Group #496085 — subgroups

Subgroups — C₁₆×#127 #496085

Order 512.

888 conjugacy classes of subgroups; only the 342 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 342 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×D₄×C₁₆	C₂	1	6	yes	maximal, complemented
2	C₈×#127	C₂	1	1	yes	maximal, characteristic
3	#13189	C₂	1	6	yes	maximal, complemented
4	#12898	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₆	C₂²	1	4	yes	—
7	C₂×C₈×D₄	C₂²	1	6	yes	—
8	C₄×#127	C₄	1	1	yes	characteristic
9	D₄×C₁₆	—	2	24	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	6	yes	—
11	C₂×C₂²⋊C₁₆	C₂²	1	12	yes	—
12	C₂×C₄×C₁₆	C₂²	1	3	yes	—
13	C₂×C₄×C₁₆	C₂²	1	3	yes	—
14	#1954	C₄	1	6	yes	—
15	#1842	C₄	1	1	yes	characteristic
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂³×C₈	C₂³	1	4	yes	—
18	C₂×#127	C₈	1	1	yes	characteristic
19	C₂×C₄×D₄	C₂×C₄	1	6	yes	—
20	C₂²×C₁₆	—	2	24	no	—
21	C₂²×C₁₆	C₂³	1	4	yes	—
22	C₂²×C₁₆	C₂³	1	3	yes	—
23	C₂²×C₁₆	C₂³	1	1	yes	characteristic
24	C₈×D₄	—	2	12	no	—
25	C₈×D₄	—	2	12	no	—
26	C₂×C₄⋊C₈	C₂×C₄	1	6	yes	—
27	C₂×C₂²⋊C₈	C₂×C₄	1	12	yes	—
28	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
29	C₂×C₄×C₈	C₂³	1	3	yes	—
30	#390	C₈	1	6	yes	—
31	#384	C₈	1	1	yes	characteristic
32	C₄³	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₁₆	—	2	12	no	—
34	C₂²⋊C₁₆	—	2	24	no	—
35	C₄×C₁₆	—	2	6	no	—
36	C₄×C₁₆	D₄	1	12	yes	complemented
37	C₂²×D₄	C₂×C₈	1	6	yes	—
38	C₂³×C₄	C₂²×C₄	1	4	yes	—
39	C₂²×C₈	—	2	12	no	—
40	C₂²×C₈	—	2	12	no	—
41	C₂²×C₈	C₂²×C₄	1	4	yes	—
42	C₂²×C₈	C₂²×C₄	1	3	yes	—
43	C₂²×C₈	C₂⁴	1	1	yes	characteristic
44	#127	C₁₆	1	8	yes	direct product factor
45	C₄×D₄	—	2	12	no	—
46	C₄×D₄	—	2	12	no	—
47	C₂×C₄⋊C₄	C₂×C₈	1	6	yes	—
48	C₂×C₂²⋊C₄	C₂×C₈	1	12	yes	—
49	C₂×C₄²	C₂×C₈	1	3	yes	—
50	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄²	C₂²×C₄	1	3	yes	—
52	C₂×C₁₆	—	2	6	no	—
53	C₂×C₁₆	—	2	12	no	—
54	C₂×C₁₆	—	4	16	no	—
55	C₂×C₁₆	C₂×D₄	1	6	yes	—
56	C₂×C₁₆	C₂×D₄	1	6	yes	—
57	C₄⋊C₈	—	2	12	no	—
58	C₂²⋊C₈	—	2	24	no	—
59	C₄×C₈	—	2	6	no	—
60	C₄×C₈	C₂×D₄	1	6	yes	—
61	C₄×C₈	#41	1	6	yes	—
62	C₂⁴	C₂²×C₈	1	4	yes	—
63	C₂×D₄	—	2	12	no	—
64	C₂×D₄	—	2	12	no	—
65	C₂×D₄	C₂×C₁₆	1	24	yes	complemented
66	C₂²×C₄	—	2	12	no	—
67	C₂²×C₄	—	2	12	no	—
68	C₂²×C₄	C₂²×C₈	1	4	yes	—
69	C₂²×C₄	C₂²×C₈	1	3	yes	—
70	C₂²×C₄	C₂²×C₈	1	3	yes	—
71	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₈	—	2	6	no	—
73	C₂×C₈	—	2	12	no	—
74	C₂×C₈	—	4	4	no	—
75	C₂×C₈	—	4	12	no	—
76	C₂×C₈	C₄×D₄	1	6	yes	—
77	C₂×C₈	C₂²×D₄	1	3	yes	—
78	C₂×C₈	C₂×#41	1	3	yes	—
79	C₄⋊C₄	—	2	12	no	—
80	C₂²⋊C₄	—	2	24	no	—
81	C₄²	—	2	6	no	—
82	C₄²	C₂×C₁₆	1	4	yes	complemented
83	C₄²	C₄×D₄	1	6	yes	—
84	C₄²	#131	1	6	yes	—
85	C₁₆	—	2	6	no	—
86	C₁₆	—	4	4	no	—
87	C₁₆	#127	1	4	yes	direct product factor
88	C₂³	—	2	12	no	—
89	C₂³	—	2	12	no	—
90	C₂³	C₂²×C₁₆	1	8	yes	—
91	C₂³	C₂³×C₈	1	1	yes	characteristic
92	D₄	—	2	48	no	—
93	C₂×C₄	—	2	6	no	—
94	C₂×C₄	—	2	12	no	—
95	C₂×C₄	—	4	4	no	—
96	C₂×C₄	—	4	12	no	—
97	C₂×C₄	C₈×D₄	1	6	yes	—
98	C₂×C₄	C₈×D₄	1	6	yes	—
99	C₂×C₄	C₂²×C₁₆	1	6	yes	—
100	C₂×C₄	#504	1	6	yes	—
101	C₂×C₄	C₂×C₄×D₄	1	3	yes	—
102	C₂×C₄	C₂×#131	1	3	yes	—
103	C₈	—	2	6	no	—
104	C₈	—	4	4	no	—
105	C₈	C₂×#127	1	1	yes	characteristic
106	C₈	#532	1	3	yes	—
107	C₂²	—	2	24	no	—
108	C₂²	—	4	12	no	—
109	C₂²	—	4	4	no	—
110	C₂²	C₂×C₈×D₄	1	3	yes	—
111	C₂²	C₂³×C₁₆	1	1	yes	characteristic
112	C₂²	C₂×#504	1	3	yes	—
113	C₄	—	2	6	no	—
114	C₄	—	4	4	no	—
115	C₄	D₄×C₁₆	1	12	yes	complemented
116	C₄	C₄×#127	1	1	yes	characteristic
117	C₄	#2931	1	3	yes	—
118	C₂	—	4	8	no	—
119	C₂	C₈×#127	1	1	yes	characteristic
120	C₂	C₂×D₄×C₁₆	1	3	yes	—
121	C₂	#33434	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×C₁₆

