Group #13141262 — subgroups

Subgroups — C₃×C₁₆×C₂²⋊C₉ #13141262

Order 1728.

178 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₃×C₈×C₂²⋊C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₁₆	C₃	1	1	yes	maximal, characteristic
3	C₁₆×C₂²⋊C₉	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₄×C₂²⋊C₉	C₄	1	1	yes	characteristic
5	C₃×C₉×C₁₆	—	4	1	no	maximal
6	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃²×C₁₆	—	3	1	no	—
8	C₂×C₃²×C₁₆	—	3	1	no	—
9	C₈×C₂²⋊C₉	C₂×C₃	1	3	yes	—
10	C₂×C₃×C₂²⋊C₉	C₈	1	1	yes	characteristic
11	C₃×C₈×C₉	—	4	1	no	—
12	C₂²×C₃×C₁₆	C₉	1	3	yes	complemented
13	C₂²×C₃×C₁₆	C₃²	1	1	yes	characteristic
14	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
15	C₂×C₃²×C₈	—	3	1	no	—
16	C₂×C₃²×C₈	—	3	1	no	—
17	C₄×C₂²⋊C₉	C₃×C₄	1	3	yes	—
18	C₃²×C₁₆	—	3	1	no	—
19	C₃²×C₁₆	A₄	1	1	yes	characteristic
20	C₉×C₁₆	—	4	3	no	—
21	C₃×C₂²⋊C₉	C₁₆	1	1	yes	characteristic, direct product factor
22	C₃×C₄×C₉	—	4	1	no	—
23	C₂²×C₃×C₈	C₂×C₉	1	3	yes	—
24	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
25	C₂×C₃×C₁₆	—	3	3	no	—
26	C₂×C₃×C₁₆	—	3	1	no	—
27	C₂×C₃×C₁₆	—	3	3	no	—
28	C₂×C₃×C₁₆	—	3	1	no	—
29	C₂³×C₃²	C₃×C₈	1	1	yes	characteristic
30	C₂×C₃²×C₄	—	3	1	no	—
31	C₂×C₃²×C₄	—	3	1	no	—
32	C₂×C₂²⋊C₉	C₃×C₈	1	3	yes	—
33	C₃²×C₈	—	3	1	no	—
34	C₃²×C₈	C₂×A₄	1	1	yes	characteristic
35	C₈×C₉	—	4	3	no	—
36	C₂²×C₁₆	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃×C₉	—	4	1	no	—
38	C₂²×C₃×C₄	C₄×C₉	1	3	yes	—
39	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
40	C₂×C₃×C₈	—	3	3	no	—
41	C₂×C₃×C₈	—	3	1	no	—
42	C₂×C₃×C₈	—	3	3	no	—
43	C₂×C₃×C₈	—	3	1	no	—
44	C₃×C₁₆	—	3	3	no	—
45	C₃×C₁₆	—	3	1	no	—
46	C₃×C₁₆	C₂²⋊C₉	1	3	yes	direct product factor
47	C₃×C₁₆	C₃×A₄	1	1	yes	characteristic
48	C₂²×C₃²	—	3	1	no	—
49	C₂²×C₃²	—	3	1	no	—
