Group #21458703 — subgroups

Subgroups — Q₁₆×Dic₁₂₂ #21458703

Order 7808.

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×Dic₁₂₂	C₂	1	2	yes	maximal
2	#16840167	C₂	1	1	yes	maximal, characteristic
3	#16840166	C₂	1	2	yes	maximal
4	#16840057	C₂	1	2	yes	maximal
5	#16840055	C₂	1	4	yes	maximal
6	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₆₁	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
9	#14519777	—	2	2	no	—
10	#14519777	C₂²	1	2	yes	—
11	#14519776	—	2	4	no	—
12	#14519776	C₂²	1	4	yes	—
13	Q₈×Dic₆₁	C₂²	1	4	yes	—
14	C₄×Dic₁₂₂	—	2	2	no	—
15	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
16	#14519711	C₂²	1	2	yes	—
17	#14519711	C₂²	1	2	yes	—
18	#14519710	C₂²	1	4	yes	—
19	C₄⋊#92418	C₂²	1	1	yes	characteristic
20	#14519706	C₂²	1	2	yes	—
21	#14519701	C₂²	1	4	yes	—
22	C₈×Dic₆₁	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
25	#107×C₆₁	C₂²	1	1	yes	characteristic
26	#103×C₆₁	C₂²	1	2	yes	—
27	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
29	C₂×Dic₁₂₂	—	2	2	no	—
30	C₂×Dic₁₂₂	—	2	4	no	—
31	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
32	Q₁₆×C₆₁	Q₈	1	4	yes	complemented
33	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
37	C₄²×C₆₁	—	2	2	no	—
38	C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₆₁	—	2	2	no	—
40	C₄⋊Dic₆₁	—	2	4	no	—
41	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
42	C₄⋊Dic₆₁	C₂³	1	4	yes	—
43	#11758196	—	2	4	no	—
44	#11758196	D₄	1	2	yes	—
45	C₄×Dic₆₁	—	2	4	no	—
46	C₄×Dic₆₁	C₂³	1	2	yes	—
47	C₂×#92418	C₂³	1	2	yes	—
48	Q₈×C₆₁	—	2	2	no	—
49	Q₈×C₆₁	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₆₁	—	2	2	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₆₁	—	2	4	no	—
56	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
57	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
58	Dic₁₂₂	—	8	4	no	—
59	Dic₁₂₂	—	8	2	no	—
60	Dic₁₂₂	Q₁₆	1	4	yes	direct product factor
61	C₈×C₆₁	—	2	1	no	—
62	C₈×C₆₁	C₂×Q₈	1	2	yes	—
63	#92418	—	2	2	no	—
64	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₆₁	—	2	4	no	—
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	—	4	2	no	—
68	C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
69	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
71	Dic₆₁	—	2	2	no	—
72	Dic₆₁	—	4	4	no	—
73	Dic₆₁	C₂×Q₁₆	1	4	yes	—
74	Q₈×Q₁₆	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	D₄×Q₈	1	1	yes	characteristic
76	C₂×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
77	C₂×C₆₁	#576	1	1	yes	characteristic
78	Q₈²	—	61	2	no	—
79	#500	—	61	2	no	—
80	#500	—	61	1	no	—
81	#477	—	61	2	no	—
82	#477	—	61	4	no	—
83	C₈×Q₈	—	61	1	no	—
84	C₄×Q₁₆	—	61	2	no	—
85	C₄×Q₁₆	D₆₁	1	1	yes	characteristic
86	C₆₁	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
88	C₄⋊Q₈	—	61	4	no	—
89	C₄⋊Q₈	—	61	2	no	—
90	C₄⋊Q₈	—	122	2	no	—
91	C₄⋊Q₈	—	122	4	no	—
92	C₄×Q₈	—	61	1	no	—
93	C₄×Q₈	—	61	4	no	—
94	C₄×Q₈	—	122	2	no	—
95	C₄×Q₈	C₂×D₆₁	1	2	yes	—
96	#107	—	61	2	no	—
97	#107	—	61	2	no	—
98	#107	—	61	4	no	—
99	#107	C₂×D₆₁	1	1	yes	characteristic
100	C₄⋊C₈	—	61	2	no	—
101	C₄⋊C₈	—	61	1	no	—
102	#103	—	61	4	no	—
103	#103	C₂×D₆₁	1	2	yes	—
104	C₄×C₈	—	61	2	no	—
105	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
106	C₂×Q₈	—	61	1	no	—
107	C₂×Q₈	—	122	4	no	—
108	C₂×Q₈	—	122	2	no	—
109	C₂×Q₈	C₂²×D₆₁	1	2	yes	—
110	Q₁₆	Dic₁₂₂	1	4	yes	direct product factor
111	C₂×C₈	—	61	2	no	—
112	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
113	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	61	2	no	—
116	C₄⋊C₄	—	61	1	no	—
117	C₄⋊C₄	—	61	4	no	—
118	C₄⋊C₄	—	122	4	no	—
119	C₄⋊C₄	—	122	4	no	—
120	C₄⋊C₄	—	122	2	no	—
121	C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
122	C₄²	—	2	2	no	—
123	C₄²	—	61	2	no	—
124	C₄²	—	122	4	no	—
125	C₄²	C₂²×D₆₁	1	1	yes	characteristic
126	Q₈	—	2	2	no	—
127	Q₈	—	61	4	no	—
128	Q₈	—	488	2	no	—
129	Q₈	—	488	4	no	—
130	Q₈	C₂×Dic₁₂₂	1	4	yes	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	61	2	no	—
134	C₂×C₄	—	61	2	no	—
135	C₂×C₄	—	122	4	no	—
136	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
137	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	122	2	no	—
141	C₈	C₂×Dic₁₂₂	1	2	yes	—
142	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	4	no	—
145	C₄	—	4	2	no	—
146	C₄	—	61	4	no	—
147	C₄	—	122	2	no	—
148	C₄	—	244	4	no	—
149	C₄	Q₁₆×D₆₁	1	2	yes	—
150	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
151	C₄	#14519824	1	1	yes	characteristic
152	C₂	D₄×Dic₁₂₂	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
154	C₂	#16840564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₂₂

