Group #14518402 — subgroups

Subgroups — C₂×C₃×C₉×C₂²⋊C₉ #14518402

Order 1944.

704 conjugacy classes of subgroups; only the 352 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 352 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×C₉×C₂²⋊C₉	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂³×C₃²×C₉	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₂²⋊C₉	C₃	1	3	yes	maximal
4	C₂×C₉×C₂²⋊C₉	C₃	1	9	yes	maximal, direct product factor
5	C₂×C₃×C₉²	—	4	1	no	maximal
6	C₂²×C₃²×C₉	—	3	1	no	—
7	C₂²×C₃²×C₉	—	3	1	no	—
8	C₂²×C₃²×C₉	C₂×C₃	1	1	yes	characteristic
9	C₃²×C₂²⋊C₉	C₂×C₃	1	3	yes	—
10	C₉×C₂²⋊C₉	C₂×C₃	1	9	yes	direct product factor
11	C₃×C₉²	—	4	1	no	Sylow
12	C₂³×C₃³	C₃²	1	1	yes	characteristic
13	C₂³×C₃×C₉	C₉	1	9	yes	complemented
14	C₂³×C₃×C₉	C₃²	1	3	yes	—
15	C₂×C₃×C₂²⋊C₉	C₉	1	27	yes	direct product factor
16	C₂×C₃×C₂²⋊C₉	C₃²	1	9	yes	—
17	C₂×C₃²×C₉	—	3	1	no	—
18	C₂×C₃²×C₉	—	3	1	no	—
19	C₂×C₃²×C₉	—	4	3	no	—
20	C₂×C₃²×C₉	A₄	1	1	yes	characteristic
21	C₂×C₉²	—	4	9	no	—
22	C₂²×C₃³	—	3	1	no	—
23	C₂²×C₃³	—	3	1	no	—
24	C₂²×C₃³	C₂×C₃²	1	1	yes	characteristic
25	C₂²×C₃×C₉	—	3	9	no	—
26	C₂²×C₃×C₉	—	3	3	no	—
27	C₂²×C₃×C₉	—	3	9	no	—
28	C₂²×C₃×C₉	—	3	3	no	—
29	C₂²×C₃×C₉	C₂×C₉	1	9	yes	complemented
30	C₂²×C₃×C₉	C₂×C₃²	1	3	yes	—
31	C₃×C₂²⋊C₉	C₂×C₉	1	27	yes	direct product factor
32	C₃×C₂²⋊C₉	C₂×C₃²	1	9	yes	—
33	C₃²×C₉	—	4	3	no	—
34	C₃²×C₉	C₂×A₄	1	1	yes	characteristic
35	C₉²	—	4	9	no	—
36	C₂³×C₃²	C₃×C₉	1	9	yes	—
37	C₂³×C₃²	C₃×C₉	1	3	yes	—
38	C₂³×C₃²	C₃³	1	1	yes	characteristic
39	C₂³×C₉	C₃×C₉	1	9	yes	complemented
40	C₂×C₂²⋊C₉	C₃×C₉	1	27	yes	direct product factor
41	C₂×C₃³	—	3	1	no	—
42	C₂×C₃³	—	3	1	no	—
43	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
44	C₂×C₃×C₉	—	3	9	no	—
45	C₂×C₃×C₉	—	3	9	no	—
46	C₂×C₃×C₉	—	3	3	no	—
47	C₂×C₃×C₉	—	3	3	no	—
48	C₂×C₃×C₉	—	4	27	no	—
49	C₂×C₃×C₉	—	4	9	no	—
50	C₂×C₃×C₉	C₂²⋊C₉	1	9	yes	direct product factor
51	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
52	C₂²×C₃²	—	3	3	no	—
53	C₂²×C₃²	—	3	1	no	—