50	C₂²×C₃²	C₃×C₁₆	1	1	yes	characteristic
51	C₃²×C₄	—	3	1	no	—
52	C₃²×C₄	C₄×A₄	1	1	yes	characteristic
53	C₂²⋊C₉	C₃×C₁₆	1	3	yes	direct product factor
54	C₄×C₉	—	4	3	no	—
55	C₂²×C₈	C₂×C₃×C₉	1	1	yes	characteristic
56	C₂×C₁₆	—	3	1	no	—
57	C₂×C₁₆	—	3	1	no	—
58	C₃×C₉	—	4	1	no	Sylow
59	C₂³×C₃	C₈×C₉	1	3	yes	—
60	C₂³×C₃	C₃²×C₈	1	1	yes	characteristic
61	C₂×C₃×C₄	—	3	3	no	—
62	C₂×C₃×C₄	—	3	1	no	—
63	C₂×C₃×C₄	—	3	3	no	—
64	C₂×C₃×C₄	—	3	1	no	—
65	C₃×C₈	—	3	3	no	—
66	C₃×C₈	—	3	1	no	—
67	C₃×C₈	C₂×C₂²⋊C₉	1	3	yes	—
68	C₃×C₈	C₂×C₃×A₄	1	1	yes	characteristic
69	C₂×C₃²	—	3	1	no	—
70	C₂×C₃²	—	3	1	no	—
71	C₂×C₃²	C₈×A₄	1	1	yes	characteristic
72	C₂×C₉	—	4	3	no	—
73	C₂²×C₄	C₃×C₄×C₉	1	1	yes	characteristic
74	C₂×C₈	—	3	1	no	—
75	C₂×C₈	—	3	1	no	—
76	C₁₆	—	3	1	no	—
77	C₁₆	C₃×C₂²⋊C₉	1	1	yes	characteristic, direct product factor
78	C₂²×C₃	—	3	3	no	—
79	C₂²×C₃	—	3	1	no	—
80	C₂²×C₃	—	3	3	no	—
81	C₂²×C₃	—	3	1	no	—
82	C₂²×C₃	C₉×C₁₆	1	3	yes	complemented
83	C₂²×C₃	C₃²×C₁₆	1	1	yes	characteristic
84	C₃×C₄	—	3	3	no	—
85	C₃×C₄	—	3	1	no	—
86	C₃×C₄	C₄×C₂²⋊C₉	1	3	yes	—
87	C₃×C₄	C₃×C₄×A₄	1	1	yes	characteristic
88	C₃²	A₄×C₁₆	1	1	yes	characteristic
89	C₉	—	4	3	no	—
90	C₂³	C₃×C₈×C₉	1	1	yes	characteristic
91	C₂×C₄	—	3	1	no	—
92	C₂×C₄	—	3	1	no	—
93	C₈	—	3	1	no	—
94	C₈	C₂×C₃×C₂²⋊C₉	1	1	yes	characteristic
95	C₂×C₃	—	3	3	no	—
96	C₂×C₃	—	3	3	no	—
97	C₂×C₃	—	3	1	no	—
98	C₂×C₃	—	3	1	no	—
99	C₂×C₃	C₈×C₂²⋊C₉	1	3	yes	—
100	C₂×C₃	C₃×C₈×A₄	1	1	yes	characteristic
101	C₂²	—	3	1	no	—
102	C₂²	—	3	1	no	—
103	C₂²	C₃×C₉×C₁₆	1	1	yes	characteristic, complemented
104	C₄	—	3	1	no	—
105	C₄	C₃×C₄×C₂²⋊C₉	1	1	yes	characteristic
106	C₃	C₁₆×C₂²⋊C₉	1	3	yes	direct product factor
107	C₃	C₃×A₄×C₁₆	1	1	yes	characteristic
108	C₂	—	3	1	no	—
109	C₂	—	3	1	no	—
110	C₂	C₃×C₈×C₂²⋊C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₈×C₂²⋊C₉

