Group #21447637 — subgroups

Subgroups — C₁₆×C₂²⋊D₆₁ #21447637

Order 7808.

174 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×C₂²⋊D₆₁	C₂	1	1	yes	maximal, characteristic
3	#16839580	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊#11758185	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839541	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	C₄×C₂²⋊D₆₁	C₄	1	1	yes	characteristic
10	C₂×#11758185	C₂²	1	1	yes	characteristic
11	C₂×#11758185	C₂²	1	1	yes	characteristic
12	C₁₆×D₆₁	—	2	2	no	—
13	#14519706	C₄	1	1	yes	characteristic
14	C₂²⋊#92418	C₄	1	1	yes	characteristic
15	#14519689	C₄	1	1	yes	characteristic
16	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₆₁	—	2	2	no	—
19	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₂²⋊D₆₁	C₈	1	1	yes	characteristic
23	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₁	—	2	1	no	—
25	C₂×C₈×C₆₁	—	2	1	no	—
26	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₆₁	C₈	1	1	yes	characteristic
29	#11758198	C₈	1	1	yes	characteristic
30	#11758196	C₈	1	1	yes	characteristic
31	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂×#92418	C₂×C₄	1	1	yes	characteristic
33	C₂×#92418	C₂×C₄	1	1	yes	characteristic
34	C₈×D₆₁	—	2	1	no	—
35	C₈×D₆₁	—	2	1	no	—
36	C₁₆×C₆₁	—	2	1	no	—
37	C₁₆×C₆₁	D₄	1	2	yes	complemented
38	#11758185	—	2	1	no	—
39	#11758185	—	2	1	no	—
40	C₂³×C₆₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	2	1	no	—
43	C₂×C₄×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₆₁	C₁₆	1	4	yes	direct product factor
47	C₂×Dic₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₆₁	C₂×C₈	1	1	yes	characteristic
49	C₂×Dic₆₁	C₂×C₈	1	1	yes	characteristic
50	C₄×D₆₁	—	2	1	no	—
51	C₄×D₆₁	—	2	1	no	—
52	C₈×C₆₁	—	2	1	no	—
53	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₈×C₆₁	#41	1	1	yes	characteristic
55	#92418	—	2	1	no	—
56	#92418	—	2	1	no	—
57	C₂²×C₆₁	—	2	1	no	—
58	C₂²×C₆₁	—	2	1	no	—
59	C₂²×C₆₁	C₂×C₁₆	1	2	yes	complemented
60	C₂²×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₆₁	—	2	1	no	—
62	C₂×D₆₁	—	2	1	no	—
63	C₂×D₆₁	C₂×C₁₆	1	2	yes	complemented
64	C₄×C₆₁	—	2	1	no	—
65	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
66	C₄×C₆₁	#131	1	1	yes	characteristic
67	Dic₆₁	—	2	1	no	—
68	Dic₆₁	—	2	1	no	—
69	Dic₆₁	C₂×C₁₆	1	2	yes	complemented
