroups — Q16×Dic142 #22601224

Conjugacy classes of subgroups

Complements

Subgroups — Q₁₆×Dic₁₄₂ #22601224