Group #13049331 — subgroups

Subgroups — C₂²×C₃²×C₂²⋊C₉ #13049331

Order 1296.

1032 conjugacy classes of subgroups; only the 390 normal subgroups are shown.

normal subgroups only

This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 390 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×C₃²×C₂²⋊C₉	C₂	1	3	yes	maximal, direct product factor
2	C₂⁴×C₃³	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₂²⋊C₉	C₃	1	12	yes	maximal, direct product factor
4	C₂²×C₃²×C₉	—	4	1	no	maximal
5	C₃²×C₂²⋊C₉	C₂²	1	1	yes	characteristic, direct product factor
6	C₂³×C₃³	—	3	1	no	—
7	C₂³×C₃³	—	3	3	no	—
8	C₂³×C₃³	C₂×C₃	1	3	yes	—
9	C₂×C₃×C₂²⋊C₉	C₂×C₃	1	36	yes	direct product factor
10	C₂×C₃²×C₉	—	4	3	no	—
11	C₂⁴×C₃²	C₉	1	9	yes	complemented
12	C₂⁴×C₃²	C₃²	1	4	yes	—
13	C₂²×C₂²⋊C₉	C₃²	1	9	yes	direct product factor
14	C₂²×C₃³	—	3	2	no	—
15	C₂²×C₃³	—	3	3	no	—
16	C₂²×C₃³	—	3	3	no	—
17	C₂²×C₃³	—	3	3	no	—
18	C₂²×C₃³	A₄	1	1	yes	characteristic
19	C₂²×C₃³	C₂²×C₃	1	1	yes	characteristic
20	C₂²×C₃×C₉	—	4	12	no	—
21	C₃×C₂²⋊C₉	C₂²×C₃	1	12	yes	direct product factor
22	C₃²×C₉	—	4	1	no	Sylow
23	C₂³×C₃²	—	3	27	no	—
24	C₂³×C₃²	—	3	12	no	—
25	C₂³×C₃²	—	3	9	no	—
26	C₂³×C₃²	—	3	4	no	—
27	C₂³×C₃²	C₂×C₉	1	27	yes	complemented
28	C₂³×C₃²	C₂×C₃²	1	12	yes	—
29	C₂×C₂²⋊C₉	C₂×C₃²	1	27	yes	direct product factor
30	C₂×C₃³	—	3	3	no	—
31	C₂×C₃³	—	3	1	no	—
32	C₂×C₃³	C₂×A₄	1	3	yes	—
33	C₂×C₃×C₉	—	4	36	no	—
34	C₂⁴×C₃	C₃×C₉	1	12	yes	complemented
35	C₂⁴×C₃	C₃³	1	1	yes	characteristic
36	C₂²×C₃²	—	3	27	no	—
37	C₂²×C₃²	—	3	12	no	—
38	C₂²×C₃²	—	3	8	no	—
39	C₂²×C₃²	—	3	18	no	—
40	C₂²×C₃²	—	3	27	no	—
41	C₂²×C₃²	—	3	27	no	—
42	C₂²×C₃²	—	3	12	no	—
43	C₂²×C₃²	—	3	12	no	—
44	C₂²×C₃²	C₂²⋊C₉	1	9	yes	direct product factor
45	C₂²×C₃²	C₂²×C₉	1	9	yes	complemented
46	C₂²×C₃²	C₃×A₄	1	4	yes	—
47	C₂²×C₃²	C₂²×C₃²	1	4	yes	—
48	C₂²×C₉	—	4	9	no	—
49	C₂²⋊C₉	C₂²×C₃²	1	9	yes	direct product factor
50	C₃³	C₂²×A₄	1	1	yes	characteristic
51	C₃×C₉	—	4	12	no	—
52	C₂³×C₃	—	3	36	no	—
53	C₂³×C₃	—	3	3	no	—
54	C₂³×C₃	—	3	12	no	—
55	C₂³×C₃	—	3	1	no	—
56	C₂³×C₃	C₂×C₃×C₉	1	36	yes	complemented
57	C₂³×C₃	C₂×C₃³	1	3	yes	—
58	C₂×C₃²	—	3	12	no	—
59	C₂×C₃²	—	3	4	no	—
60	C₂×C₃²	—	3	27	no	—
61	C₂×C₃²	—	3	9	no	—
62	C₂×C₃²	C₂×C₂²⋊C₉	1	27	yes	direct product factor
63	C₂×C₃²	C₂×C₃×A₄	1	12	yes	—
64	C₂×C₉	—	4	27	no	—
65	C₂⁴	C₃²×C₉	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₃	—	3	36	no	—
67	C₂²×C₃	—	3	3	no	—
68	C₂²×C₃	—	3	2	no	—
69	C₂²×C₃	—	3	24	no	—
70	C₂²×C₃	—	3	36	no	—
71	C₂²×C₃	—	3	36	no	—
72	C₂²×C₃	—	3	3	no	—
73	C₂²×C₃	—	3	3	no	—
74	C₂²×C₃	C₃×C₂²⋊C₉	1	12	yes	direct product factor
75	C₂²×C₃	C₂²×C₃×C₉	1	12	yes	complemented
76	C₂²×C₃	C₃²×A₄	1	1	yes	characteristic
77	C₂²×C₃	C₂²×C₃³	1	1	yes	characteristic
78	C₃²	C₂²×C₂²⋊C₉	1	9	yes	direct product factor
79	C₃²	C₂²×C₃×A₄	1	4	yes	—
80	C₉	—	4	9	no	—
81	C₂³	—	3	3	no	—
82	C₂³	—	3	1	no	—
83	C₂³	C₂×C₃²×C₉	1	3	yes	complemented
84	C₂×C₃	—	3	1	no	—
85	C₂×C₃	—	3	3	no	—
86	C₂×C₃	—	3	12	no	—
87	C₂×C₃	—	3	36	no	—
88	C₂×C₃	C₂×C₃×C₂²⋊C₉	1	36	yes	direct product factor
89	C₂×C₃	C₂×C₃²×A₄	1	3	yes	—
90	C₂²	—	3	3	no	—
91	C₂²	—	3	2	no	—
92	C₂²	—	3	3	no	—
93	C₂²	—	3	3	no	—
94	C₂²	C₃²×C₂²⋊C₉	1	1	yes	characteristic, direct product factor
95	C₂²	C₂²×C₃²×C₉	1	1	yes	characteristic, complemented
96	C₃	C₂²×C₃×C₂²⋊C₉	1	12	yes	direct product factor
97	C₃	C₂²×C₃²×A₄	1	1	yes	characteristic
98	C₂	—	3	1	no	—
99	C₂	—	3	3	no	—
100	C₂	C₂×C₃²×C₂²⋊C₉	1	3	yes	direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃²×C₂²⋊C₉

