Group #13142390 — subgroups

Subgroups — C₄²×C₉×Dic₃ #13142390

Order 1728.

882 conjugacy classes of subgroups; only the 549 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 549 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×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₉×Dic₃	C₂	1	6	yes	maximal
3	C₃×C₄²×Dic₃	C₃	1	1	yes	maximal, characteristic
4	C₄³×C₉	—	3	1	no	maximal
5	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
6	C₂²×C₉×Dic₃	C₂²	1	4	yes	—
7	C₃×C₄²×C₉	C₄	1	4	yes	complemented
8	C₄×C₉×Dic₃	C₄	1	24	yes	direct product factor
9	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₃	C₂×C₃	1	6	yes	—
11	C₂×C₄²×C₉	—	2	1	no	—
12	C₂×C₄²×C₉	—	3	6	no	—
13	C₂×C₄²×C₉	D₃	1	1	yes	characteristic
14	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
15	C₂×C₃×C₄×C₉	C₂×C₄	1	12	yes	—
16	C₂×C₃×C₄×C₉	C₂×C₄	1	6	yes	—
17	C₂×C₉×Dic₃	C₂×C₄	1	24	yes	—
18	C₃×C₄³	—	3	1	no	—
19	C₄²×Dic₃	C₉	1	1	yes	characteristic, direct product factor
20	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
21	C₂²×C₃×Dic₃	C₂²×C₃	1	4	yes	—
22	C₃²×C₄²	C₃×C₄	1	4	yes	—
23	C₃×C₄×Dic₃	C₃×C₄	1	24	yes	—
24	C₂²×C₄×C₉	—	2	3	no	—
25	C₂²×C₄×C₉	—	3	4	no	—
26	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
27	C₄²×C₉	—	2	4	no	—
28	C₄²×C₉	—	3	24	no	—
29	C₄²×C₉	Dic₃	1	4	yes	direct product factor
30	C₂²×C₃×C₉	C₂²×C₄	1	6	yes	—
31	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
32	C₃×C₄×C₉	C₄²	1	12	yes	complemented
33	C₉×Dic₃	C₄²	1	16	yes	direct product factor
34	C₂×C₃×C₄²	—	2	1	no	—
35	C₂×C₃×C₄²	—	3	6	no	—
36	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic
37	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
38	C₂×C₄×Dic₃	C₂×C₉	1	6	yes	—
39	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
40	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
41	C₂×C₃²×C₄	C₂×C₃×C₄	1	6	yes	—
42	C₂×C₃×Dic₃	C₂×C₃×C₄	1	24	yes	—
43	C₂³×C₉	—	2	1	no	—
44	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
45	C₂×C₄×C₉	—	2	12	no	—
46	C₂×C₄×C₉	—	2	6	no	—
47	C₂×C₄×C₉	—	3	24	no	—
48	C₂×C₄×C₉	C₄×D₃	1	12	yes	—
49	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
50	C₄³	—	3	1	no	Sylow
51	C₂×C₃×C₉	C₂×C₄²	1	4	yes	—
52	C₂×C₃×C₉	C₂×C₄²	1	3	yes	—
53	C₂²×C₃×C₄	—	2	3	no	—
54	C₂²×C₃×C₄	—	3	4	no	—
55	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
56	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
57	C₂²×Dic₃	C₂²×C₉	1	4	yes	—
58	C₃×C₄²	—	2	4	no	—
59	C₃×C₄²	—	3	24	no	—
60	C₃×C₄²	C₄×C₉	1	4	yes	complemented
61	C₃×C₄²	C₃×Dic₃	1	4	yes	—
62	C₄×Dic₃	C₄×C₉	1	24	yes	direct product factor
63	C₂²×C₃²	C₂²×C₃×C₄	1	6	yes	—
64	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
65	C₃²×C₄	C₃×C₄²	1	12	yes	—
66	C₃×Dic₃	C₃×C₄²	1	16	yes	—
67	C₂²×C₉	—	2	6	no	—
68	C₂²×C₉	—	2	1	no	—
69	C₂²×C₉	C₂×C₄×D₃	1	6	yes	—
70	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
71	C₄×C₉	—	2	12	no	—
72	C₄×C₉	—	3	16	no	—
73	C₄×C₉	C₄×Dic₃	1	12	yes	direct product factor
74	C₂×C₄²	—	3	6	no	—
75	C₂×C₄²	D₃×C₉	1	1	yes	characteristic
76	C₃×C₉	C₄³	1	1	yes	characteristic, complemented, Sylow
77	C₂³×C₃	—	2	1	no	—
78	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
79	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
80	C₂×C₃×C₄	—	2	12	no	—
81	C₂×C₃×C₄	—	2	6	no	—
82	C₂×C₃×C₄	—	3	24	no	—
83	C₂×C₃×C₄	C₂×C₄×C₉	1	12	yes	—
84	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
85	C₂×C₃×C₄	C₃×C₄×D₃	1	12	yes	—
86	C₂×C₃×C₄	C₂×C₃×Dic₃	1	6	yes	—
87	C₂×Dic₃	C₂×C₄×C₉	1	24	yes	—
88	C₂×C₃²	C₂×C₃×C₄²	1	3	yes	—
89	C₂×C₃²	C₂×C₃×C₄²	1	4	yes	—
90	C₂×C₉	—	2	3	no	—
91	C₂×C₉	—	2	4	no	—
92	C₂×C₉	C₄²×D₃	1	4	yes	—
93	C₂×C₉	C₂×C₄×Dic₃	1	3	yes	—
94	C₂²×C₄	—	3	4	no	—
95	C₂²×C₄	C₂×D₃×C₉	1	3	yes	—
96	C₄²	—	3	24	no	—
97	C₄²	C₉×Dic₃	1	4	yes	direct product factor
98	C₂²×C₃	—	2	6	no	—
99	C₂²×C₃	—	2	1	no	—
100	C₂²×C₃	C₂²×C₄×C₉	1	6	yes	—
101	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
102	C₂²×C₃	C₂×C₃×C₄×D₃	1	6	yes	—
103	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
104	C₃×C₄	—	2	12	no	—
105	C₃×C₄	—	3	16	no	—
106	C₃×C₄	C₄²×C₉	1	12	yes	complemented
107	C₃×C₄	C₃×C₄×Dic₃	1	12	yes	—
108	Dic₃	C₄²×C₉	1	16	yes	direct product factor
109	C₃²	C₃×C₄³	1	1	yes	characteristic
110	C₉	—	2	1	no	—
111	C₉	C₄²×Dic₃	1	1	yes	characteristic, direct product factor
112	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
113	C₂×C₄	—	3	24	no	—
114	C₂×C₄	C₄×D₃×C₉	1	12	yes	—
115	C₂×C₄	C₂×C₉×Dic₃	1	6	yes	—
116	C₂×C₃	—	2	3	no	—
117	C₂×C₃	—	2	4	no	—
118	C₂×C₃	C₂×C₄²×C₉	1	3	yes	—
119	C₂×C₃	C₂×C₄²×C₉	1	4	yes	—
120	C₂×C₃	C₃×C₄²×D₃	1	4	yes	—
121	C₂×C₃	C₂×C₃×C₄×Dic₃	1	3	yes	—
122	C₂²	C₂×C₄×D₃×C₉	1	6	yes	—
123	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
124	C₄	—	3	16	no	—
125	C₄	C₄×C₉×Dic₃	1	12	yes	direct product factor
126	C₃	—	2	1	no	—
127	C₃	C₄³×C₉	1	1	yes	characteristic, complemented
128	C₃	C₃×C₄²×Dic₃	1	1	yes	characteristic
129	C₂	C₄²×D₃×C₉	1	4	yes	—
130	C₂	C₂×C₄×C₉×Dic₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄²×C₉

