Group #13164964 — subgroups

Subgroups — C₂⁴×C₉×Dic₃ #13164964

Order 1728.

4834 conjugacy classes of subgroups; only the 3165 normal subgroups are shown.

normal subgroups only

This group has 4834 conjugacy classes of subgroups — too many to draw. Only its 3165 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	C₂⁵×C₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₉×Dic₃	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃×Dic₃	C₃	1	1	yes	maximal, characteristic
4	C₂⁴×C₄×C₉	—	3	1	no	maximal
5	C₂⁴×C₃×C₉	C₄	1	16	yes	complemented
6	C₂⁴×C₃×C₉	C₂²	1	15	yes	—
7	C₂²×C₉×Dic₃	C₂²	1	140	yes	direct product factor
8	C₂⁵×C₃²	C₂×C₃	1	1	yes	characteristic
9	C₂³×C₃×Dic₃	C₂×C₃	1	30	yes	—
10	C₂⁵×C₉	—	2	1	no	—
11	C₂⁵×C₉	D₃	1	1	yes	characteristic
12	C₂³×C₄×C₉	—	3	30	no	—
13	C₂³×C₃×C₉	C₂×C₄	1	120	yes	complemented
14	C₂³×C₃×C₉	C₂³	1	35	yes	—
15	C₂×C₉×Dic₃	C₂³	1	120	yes	direct product factor
16	C₂⁴×C₃×C₄	—	3	1	no	—
17	C₂⁴×Dic₃	C₉	1	1	yes	characteristic, direct product factor
18	C₂⁴×C₃²	C₃×C₄	1	16	yes	—
19	C₂⁴×C₃²	C₂²×C₃	1	15	yes	—
20	C₂²×C₃×Dic₃	C₂²×C₃	1	140	yes	—
21	C₂⁴×C₉	—	2	16	no	—
22	C₂⁴×C₉	—	2	15	no	—
23	C₂⁴×C₉	Dic₃	1	16	yes	direct product factor
24	C₂⁴×C₉	C₂×D₃	1	15	yes	—
25	C₂²×C₄×C₉	—	3	140	no	—
26	C₂²×C₃×C₉	C₂²×C₄	1	140	yes	complemented
27	C₂²×C₃×C₉	C₂⁴	1	15	yes	—
28	C₉×Dic₃	C₂⁴	1	16	yes	direct product factor
29	C₂⁵×C₃	—	2	1	no	—
30	C₂⁵×C₃	C₂×C₉	1	1	yes	characteristic
31	C₂⁵×C₃	C₃×D₃	1	1	yes	characteristic
32	C₂³×C₃×C₄	—	3	30	no	—
33	C₂³×Dic₃	C₂×C₉	1	30	yes	direct product factor
34	C₂³×C₃²	C₂×C₃×C₄	1	120	yes	—
35	C₂³×C₃²	C₂³×C₃	1	35	yes	—
36	C₂×C₃×Dic₃	C₂³×C₃	1	120	yes	—
37	C₂³×C₉	—	2	120	no	—
38	C₂³×C₉	—	2	35	no	—
39	C₂³×C₉	C₂×Dic₃	1	120	yes	direct product factor
40	C₂³×C₉	C₂²×D₃	1	35	yes	—
41	C₂×C₄×C₉	—	3	120	no	—
42	C₂⁴×C₄	—	3	1	no	Sylow
43	C₂×C₃×C₉	C₂³×C₄	1	30	yes	complemented
44	C₂×C₃×C₉	C₂⁵	1	1	yes	characteristic
45	C₂⁴×C₃	—	2	16	no	—
46	C₂⁴×C₃	—	2	15	no	—
47	C₂⁴×C₃	C₄×C₉	1	16	yes	complemented
48	C₂⁴×C₃	C₂²×C₉	1	15	yes	—
49	C₂⁴×C₃	C₃×Dic₃	1	16	yes	—
50	C₂⁴×C₃	C₂×C₃×D₃	1	15	yes	—