70	D₄×C₁₆	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	—	2	2	no	—
72	C₂×C₆₁	C₈×D₄	1	1	yes	characteristic
73	C₂×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
74	C₂×C₆₁	#504	1	1	yes	characteristic
75	D₆₁	—	2	2	no	—
76	C₂²×C₁₆	—	61	1	no	—
77	C₂²×C₁₆	D₆₁	1	1	yes	characteristic, complemented
78	C₈×D₄	—	61	1	no	—
79	C₄⋊C₁₆	—	61	1	no	—
80	C₂²⋊C₁₆	—	61	1	no	—
81	C₂²⋊C₁₆	—	61	1	no	—
82	C₄×C₁₆	—	61	1	no	—
83	C₆₁	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	61	1	no	—
85	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
86	C₄×D₄	—	61	1	no	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	61	1	no	—
89	C₂×C₁₆	—	61	1	no	—
90	C₂×C₁₆	—	122	2	no	—
91	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
92	C₂×C₁₆	C₂×D₆₁	1	1	yes	characteristic
93	C₄⋊C₈	—	61	1	no	—
94	C₂²⋊C₈	—	61	1	no	—
95	C₂²⋊C₈	—	61	1	no	—
96	C₄×C₈	—	61	1	no	—
97	C₂×D₄	—	61	1	no	—
98	C₂²×C₄	—	61	1	no	—
99	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	61	1	no	—
103	C₂×C₈	—	61	1	no	—
104	C₂×C₈	—	122	1	no	—
105	C₂×C₈	—	122	1	no	—
106	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
108	C₄⋊C₄	—	61	1	no	—
109	C₂²⋊C₄	—	61	1	no	—
110	C₂²⋊C₄	—	61	1	no	—
111	C₄²	—	61	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	122	1	no	—
114	C₁₆	—	122	1	no	—
115	C₁₆	C₂²⋊D₆₁	1	2	yes	direct product factor
116	C₂³	—	61	1	no	—
117	C₂³	C₈×D₆₁	1	1	yes	characteristic
118	D₄	—	61	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	61	1	no	—
122	C₂×C₄	—	61	1	no	—
123	C₂×C₄	—	61	1	no	—
124	C₂×C₄	—	122	1	no	—
125	C₂×C₄	—	122	1	no	—
126	C₂×C₄	C₈×D₆₁	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	122	1	no	—
130	C₈	—	122	1	no	—
131	C₈	#11758222	1	1	yes	characteristic
132	C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	61	2	no	—
136	C₂²	—	122	1	no	—
137	C₂²	—	122	1	no	—
138	C₂²	C₁₆×D₆₁	1	2	yes	complemented
139	C₂²	C₂×C₈×D₆₁	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	61	2	no	—
142	C₄	—	122	1	no	—
143	C₄	—	122	1	no	—
144	C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
145	C₄	#14519783	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	—	122	2	no	—
148	C₂	C₈×C₂²⋊D₆₁	1	1	yes	characteristic
149	C₂	C₂×C₁₆×D₆₁	1	1	yes	characteristic
150	C₂	#16840178	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×D₆₁