C₂

yes

maximal, characteristic

C₂²×C₃²×C₁₆

C₃

yes

maximal, characteristic

C₁₆×C₂²⋊C₉

C₃

yes

maximal, direct product factor

C₃×C₄×C₂²⋊C₉

C₄

yes

characteristic

C₃×C₉×C₁₆

—

maximal

C₂²×C₃²×C₈

C₂×C₃

yes

characteristic

C₂×C₃²×C₁₆

—

C₂×C₃²×C₁₆

—

C₈×C₂²⋊C₉

C₂×C₃

yes

—

C₂×C₃×C₂²⋊C₉

C₈

yes

characteristic

C₃×C₈×C₉

—

C₂²×C₃×C₁₆

C₉

yes

complemented

C₂²×C₃×C₁₆

C₃²

yes

characteristic

C₂²×C₃²×C₄

C₃×C₄

yes

characteristic

C₂×C₃²×C₈

—

C₂×C₃²×C₈

—

C₄×C₂²⋊C₉

C₃×C₄

yes

—

C₃²×C₁₆

—

C₃²×C₁₆

A₄

yes

characteristic

C₉×C₁₆

—

C₃×C₂²⋊C₉

C₁₆

yes

characteristic, direct product factor

C₃×C₄×C₉

—

C₂²×C₃×C₈

C₂×C₉

yes

—

C₂²×C₃×C₈

C₂×C₃²

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₂³×C₃²

C₃×C₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₂²⋊C₉

C₃×C₈

yes

—

C₃²×C₈

—

C₃²×C₈

C₂×A₄

yes

characteristic

C₈×C₉

—

C₂²×C₁₆

C₃×C₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₉

—

C₂²×C₃×C₄

C₄×C₉

yes

—

C₂²×C₃×C₄

C₃²×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

C₂²⋊C₉

yes

direct product factor

C₃×C₁₆

C₃×A₄

yes

characteristic

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×C₁₆

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄×A₄

yes

characteristic

C₂²⋊C₉

C₃×C₁₆

yes

direct product factor

C₄×C₉

—

C₂²×C₈

C₂×C₃×C₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₃×C₉

—

Sylow

C₂³×C₃

C₈×C₉

yes

—

C₂³×C₃

C₃²×C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×C₂²⋊C₉

yes

—

C₃×C₈

C₂×C₃×A₄

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₈×A₄

yes

characteristic

C₂×C₉

—

C₂²×C₄

C₃×C₄×C₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₁₆

—

C₁₆

C₃×C₂²⋊C₉

yes

characteristic, direct product factor

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₉×C₁₆

yes

complemented

C₂²×C₃

C₃²×C₁₆

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×C₂²⋊C₉

yes

—

C₃×C₄

C₃×C₄×A₄

yes

characteristic

C₃²

A₄×C₁₆

yes

characteristic

C₉

—

C₂³

C₃×C₈×C₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

C₂×C₃×C₂²⋊C₉

yes

characteristic

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₈×C₂²⋊C₉

yes

—

100

C₂×C₃

C₃×C₈×A₄

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

C₃×C₉×C₁₆

yes

characteristic, complemented

104

C₄

—

105

C₄

C₃×C₄×C₂²⋊C₉

yes

characteristic

106

C₃

C₁₆×C₂²⋊C₉

yes

direct product factor

107

C₃

C₃×A₄×C₁₆

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

C₃×C₈×C₂²⋊C₉

yes

characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₁₆	A₄
commutator, hypocenter	C₂²	C₃×C₉×C₁₆
Frattini	C₃×C₈	C₂×C₃×A₄
Fitting	C₂²×C₃²×C₁₆	C₃
socle, prime radical	C₂³×C₃²	C₃×C₈
cosocle	C₂²×C₃×C₈	C₂×C₃²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃²×C₁₆

subgroup

G/H

center, hypercenter

C₃²×C₁₆

A₄

commutator, hypocenter

C₂²

C₃×C₉×C₁₆

Frattini

C₃×C₈

C₂×C₃×A₄

Fitting

C₂²×C₃²×C₁₆

C₃

socle, prime radical

C₂³×C₃²

C₃×C₈

cosocle

C₂²×C₃×C₈

C₂×C₃²

LCS

G ▷ C₂²

UCS

C₁ ◁ C₃²×C₁₆

N	H ≅ G/N	type
3. C₁₆×C₂²⋊C₉	C₃	direct (N × H)
12. C₂²×C₃×C₁₆	C₉	semidirect
21. C₃×C₂²⋊C₉	C₁₆	direct (N × H)
36. C₂²×C₁₆	C₃×C₉	semidirect
46. C₃×C₁₆	C₂²⋊C₉	direct (N × H)
53. C₂²⋊C₉	C₃×C₁₆	direct (N × H)
77. C₁₆	C₃×C₂²⋊C₉	direct (N × H)
82. C₂²×C₃	C₉×C₁₆	semidirect
103. C₂²	C₃×C₉×C₁₆	semidirect
106. C₃	C₁₆×C₂²⋊C₉	direct (N × H)

H ≅ G/N

type

3. C₁₆×C₂²⋊C₉

C₃

direct (N × H)

12. C₂²×C₃×C₁₆

C₉

semidirect

21. C₃×C₂²⋊C₉

C₁₆

direct (N × H)

36. C₂²×C₁₆

C₃×C₉

semidirect

46. C₃×C₁₆

C₂²⋊C₉

direct (N × H)

53. C₂²⋊C₉

C₃×C₁₆

direct (N × H)

77. C₁₆

C₃×C₂²⋊C₉

direct (N × H)

82. C₂²×C₃

C₉×C₁₆

semidirect

103. C₂²

C₃×C₉×C₁₆

semidirect

106. C₃

C₁₆×C₂²⋊C₉

direct (N × H)

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

subgroup	index
C₃×C₈×C₂²⋊C₉	2
C₂²×C₃²×C₁₆	3
C₁₆×C₂²⋊C₉	3
C₃×C₉×C₁₆	4

subgroup

index

C₃×C₈×C₂²⋊C₉

C₂²×C₃²×C₁₆

C₁₆×C₂²⋊C₉

C₃×C₉×C₁₆

Subgroups — C₃×C₁₆×C₂²⋊C₉ #13141262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C16×C22⋊C9 #13141262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₆×C₂²⋊C₉ #13141262