51	C₂²×C₃×C₄	—	3	140	no	—
52	C₂²×Dic₃	C₂²×C₉	1	140	yes	direct product factor
53	C₂²×C₃²	C₂²×C₃×C₄	1	140	yes	—
54	C₂²×C₃²	C₂⁴×C₃	1	15	yes	—
55	C₃×Dic₃	C₂⁴×C₃	1	16	yes	—
56	C₂²×C₉	—	2	15	no	—
57	C₂²×C₉	—	2	140	no	—
58	C₂²×C₉	C₂²×Dic₃	1	140	yes	direct product factor
59	C₂²×C₉	C₂³×D₃	1	15	yes	—
60	C₄×C₉	—	3	16	no	—
61	C₂⁵	D₃×C₉	1	1	yes	characteristic
62	C₂³×C₄	—	3	30	no	—
63	C₃×C₉	C₂⁴×C₄	1	1	yes	characteristic, complemented, Sylow
64	C₂³×C₃	—	2	120	no	—
65	C₂³×C₃	—	2	35	no	—
66	C₂³×C₃	C₂×C₄×C₉	1	120	yes	complemented
67	C₂³×C₃	C₂³×C₉	1	35	yes	—
68	C₂³×C₃	C₂×C₃×Dic₃	1	120	yes	—
69	C₂³×C₃	C₂²×C₃×D₃	1	35	yes	—
70	C₂×C₃×C₄	—	3	120	no	—
71	C₂×Dic₃	C₂³×C₉	1	120	yes	direct product factor
72	C₂×C₃²	C₂³×C₃×C₄	1	30	yes	—
73	C₂×C₃²	C₂⁵×C₃	1	1	yes	characteristic
74	C₂×C₉	—	2	30	no	—
75	C₂×C₉	—	2	1	no	—
76	C₂×C₉	C₂³×Dic₃	1	30	yes	direct product factor
77	C₂×C₉	C₂⁴×D₃	1	1	yes	characteristic
78	C₂⁴	C₉×Dic₃	1	16	yes	direct product factor
79	C₂⁴	C₂×D₃×C₉	1	15	yes	—
80	C₂²×C₄	—	3	140	no	—
81	C₂²×C₃	—	2	15	no	—
82	C₂²×C₃	—	2	140	no	—
83	C₂²×C₃	C₂²×C₄×C₉	1	140	yes	complemented
84	C₂²×C₃	C₂⁴×C₉	1	15	yes	—
85	C₂²×C₃	C₂²×C₃×Dic₃	1	140	yes	—
86	C₂²×C₃	C₂³×C₃×D₃	1	15	yes	—
87	C₃×C₄	—	3	16	no	—
88	Dic₃	C₂⁴×C₉	1	16	yes	direct product factor
89	C₃²	C₂⁴×C₃×C₄	1	1	yes	characteristic
90	C₉	—	2	1	no	—
91	C₉	C₂⁴×Dic₃	1	1	yes	characteristic, direct product factor
92	C₂³	C₂×C₉×Dic₃	1	120	yes	direct product factor
93	C₂³	C₂²×D₃×C₉	1	35	yes	—
94	C₂×C₄	—	3	120	no	—
95	C₂×C₃	—	2	1	no	—
96	C₂×C₃	—	2	30	no	—
97	C₂×C₃	C₂³×C₄×C₉	1	30	yes	complemented
98	C₂×C₃	C₂⁵×C₉	1	1	yes	characteristic
99	C₂×C₃	C₂³×C₃×Dic₃	1	30	yes	—
100	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
101	C₂²	C₂²×C₉×Dic₃	1	140	yes	direct product factor
102	C₂²	C₂³×D₃×C₉	1	15	yes	—
103	C₄	—	3	16	no	—
104	C₃	—	2	1	no	—
105	C₃	C₂⁴×C₄×C₉	1	1	yes	characteristic, complemented
106	C₃	C₂⁴×C₃×Dic₃	1	1	yes	characteristic
107	C₂	C₂³×C₉×Dic₃	1	30	yes	direct product factor
108	C₂	C₂⁴×D₃×C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁵×C₃×C₉