54	C₂²×C₃²	—	3	9	no	—
55	C₂²×C₃²	—	3	9	no	—
56	C₂²×C₃²	—	3	3	no	—
57	C₂²×C₃²	—	3	1	no	—
58	C₂²×C₃²	C₂×C₃×C₉	1	9	yes	—
59	C₂²×C₃²	C₂×C₃×C₉	1	3	yes	—
60	C₂²×C₃²	C₂×C₃³	1	1	yes	characteristic
61	C₂²×C₉	—	3	9	no	—
62	C₂²×C₉	—	3	9	no	—
63	C₂²×C₉	C₂×C₃×C₉	1	9	yes	complemented
64	C₂²⋊C₉	C₂×C₃×C₉	1	27	yes	direct product factor
65	C₃³	C₂×C₃×A₄	1	1	yes	characteristic
66	C₃×C₉	—	4	27	no	—
67	C₃×C₉	—	4	9	no	—
68	C₃×C₉	C₂×C₂²⋊C₉	1	9	yes	direct product factor
69	C₃×C₉	C₂×C₃×A₄	1	3	yes	—
70	C₂³×C₃	C₉²	1	9	yes	complemented
71	C₂³×C₃	C₃²×C₉	1	3	yes	—
72	C₂³×C₃	C₃²×C₉	1	1	yes	characteristic
73	C₂×C₃²	—	3	9	no	—
74	C₂×C₃²	—	3	9	no	—
75	C₂×C₃²	—	3	3	no	—
76	C₂×C₃²	—	3	3	no	—
77	C₂×C₃²	—	3	1	no	—
78	C₂×C₃²	—	3	1	no	—
79	C₂×C₃²	C₉×A₄	1	9	yes	—
80	C₂×C₃²	C₃×C₂²⋊C₉	1	3	yes	—
81	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
82	C₂×C₉	—	3	9	no	—
83	C₂×C₉	—	3	9	no	—
84	C₂×C₉	—	4	27	no	—
85	C₂×C₉	C₃×C₂²⋊C₉	1	9	yes	direct product factor
86	C₂²×C₃	—	3	9	no	—
87	C₂²×C₃	—	3	1	no	—
88	C₂²×C₃	—	3	3	no	—
89	C₂²×C₃	—	3	9	no	—
90	C₂²×C₃	—	3	3	no	—
91	C₂²×C₃	—	3	1	no	—
92	C₂²×C₃	C₂×C₉²	1	9	yes	complemented
93	C₂²×C₃	C₂×C₃²×C₉	1	3	yes	—
94	C₂²×C₃	C₂×C₃²×C₉	1	1	yes	characteristic
95	C₃²	C₂×C₉×A₄	1	9	yes	—
96	C₃²	C₂×C₃×C₂²⋊C₉	1	3	yes	—
97	C₃²	C₂×C₃²×A₄	1	1	yes	characteristic
98	C₉	—	4	27	no	—
99	C₉	C₂×C₃×C₂²⋊C₉	1	9	yes	direct product factor
100	C₂³	C₃×C₉²	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₃	—	3	9	no	—
102	C₂×C₃	—	3	9	no	—
103	C₂×C₃	—	3	3	no	—
104	C₂×C₃	—	3	3	no	—
105	C₂×C₃	—	3	1	no	—
106	C₂×C₃	—	3	1	no	—
107	C₂×C₃	C₉×C₂²⋊C₉	1	9	yes	direct product factor
108	C₂×C₃	C₃×C₉×A₄	1	3	yes	—
109	C₂×C₃	C₃²×C₂²⋊C₉	1	1	yes	characteristic
110	C₂²	—	3	1	no	—
111	C₂²	—	3	1	no	—
112	C₂²	C₂×C₃×C₉²	1	1	yes	characteristic, complemented
113	C₃	C₂×C₉×C₂²⋊C₉	1	9	yes	direct product factor
114	C₃	C₂×C₃×C₉×A₄	1	3	yes	—
115	C₃	C₂×C₃²×C₂²⋊C₉	1	1	yes	characteristic
116	C₂	—	3	1	no	—
117	C₂	—	3	1	no	—
118	C₂	C₃×C₉×C₂²⋊C₉	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×C₂²⋊C₉