C₂

yes

maximal, direct product factor

C₂⁴×C₃³

C₃

yes

maximal, characteristic

C₂²×C₃×C₂²⋊C₉

C₃

yes

maximal, direct product factor

C₂²×C₃²×C₉

—

maximal

C₃²×C₂²⋊C₉

C₂²

yes

characteristic, direct product factor

C₂³×C₃³

—

C₂³×C₃³

—

C₂³×C₃³

C₂×C₃

yes

—

C₂×C₃×C₂²⋊C₉

C₂×C₃

yes

direct product factor

C₂×C₃²×C₉

—

C₂⁴×C₃²

C₉

yes

complemented

C₂⁴×C₃²

C₃²

yes

—

C₂²×C₂²⋊C₉

C₃²

yes

direct product factor

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

A₄

yes

characteristic

C₂²×C₃³

C₂²×C₃

yes

characteristic

C₂²×C₃×C₉

—

C₃×C₂²⋊C₉

C₂²×C₃

yes

direct product factor

C₃²×C₉

—

Sylow

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

—

C₂³×C₃²

C₂×C₉

yes

complemented

C₂³×C₃²

C₂×C₃²

yes

—

C₂×C₂²⋊C₉

C₂×C₃²

yes

direct product factor

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₂×A₄

yes

—

C₂×C₃×C₉

—

C₂⁴×C₃

C₃×C₉

yes

complemented

C₂⁴×C₃

C₃³

yes

characteristic

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂²⋊C₉

yes

direct product factor

C₂²×C₃²

C₂²×C₉

yes

complemented

C₂²×C₃²

C₃×A₄

yes

—

C₂²×C₃²

yes

—

C₂²×C₉

—

C₂²⋊C₉

C₂²×C₃²

yes

direct product factor

C₃³

C₂²×A₄

yes

characteristic

C₃×C₉

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×C₃×C₉

yes

complemented

C₂³×C₃

C₂×C₃³

yes

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×C₂²⋊C₉

yes

direct product factor

C₂×C₃²

C₂×C₃×A₄

yes

—

C₂×C₉

—

C₂⁴

C₃²×C₉

yes

characteristic, complemented, Sylow

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₃×C₂²⋊C₉

yes

direct product factor

C₂²×C₃

C₂²×C₃×C₉

yes

complemented

C₂²×C₃

C₃²×A₄

yes

characteristic

C₂²×C₃

C₂²×C₃³

yes

characteristic

C₃²

C₂²×C₂²⋊C₉

yes

direct product factor

C₃²

C₂²×C₃×A₄

yes

—

C₉

—

C₂³

—

C₂³

—

C₂³

C₂×C₃²×C₉

yes

complemented

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂×C₃×C₂²⋊C₉

yes

direct product factor

C₂×C₃

C₂×C₃²×A₄

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₃²×C₂²⋊C₉

yes

characteristic, direct product factor

C₂²

C₂²×C₃²×C₉

yes

characteristic, complemented

C₃

C₂²×C₃×C₂²⋊C₉

yes

direct product factor

C₃

C₂²×C₃²×A₄

yes

characteristic

C₂

—

C₂

—

100

C₂

C₂×C₃²×C₂²⋊C₉

yes

direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃³	A₄
commutator, hypocenter	C₂²	C₂²×C₃²×C₉
Frattini	C₃	C₂²×C₃²×A₄
Fitting, socle, prime radical	C₂⁴×C₃³	C₃
cosocle	C₂²×C₃	C₂²×C₃³
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂²×C₃³