C₂

yes

maximal, characteristic

C₂³×C₉×Dic₃

C₂

yes

maximal, direct product factor

C₂⁴×C₃×Dic₃

C₃

yes

maximal, characteristic

C₂⁴×C₄×C₉

—

maximal

C₂⁴×C₃×C₉

C₄

yes

complemented

C₂⁴×C₃×C₉

C₂²

yes

—

C₂²×C₉×Dic₃

C₂²

140

yes

direct product factor

C₂⁵×C₃²

C₂×C₃

yes

characteristic

C₂³×C₃×Dic₃

C₂×C₃

yes

—

C₂⁵×C₉

—

C₂⁵×C₉

D₃

yes

characteristic

C₂³×C₄×C₉

—

C₂³×C₃×C₉

C₂×C₄

120

yes

complemented

C₂³×C₃×C₉

C₂³

yes

—

C₂×C₉×Dic₃

C₂³

120

yes

direct product factor

C₂⁴×C₃×C₄

—

C₂⁴×Dic₃

C₉

yes

characteristic, direct product factor

C₂⁴×C₃²

C₃×C₄

yes

—

C₂⁴×C₃²

C₂²×C₃

yes

—

C₂²×C₃×Dic₃

C₂²×C₃

140

yes

—

C₂⁴×C₉

—

C₂⁴×C₉

—

C₂⁴×C₉

Dic₃

yes

direct product factor

C₂⁴×C₉

C₂×D₃

yes

—

C₂²×C₄×C₉

—

140

—

C₂²×C₃×C₉

C₂²×C₄

140

yes

complemented

C₂²×C₃×C₉

C₂⁴

yes

—

C₉×Dic₃

C₂⁴

yes

direct product factor

C₂⁵×C₃

—

C₂⁵×C₃

C₂×C₉

yes

characteristic

C₂⁵×C₃

C₃×D₃

yes

characteristic

C₂³×C₃×C₄

—

C₂³×Dic₃

C₂×C₉

yes

direct product factor

C₂³×C₃²

C₂×C₃×C₄

120

yes

—

C₂³×C₃²

C₂³×C₃

yes

—

C₂×C₃×Dic₃

C₂³×C₃

120

yes

—

C₂³×C₉

—

120

—

C₂³×C₉

—

C₂³×C₉

C₂×Dic₃

120

yes

direct product factor

C₂³×C₉

C₂²×D₃

yes

—

C₂×C₄×C₉

—

120

—

C₂⁴×C₄

—

Sylow

C₂×C₃×C₉

C₂³×C₄

yes

complemented

C₂×C₃×C₉

C₂⁵

yes

characteristic

C₂⁴×C₃

—

C₂⁴×C₃

—

C₂⁴×C₃

C₄×C₉

yes

complemented

C₂⁴×C₃

C₂²×C₉

yes

—

C₂⁴×C₃

C₃×Dic₃

yes

—

C₂⁴×C₃

C₂×C₃×D₃

yes

—

C₂²×C₃×C₄

—

140

—

C₂²×Dic₃

C₂²×C₉

140

yes

direct product factor

C₂²×C₃²

C₂²×C₃×C₄

140

yes

—

C₂²×C₃²

C₂⁴×C₃

yes

—

C₃×Dic₃

C₂⁴×C₃

yes

—

C₂²×C₉

—

C₂²×C₉

—

140

—

C₂²×C₉

C₂²×Dic₃

140

yes

direct product factor

C₂²×C₉

C₂³×D₃

yes

—