C₂

yes

maximal, complemented

C₈×#127

C₂

yes

maximal, characteristic

#13189

C₂

yes

maximal, complemented

#12898

C₂

yes

maximal, characteristic, complemented

C₄²×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂³×C₁₆

C₂²

yes

—

C₂×C₈×D₄

C₂²

yes

—

C₄×#127

C₄

yes

characteristic

D₄×C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₂²⋊C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

#1954

C₄

yes

—

#1842

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂³×C₈

C₂³

yes

—

C₂×#127

C₈

yes

characteristic

C₂×C₄×D₄

C₂×C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

C₈×D₄

—

C₈×D₄

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₂²⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

#390

C₈

yes

—

#384

C₈

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

complemented

C₂²×D₄

C₂×C₈

yes

—

C₂³×C₄

C₂²×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

—

C₂²×C₈

C₂⁴

yes

characteristic

#127

C₁₆

yes

direct product factor

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₂²⋊C₄

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂⁴

C₂²×C₈

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₁₆

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₈

yes

—

C₂²×C₄

C₂²×C₈

yes

—

C₂²×C₄

C₂²×C₈

yes

—

C₂²×C₄

C₂³×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

C₂²×D₄

yes

—

C₂×C₈

C₂×#41

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×C₁₆

yes

complemented

C₄²

C₄×D₄

yes

—

C₄²

#131

yes

—

C₁₆

—

C₁₆

—

C₁₆

#127

yes

direct product factor

C₂³

—

C₂³

—

C₂³

C₂²×C₁₆

yes

—

C₂³

C₂³×C₈

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₄

yes

—

C₂×C₄

C₈×D₄

yes

—

C₂×C₄

C₂²×C₁₆

yes

—

100

C₂×C₄

#504

yes

—

101

C₂×C₄

C₂×C₄×D₄

yes

—

102

C₂×C₄

C₂×#131

yes

—

103

C₈

—

104

C₈

—

105

C₈

C₂×#127

yes

characteristic

106

C₈

#532

yes

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₈×D₄

yes

—

111

C₂²

C₂³×C₁₆

yes

characteristic

112

C₂²

C₂×#504

yes

—

113

C₄

—

114

C₄

—

115

C₄

D₄×C₁₆

yes

complemented

116

C₄

C₄×#127

yes

characteristic

117

C₄

#2931

yes

—

118

C₂

—

119

C₂

C₈×#127

yes

characteristic

120

C₂

C₂×D₄×C₁₆

yes

—

121

C₂

#33434

yes

—

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂³×C₈
prime radical	C₂×#127	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₁₆

C₂³

commutator

C₂²

C₂³×C₁₆

Frattini, cosocle

C₂²×C₈

C₂⁴

socle

C₂³

C₂³×C₈

C₈

G ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₁₆ ◁ G

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×D₄×C₁₆	C₂	semidirect
3. #13189	C₂	semidirect
4. #12898	C₂	semidirect
5. C₄²×C₁₆	C₂	semidirect
36. C₄×C₁₆	D₄	semidirect
44. #127	C₁₆	direct (N × H)
65. C₂×D₄	C₂×C₁₆	semidirect
82. C₄²	C₂×C₁₆	semidirect
87. C₁₆	#127	direct (N × H)
115. C₄	D₄×C₁₆	semidirect

H ≅ G/N

type

1. C₂×D₄×C₁₆

C₂

semidirect

3. #13189

C₂

semidirect

4. #12898

C₂

semidirect

5. C₄²×C₁₆

C₂

semidirect

36. C₄×C₁₆

D₄

semidirect

44. #127

C₁₆

direct (N × H)

65. C₂×D₄

C₂×C₁₆

semidirect

82. C₄²

C₂×C₁₆

semidirect

87. C₁₆

#127

direct (N × H)

115. C₄

D₄×C₁₆

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄×C₁₆	2
C₈×#127	2
#13189	2
#12898	2
C₄²×C₁₆	2

subgroup

index

C₂×D₄×C₁₆

C₈×#127

#13189

#12898

C₄²×C₁₆

Subgroups — C₁₆×#127 #496085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C16×#127 #496085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#127 #496085