subgroup

G/H

center, hypercenter

C₂²×C₃³

A₄

commutator, hypocenter

C₂²

C₂²×C₃²×C₉

Frattini

C₃

C₂²×C₃²×A₄

Fitting, socle, prime radical

C₂⁴×C₃³

C₃

cosocle

C₂²×C₃

C₂²×C₃³

LCS

G ▷ C₂²

UCS

C₁ ◁ C₂²×C₃³

N	H ≅ G/N	type
1. C₂×C₃²×C₂²⋊C₉	C₂	direct (N × H)
3. C₂²×C₃×C₂²⋊C₉	C₃	direct (N × H)
5. C₃²×C₂²⋊C₉	C₂²	direct (N × H)
9. C₂×C₃×C₂²⋊C₉	C₂×C₃	direct (N × H)
11. C₂⁴×C₃²	C₉	semidirect
13. C₂²×C₂²⋊C₉	C₃²	direct (N × H)
21. C₃×C₂²⋊C₉	C₂²×C₃	direct (N × H)
27. C₂³×C₃²	C₂×C₉	semidirect
29. C₂×C₂²⋊C₉	C₂×C₃²	direct (N × H)
34. C₂⁴×C₃	C₃×C₉	semidirect
44. C₂²×C₃²	C₂²⋊C₉	direct (N × H)
45. C₂²×C₃²	C₂²×C₉	semidirect
49. C₂²⋊C₉	C₂²×C₃²	direct (N × H)
56. C₂³×C₃	C₂×C₃×C₉	semidirect
62. C₂×C₃²	C₂×C₂²⋊C₉	direct (N × H)
65. C₂⁴	C₃²×C₉	semidirect
74. C₂²×C₃	C₃×C₂²⋊C₉	direct (N × H)
75. C₂²×C₃	C₂²×C₃×C₉	semidirect
78. C₃²	C₂²×C₂²⋊C₉	direct (N × H)
83. C₂³	C₂×C₃²×C₉	semidirect
88. C₂×C₃	C₂×C₃×C₂²⋊C₉	direct (N × H)
94. C₂²	C₃²×C₂²⋊C₉	direct (N × H)
95. C₂²	C₂²×C₃²×C₉	semidirect
96. C₃	C₂²×C₃×C₂²⋊C₉	direct (N × H)
100. C₂	C₂×C₃²×C₂²⋊C₉	direct (N × H)

H ≅ G/N

type

1. C₂×C₃²×C₂²⋊C₉

C₂

direct (N × H)

3. C₂²×C₃×C₂²⋊C₉

C₃

direct (N × H)

5. C₃²×C₂²⋊C₉

C₂²

direct (N × H)

9. C₂×C₃×C₂²⋊C₉

C₂×C₃

direct (N × H)

11. C₂⁴×C₃²

C₉

semidirect

13. C₂²×C₂²⋊C₉

C₃²

direct (N × H)

21. C₃×C₂²⋊C₉

C₂²×C₃

direct (N × H)

27. C₂³×C₃²

C₂×C₉

semidirect

29. C₂×C₂²⋊C₉

C₂×C₃²

direct (N × H)

34. C₂⁴×C₃

C₃×C₉

semidirect

44. C₂²×C₃²

C₂²⋊C₉

direct (N × H)

45. C₂²×C₃²

C₂²×C₉

semidirect

49. C₂²⋊C₉

C₂²×C₃²

direct (N × H)

56. C₂³×C₃

C₂×C₃×C₉

semidirect

62. C₂×C₃²

C₂×C₂²⋊C₉

direct (N × H)

65. C₂⁴

C₃²×C₉

semidirect

74. C₂²×C₃

C₃×C₂²⋊C₉

direct (N × H)

75. C₂²×C₃

C₂²×C₃×C₉

semidirect

78. C₃²

C₂²×C₂²⋊C₉

direct (N × H)

83. C₂³

C₂×C₃²×C₉

semidirect

88. C₂×C₃

C₂×C₃×C₂²⋊C₉

direct (N × H)

94. C₂²

C₃²×C₂²⋊C₉

direct (N × H)

95. C₂²

C₂²×C₃²×C₉

semidirect

96. C₃

C₂²×C₃×C₂²⋊C₉

direct (N × H)

100. C₂

C₂×C₃²×C₂²⋊C₉

direct (N × H)

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

subgroup	index
C₂×C₃²×C₂²⋊C₉	2
C₂⁴×C₃³	3
C₂²×C₃×C₂²⋊C₉	3
C₂²×C₃²×C₉	4

subgroup

index

C₂×C₃²×C₂²⋊C₉

C₂⁴×C₃³

C₂²×C₃×C₂²⋊C₉

C₂²×C₃²×C₉

Subgroups — C₂²×C₃²×C₂²⋊C₉ #13049331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C32×C22⋊C9 #13049331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×C₂²⋊C₉ #13049331