C₄×C₉

—

C₂⁵

D₃×C₉

yes

characteristic

C₂³×C₄

—

C₃×C₉

C₂⁴×C₄

yes

characteristic, complemented, Sylow

C₂³×C₃

—

120

—

C₂³×C₃

—

C₂³×C₃

C₂×C₄×C₉

120

yes

complemented

C₂³×C₃

C₂³×C₉

yes

—

C₂³×C₃

C₂×C₃×Dic₃

120

yes

—

C₂³×C₃

C₂²×C₃×D₃

yes

—

C₂×C₃×C₄

—

120

—

C₂×Dic₃

C₂³×C₉

120

yes

direct product factor

C₂×C₃²

C₂³×C₃×C₄

yes

—

C₂×C₃²

C₂⁵×C₃

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂³×Dic₃

yes

direct product factor

C₂×C₉

C₂⁴×D₃

yes

characteristic

C₂⁴

C₉×Dic₃

yes

direct product factor

C₂⁴

C₂×D₃×C₉

yes

—

C₂²×C₄

—

140

—

C₂²×C₃

—

C₂²×C₃

—

140

—

C₂²×C₃

C₂²×C₄×C₉

140

yes

complemented

C₂²×C₃

C₂⁴×C₉

yes

—

C₂²×C₃

C₂²×C₃×Dic₃

140

yes

—

C₂²×C₃

C₂³×C₃×D₃

yes

—

C₃×C₄

—

Dic₃

C₂⁴×C₉

yes

direct product factor

C₃²

C₂⁴×C₃×C₄

yes

characteristic

C₉

—

C₉

C₂⁴×Dic₃

yes

characteristic, direct product factor

C₂³

C₂×C₉×Dic₃

120

yes

direct product factor

C₂³

C₂²×D₃×C₉

yes

—

C₂×C₄

—

120

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂³×C₄×C₉

yes

complemented

C₂×C₃

C₂⁵×C₉

yes

characteristic

C₂×C₃

C₂³×C₃×Dic₃

yes

—

100

C₂×C₃

C₂⁴×C₃×D₃

yes

characteristic

101

C₂²

C₂²×C₉×Dic₃

140

yes

direct product factor

102

C₂²

C₂³×D₃×C₉

yes

—

103

C₄

—

104

C₃

—

105

C₃

C₂⁴×C₄×C₉

yes

characteristic, complemented

106

C₃

C₂⁴×C₃×Dic₃

yes

characteristic

107

C₂

C₂³×C₉×Dic₃

yes

direct product factor

108

C₂

C₂⁴×D₃×C₉

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂⁵×C₉	D₃
commutator, hypocenter	C₃	C₂⁴×C₄×C₉
Frattini	C₂×C₃	C₂⁴×C₃×D₃
Fitting	C₂⁵×C₃×C₉	C₂
socle, prime radical	C₂⁵×C₃²	C₂×C₃
cosocle	C₂×C₃²	C₂⁵×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁵×C₉