C₂

yes

maximal, characteristic, direct product factor

C₂³×C₃²×C₉

C₃

yes

maximal, characteristic

C₂×C₃²×C₂²⋊C₉

C₃

yes

maximal

C₂×C₉×C₂²⋊C₉

C₃

yes

maximal, direct product factor

C₂×C₃×C₉²

—

maximal

C₂²×C₃²×C₉

—

C₂²×C₃²×C₉

—

C₂²×C₃²×C₉

C₂×C₃

yes

characteristic

C₃²×C₂²⋊C₉

C₂×C₃

yes

—

C₉×C₂²⋊C₉

C₂×C₃

yes

direct product factor

C₃×C₉²

—

Sylow

C₂³×C₃³

C₃²

yes

characteristic

C₂³×C₃×C₉

C₉

yes

complemented

C₂³×C₃×C₉

C₃²

yes

—

C₂×C₃×C₂²⋊C₉

C₉

yes

direct product factor

C₂×C₃×C₂²⋊C₉

C₃²

yes

—

C₂×C₃²×C₉

—

C₂×C₃²×C₉

—

C₂×C₃²×C₉

—

C₂×C₃²×C₉

A₄

yes

characteristic

C₂×C₉²

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

C₂×C₃²

yes

characteristic

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×C₉

yes

complemented

C₂²×C₃×C₉

C₂×C₃²

yes

—

C₃×C₂²⋊C₉

C₂×C₉

yes

direct product factor

C₃×C₂²⋊C₉

C₂×C₃²

yes

—

C₃²×C₉

—

C₃²×C₉

C₂×A₄

yes

characteristic

C₉²

—

C₂³×C₃²

C₃×C₉

yes

—

C₂³×C₃²

C₃×C₉

yes

—

C₂³×C₃²

C₃³

yes

characteristic

C₂³×C₉

C₃×C₉

yes

complemented

C₂×C₂²⋊C₉

C₃×C₉

yes

direct product factor

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₃×A₄

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²⋊C₉

yes

direct product factor

C₂×C₃×C₉

C₃×A₄

yes

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃×C₉

yes

—

C₂²×C₃²

C₂×C₃×C₉

yes

—

C₂²×C₃²

C₂×C₃³

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×C₉

yes

complemented

C₂²⋊C₉

C₂×C₃×C₉

yes

direct product factor

C₃³

C₂×C₃×A₄

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₂×C₂²⋊C₉

yes

direct product factor

C₃×C₉

C₂×C₃×A₄

yes

—

C₂³×C₃

C₉²

yes

complemented

C₂³×C₃

C₃²×C₉

yes

—

C₂³×C₃

C₃²×C₉

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₉×A₄

yes

—

C₂×C₃²

C₃×C₂²⋊C₉

yes

—

C₂×C₃²

C₃²×A₄

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₃×C₂²⋊C₉

yes

direct product factor

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₉²

yes

complemented

C₂²×C₃

C₂×C₃²×C₉

yes

—

C₂²×C₃

C₂×C₃²×C₉

yes

characteristic

C₃²

C₂×C₉×A₄

yes

—

C₃²

C₂×C₃×C₂²⋊C₉

yes

—

C₃²

C₂×C₃²×A₄

yes

characteristic

C₉

—

C₉

C₂×C₃×C₂²⋊C₉

yes

direct product factor

100

C₂³

C₃×C₉²

yes

characteristic, complemented, Sylow

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

—

104

C₂×C₃

—

105

C₂×C₃

—

106

C₂×C₃

—

107

C₂×C₃

C₉×C₂²⋊C₉

yes

direct product factor

108

C₂×C₃

C₃×C₉×A₄

yes

—

109

C₂×C₃

C₃²×C₂²⋊C₉

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

C₂×C₃×C₉²

yes

characteristic, complemented

113

C₃

C₂×C₉×C₂²⋊C₉

yes

direct product factor

114

C₃

C₂×C₃×C₉×A₄

yes

—

115

C₃

C₂×C₃²×C₂²⋊C₉

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

C₃×C₉×C₂²⋊C₉

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₉	A₄
commutator, hypocenter	C₂²	C₂×C₃×C₉²
Frattini	C₃²	C₂×C₃²×A₄
Fitting	C₂³×C₃²×C₉	C₃
socle, prime radical	C₂³×C₃³	C₃²
cosocle	C₂²×C₃²	C₂×C₃³
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂×C₃²×C₉