C₂

yes

maximal

#16840167

C₂

yes

maximal, characteristic

#16840166

C₂

yes

maximal

#16840057

C₂

yes

maximal

#16840055

C₂

yes

maximal

C₈×Dic₁₂₂

C₂

yes

maximal, characteristic

Q₁₆×Dic₆₁

C₂

yes

maximal

C₄×Q₁₆×C₆₁

C₂

yes

maximal, characteristic

#14519777

—

#14519777

C₂²

yes

—

#14519776

—

#14519776

C₂²

yes

—

Q₈×Dic₆₁

C₂²

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519711

C₂²

yes

—

#14519710

C₂²

yes

—

C₄⋊#92418

C₂²

yes

characteristic

#14519706

C₂²

yes

—

#14519701

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

C₂²

yes

—

#107×C₆₁

C₂²

yes

characteristic

#103×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

D₄

yes

characteristic

Q₁₆×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

D₄

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

Q₁₆

yes

direct product factor

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₈

yes

—

#92418

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₁₆

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×Q₁₆

yes

—

Q₈×Q₁₆

—

maximal, Sylow

C₂×C₆₁

D₄×Q₈

yes

characteristic

C₂×C₆₁

C₂²×Q₁₆

yes

characteristic

C₂×C₆₁

#576

yes

characteristic

Q₈²

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

D₆₁

yes

characteristic

C₆₁

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₄×Q₈

C₂×D₆₁

yes

—

—

—

—

C₂×D₆₁

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#103

—

103

#103

C₂×D₆₁

yes

—

104

C₄×C₈

—

105

C₄×C₈

C₂×D₆₁

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

122

—

108

C₂×Q₈

—

122

—

109

C₂×Q₈

C₂²×D₆₁

yes

—

110

Q₁₆

Dic₁₂₂

yes

direct product factor

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₆₁

yes

characteristic

113

C₂×C₈

C₂²×D₆₁

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

122

—

119

C₄⋊C₄

—

122

—

120

C₄⋊C₄

—

122

—

121

C₄⋊C₄

C₂²×D₆₁

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

122

—

125

C₄²

C₂²×D₆₁

yes

characteristic

126

Q₈

—

127

Q₈

—

128

Q₈

—

488

—

129

Q₈

—

488

—

130

Q₈

C₂×Dic₁₂₂

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

122

—

136

C₂×C₄

D₄×D₆₁

yes

characteristic

137

C₂×C₄

D₄×D₆₁

yes

characteristic

138

C₂×C₄

C₂³×D₆₁

yes

characteristic

139

C₈

—

140

C₈

—

122

—

141

C₈

C₂×Dic₁₂₂

yes

—

142

C₂²

C₂×D₄×D₆₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

122

—

148

C₄

—

244

—

149

C₄

Q₁₆×D₆₁

yes

—

150

C₄

C₂²×Dic₁₂₂

yes

characteristic

151

C₄

#14519824

yes

characteristic

152

C₂

D₄×Dic₁₂₂

yes

characteristic

153

C₂

C₂×Q₁₆×D₆₁

yes

characteristic

154

C₂

#16840564

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₄×Q₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₄×Q₁₆	D₆₁
hypocenter	C₆₁	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₄×Q₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₄×Q₁₆

D₆₁

hypocenter

C₆₁

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
32. Q₁₆×C₆₁	Q₈	semidirect
60. Dic₁₂₂	Q₁₆	direct (N × H)
86. C₆₁	Q₈×Q₁₆	semidirect
110. Q₁₆	Dic₁₂₂	direct (N × H)

H ≅ G/N

type

32. Q₁₆×C₆₁

Q₈

semidirect

60. Dic₁₂₂

Q₁₆

direct (N × H)

86. C₆₁

Q₈×Q₁₆

semidirect

110. Q₁₆

Dic₁₂₂

direct (N × H)

Subgroups — Q₁₆×Dic₁₂₂ #21458703

Conjugacy classes of subgroups

Complements

Subgroups — Q16×Dic122 #21458703

Conjugacy classes of subgroups

Complements

Subgroups — Q₁₆×Dic₁₂₂ #21458703