subgroup

G/H

center, hypercenter

C₂⁵×C₉

D₃

commutator, hypocenter

C₃

C₂⁴×C₄×C₉

Frattini

C₂×C₃

C₂⁴×C₃×D₃

Fitting

C₂⁵×C₃×C₉

C₂

socle, prime radical

C₂⁵×C₃²

C₂×C₃

cosocle

C₂×C₃²

C₂⁵×C₃

LCS

G ▷ C₃

UCS

C₁ ◁ C₂⁵×C₉

N	H ≅ G/N	type
2. C₂³×C₉×Dic₃	C₂	direct (N × H)
5. C₂⁴×C₃×C₉	C₄	semidirect
7. C₂²×C₉×Dic₃	C₂²	direct (N × H)
13. C₂³×C₃×C₉	C₂×C₄	semidirect
15. C₂×C₉×Dic₃	C₂³	direct (N × H)
17. C₂⁴×Dic₃	C₉	direct (N × H)
23. C₂⁴×C₉	Dic₃	direct (N × H)
26. C₂²×C₃×C₉	C₂²×C₄	semidirect
28. C₉×Dic₃	C₂⁴	direct (N × H)
33. C₂³×Dic₃	C₂×C₉	direct (N × H)
39. C₂³×C₉	C₂×Dic₃	direct (N × H)
43. C₂×C₃×C₉	C₂³×C₄	semidirect
47. C₂⁴×C₃	C₄×C₉	semidirect
52. C₂²×Dic₃	C₂²×C₉	direct (N × H)
58. C₂²×C₉	C₂²×Dic₃	direct (N × H)
63. C₃×C₉	C₂⁴×C₄	semidirect
66. C₂³×C₃	C₂×C₄×C₉	semidirect
71. C₂×Dic₃	C₂³×C₉	direct (N × H)
76. C₂×C₉	C₂³×Dic₃	direct (N × H)
78. C₂⁴	C₉×Dic₃	direct (N × H)
83. C₂²×C₃	C₂²×C₄×C₉	semidirect
88. Dic₃	C₂⁴×C₉	direct (N × H)
91. C₉	C₂⁴×Dic₃	direct (N × H)
92. C₂³	C₂×C₉×Dic₃	direct (N × H)
97. C₂×C₃	C₂³×C₄×C₉	semidirect
101. C₂²	C₂²×C₉×Dic₃	direct (N × H)
105. C₃	C₂⁴×C₄×C₉	semidirect
107. C₂	C₂³×C₉×Dic₃	direct (N × H)

H ≅ G/N

type

2. C₂³×C₉×Dic₃

C₂

direct (N × H)

5. C₂⁴×C₃×C₉

C₄

semidirect

7. C₂²×C₉×Dic₃

C₂²

direct (N × H)

13. C₂³×C₃×C₉

C₂×C₄

semidirect

15. C₂×C₉×Dic₃

C₂³

direct (N × H)

17. C₂⁴×Dic₃

C₉

direct (N × H)

23. C₂⁴×C₉

Dic₃

direct (N × H)

26. C₂²×C₃×C₉

C₂²×C₄

semidirect

28. C₉×Dic₃

C₂⁴

direct (N × H)

33. C₂³×Dic₃

C₂×C₉

direct (N × H)

39. C₂³×C₉

C₂×Dic₃

direct (N × H)

43. C₂×C₃×C₉

C₂³×C₄

semidirect

47. C₂⁴×C₃

C₄×C₉

semidirect

52. C₂²×Dic₃

C₂²×C₉

direct (N × H)

58. C₂²×C₉

C₂²×Dic₃

direct (N × H)

63. C₃×C₉

C₂⁴×C₄

semidirect

66. C₂³×C₃

C₂×C₄×C₉

semidirect

71. C₂×Dic₃

C₂³×C₉

direct (N × H)

76. C₂×C₉

C₂³×Dic₃

direct (N × H)

78. C₂⁴

C₉×Dic₃

direct (N × H)

83. C₂²×C₃

C₂²×C₄×C₉

semidirect

88. Dic₃

C₂⁴×C₉

direct (N × H)

91. C₉

C₂⁴×Dic₃

direct (N × H)

92. C₂³

C₂×C₉×Dic₃

direct (N × H)

97. C₂×C₃

C₂³×C₄×C₉

semidirect

101. C₂²

C₂²×C₉×Dic₃

direct (N × H)

105. C₃

C₂⁴×C₄×C₉

semidirect

107. C₂

C₂³×C₉×Dic₃

direct (N × H)

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

subgroup	index
C₂⁵×C₃×C₉	2
C₂³×C₉×Dic₃	2
C₂⁴×C₃×Dic₃	3
C₂⁴×C₄×C₉	3

subgroup

index

C₂⁵×C₃×C₉

C₂³×C₉×Dic₃

C₂⁴×C₃×Dic₃

C₂⁴×C₄×C₉

Subgroups — C₂⁴×C₉×Dic₃ #13164964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C24×C9×Dic3 #13164964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₉×Dic₃ #13164964