C₂

yes

maximal, characteristic, complemented

C₈×C₂²⋊D₆₁

C₂

yes

maximal, characteristic

#16839580

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11758185

C₂

yes

maximal, characteristic, complemented

#16839541

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₆₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₆₁

C₂²

yes

characteristic

C₄×C₂²⋊D₆₁

C₄

yes

characteristic

C₂×#11758185

C₂²

yes

characteristic

C₂×#11758185

C₂²

yes

characteristic

C₁₆×D₆₁

—

#14519706

C₄

yes

characteristic

C₂²⋊#92418

C₄

yes

characteristic

#14519689

C₄

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₆₁

C₈

yes

characteristic

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

C₈

yes

characteristic

#11758198

C₈

yes

characteristic

#11758196

C₈

yes

characteristic

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

—

C₁₆×C₆₁

D₄

yes

complemented

#11758185

—

#11758185

—

C₂³×C₆₁

C₂×C₈

yes

characteristic

C₂²×D₆₁

C₂×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²⋊D₆₁

C₁₆

yes

direct product factor

C₂×Dic₆₁

C₂×C₈

yes

characteristic

C₂×Dic₆₁

C₂×C₈

yes

characteristic

C₂×Dic₆₁

C₂×C₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×D₄

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₁₆

yes

complemented

C₂²×C₆₁

C₂²×C₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂×C₁₆

yes

complemented

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

#131

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×C₁₆

yes

complemented

D₄×C₁₆

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₈×D₄

yes

characteristic

C₂×C₆₁

C₂²×C₁₆

yes

characteristic

C₂×C₆₁

#504

yes

characteristic

D₆₁

—

C₂²×C₁₆

—

C₂²×C₁₆

D₆₁

yes

characteristic, complemented

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₆₁

D₄×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×D₆₁

yes

characteristic

C₂×C₁₆

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₁

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

122

—

105

C₂×C₈

—

122

—

106

C₂×C₈

C₄×D₆₁

yes

characteristic

107

C₂×C₈

C₂²×D₆₁

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

122

—

114

C₁₆

—

122

—

115

C₁₆

C₂²⋊D₆₁

yes

direct product factor

116

C₂³

—

117

C₂³

C₈×D₆₁

yes

characteristic

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

122

—

125

C₂×C₄

—

122

—

126

C₂×C₄

C₈×D₆₁

yes

characteristic

127

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

128

C₈

—

129

C₈

—

122

—

130

C₈

—

122

—

131

C₈

#11758222

yes

characteristic

132

C₈

C₂×C₂²⋊D₆₁

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

122

—

137

C₂²

—

122

—

138

C₂²

C₁₆×D₆₁

yes

complemented

139

C₂²

C₂×C₈×D₆₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

122

—

143

C₄

—

122

—

144

C₄

C₄×C₂²⋊D₆₁

yes

characteristic

145

C₄

#14519783

yes

characteristic

146

C₂

—

147

C₂

—

122

—

148

C₂

C₈×C₂²⋊D₆₁

yes

characteristic

149

C₂

C₂×C₁₆×D₆₁

yes

characteristic

150

C₂

#16840178

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₆₁
commutator	C₂×C₆₁	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₂²×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×C₈
hypercenter	C₂²×C₁₆	D₆₁
hypocenter	C₆₁	D₄×C₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₁₆

C₂×D₆₁

commutator

C₂×C₆₁

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₂²×C₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×C₈

hypercenter

C₂²×C₁₆

D₆₁

hypocenter

C₆₁

D₄×C₁₆

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂×C₁₆×D₆₁	C₂	semidirect
3. #16839580	C₂	semidirect
4. C₂²⋊#11758185	C₂	semidirect
5. #16839541	C₂	semidirect
6. C₁₆×Dic₆₁	C₂	semidirect
7. C₂²×C₁₆×C₆₁	C₂	semidirect
37. C₁₆×C₆₁	D₄	semidirect
46. C₂²⋊D₆₁	C₁₆	direct (N × H)
59. C₂²×C₆₁	C₂×C₁₆	semidirect
63. C₂×D₆₁	C₂×C₁₆	semidirect
69. Dic₆₁	C₂×C₁₆	semidirect
77. C₂²×C₁₆	D₆₁	semidirect
83. C₆₁	D₄×C₁₆	semidirect
115. C₁₆	C₂²⋊D₆₁	direct (N × H)
138. C₂²	C₁₆×D₆₁	semidirect

H ≅ G/N

type

1. C₂×C₁₆×D₆₁

C₂

semidirect

3. #16839580

C₂

semidirect

4. C₂²⋊#11758185

C₂

semidirect

5. #16839541

C₂

semidirect

6. C₁₆×Dic₆₁

C₂

semidirect

7. C₂²×C₁₆×C₆₁

C₂

semidirect

37. C₁₆×C₆₁

D₄

semidirect

46. C₂²⋊D₆₁

C₁₆

direct (N × H)

59. C₂²×C₆₁

C₂×C₁₆

semidirect

63. C₂×D₆₁

C₂×C₁₆

semidirect

69. Dic₆₁

C₂×C₁₆

semidirect

77. C₂²×C₁₆

D₆₁

semidirect

83. C₆₁

D₄×C₁₆

semidirect

115. C₁₆

C₂²⋊D₆₁

direct (N × H)

138. C₂²

C₁₆×D₆₁

semidirect

Subgroups — C₁₆×C₂²⋊D₆₁ #21447637

Conjugacy classes of subgroups

Complements

Subgroups — C16×C22⋊D61 #21447637

Conjugacy classes of subgroups

Complements

Subgroups — C₁₆×C₂²⋊D₆₁ #21447637