subgroup

G/H

center, hypercenter

C₂×C₃²×C₉

A₄

commutator, hypocenter

C₂²

C₂×C₃×C₉²

Frattini

C₃²

C₂×C₃²×A₄

Fitting

C₂³×C₃²×C₉

C₃

socle, prime radical

C₂³×C₃³

C₃²

cosocle

C₂²×C₃²

C₂×C₃³

LCS

G ▷ C₂²

UCS

C₁ ◁ C₂×C₃²×C₉

N	H ≅ G/N	type
1. C₃×C₉×C₂²⋊C₉	C₂	direct (N × H)
4. C₂×C₉×C₂²⋊C₉	C₃	direct (N × H)
10. C₉×C₂²⋊C₉	C₂×C₃	direct (N × H)
13. C₂³×C₃×C₉	C₉	semidirect
15. C₂×C₃×C₂²⋊C₉	C₉	direct (N × H)
29. C₂²×C₃×C₉	C₂×C₉	semidirect
31. C₃×C₂²⋊C₉	C₂×C₉	direct (N × H)
39. C₂³×C₉	C₃×C₉	semidirect
40. C₂×C₂²⋊C₉	C₃×C₉	direct (N × H)
50. C₂×C₃×C₉	C₂²⋊C₉	direct (N × H)
63. C₂²×C₉	C₂×C₃×C₉	semidirect
64. C₂²⋊C₉	C₂×C₃×C₉	direct (N × H)
68. C₃×C₉	C₂×C₂²⋊C₉	direct (N × H)
70. C₂³×C₃	C₉²	semidirect
85. C₂×C₉	C₃×C₂²⋊C₉	direct (N × H)
92. C₂²×C₃	C₂×C₉²	semidirect
99. C₉	C₂×C₃×C₂²⋊C₉	direct (N × H)
100. C₂³	C₃×C₉²	semidirect
107. C₂×C₃	C₉×C₂²⋊C₉	direct (N × H)
112. C₂²	C₂×C₃×C₉²	semidirect
113. C₃	C₂×C₉×C₂²⋊C₉	direct (N × H)
118. C₂	C₃×C₉×C₂²⋊C₉	direct (N × H)

H ≅ G/N

type

1. C₃×C₉×C₂²⋊C₉

C₂

direct (N × H)

4. C₂×C₉×C₂²⋊C₉

C₃

direct (N × H)

10. C₉×C₂²⋊C₉

C₂×C₃

direct (N × H)

13. C₂³×C₃×C₉

C₉

semidirect

15. C₂×C₃×C₂²⋊C₉

C₉

direct (N × H)

29. C₂²×C₃×C₉

C₂×C₉

semidirect

31. C₃×C₂²⋊C₉

C₂×C₉

direct (N × H)

39. C₂³×C₉

C₃×C₉

semidirect

40. C₂×C₂²⋊C₉

C₃×C₉

direct (N × H)

50. C₂×C₃×C₉

C₂²⋊C₉

direct (N × H)

63. C₂²×C₉

C₂×C₃×C₉

semidirect

64. C₂²⋊C₉

C₂×C₃×C₉

direct (N × H)

68. C₃×C₉

C₂×C₂²⋊C₉

direct (N × H)

70. C₂³×C₃

C₉²

semidirect

85. C₂×C₉

C₃×C₂²⋊C₉

direct (N × H)

92. C₂²×C₃

C₂×C₉²

semidirect

99. C₉

C₂×C₃×C₂²⋊C₉

direct (N × H)

100. C₂³

C₃×C₉²

semidirect

107. C₂×C₃

C₉×C₂²⋊C₉

direct (N × H)

112. C₂²

C₂×C₃×C₉²

semidirect

113. C₃

C₂×C₉×C₂²⋊C₉

direct (N × H)

118. C₂

C₃×C₉×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×C₉×C₂²⋊C₉	2
C₂³×C₃²×C₉	3
C₂×C₃²×C₂²⋊C₉	3
C₂×C₉×C₂²⋊C₉	3
C₂×C₃×C₉²	4

subgroup

index

C₃×C₉×C₂²⋊C₉

C₂³×C₃²×C₉

C₂×C₃²×C₂²⋊C₉

C₂×C₉×C₂²⋊C₉

C₂×C₃×C₉²

Subgroups — C₂×C₃×C₉×C₂²⋊C₉ #14518402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C3×C9×C22⋊C9 #14518402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₉×C₂²⋊C₉ #14518402