C₂

yes

maximal, characteristic

C₂×C₄×C₉×Dic₃

C₂

yes

maximal

C₃×C₄²×Dic₃

C₃

yes

maximal, characteristic

C₄³×C₉

—

maximal

C₂²×C₃×C₄×C₉

C₂²

yes

—

C₂²×C₉×Dic₃

C₂²

yes

—

C₃×C₄²×C₉

C₄

yes

complemented

C₄×C₉×Dic₃

C₄

yes

direct product factor

C₂×C₃²×C₄²

C₂×C₃

yes

characteristic

C₂×C₃×C₄×Dic₃

C₂×C₃

yes

—

C₂×C₄²×C₉

—

C₂×C₄²×C₉

—

C₂×C₄²×C₉

D₃

yes

characteristic

C₂³×C₃×C₉

C₂³

yes

characteristic

C₂×C₃×C₄×C₉

C₂×C₄

yes

—

C₂×C₃×C₄×C₉

C₂×C₄

yes

—

C₂×C₉×Dic₃

C₂×C₄

yes

—

C₃×C₄³

—

C₄²×Dic₃

C₉

yes

characteristic, direct product factor

C₂²×C₃²×C₄

C₂²×C₃

yes

—

C₂²×C₃×Dic₃

C₂²×C₃

yes

—

C₃²×C₄²

C₃×C₄

yes

—

C₃×C₄×Dic₃

C₃×C₄

yes

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂×D₃

yes

—

C₄²×C₉

—

C₄²×C₉

—

C₄²×C₉

Dic₃

yes

direct product factor

C₂²×C₃×C₉

C₂²×C₄

yes

—

C₂²×C₃×C₉

C₂²×C₄

yes

characteristic

C₃×C₄×C₉

C₄²

yes

complemented

C₉×Dic₃

C₄²

yes

direct product factor

C₂×C₃×C₄²

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂×C₉

yes

characteristic

C₂×C₃×C₄²

C₃×D₃

yes

characteristic

C₂×C₄×Dic₃

C₂×C₉

yes

—

C₂³×C₃²

C₂³×C₃

yes

characteristic

C₂×C₃²×C₄

C₂×C₃×C₄

yes

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

—

C₂×C₃×Dic₃

C₂×C₃×C₄

yes

—

C₂³×C₉

—

C₂³×C₉

C₂²×D₃

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₄×D₃

yes

—

C₂×C₄×C₉

C₂×Dic₃

yes

—

C₄³

—

Sylow

C₂×C₃×C₉

C₂×C₄²

yes

—

C₂×C₃×C₉

C₂×C₄²

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²×C₉

yes

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

C₂²×C₉

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₄×C₉

yes

complemented

C₃×C₄²

C₃×Dic₃

yes

—

C₄×Dic₃

C₄×C₉

yes

direct product factor

C₂²×C₃²

C₂²×C₃×C₄

yes

—

C₂²×C₃²

C₂²×C₃×C₄

yes

characteristic

C₃²×C₄

C₃×C₄²

yes

—

C₃×Dic₃

C₃×C₄²

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₄×D₃

yes

—

C₂²×C₉

C₂²×Dic₃

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₄×Dic₃

yes

direct product factor

C₂×C₄²

—

C₂×C₄²

D₃×C₉

yes

characteristic

C₃×C₉

C₄³

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂³×C₉

yes

characteristic

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₄×C₉

yes

—

C₂×C₃×C₄

C₂×C₄×C₉

yes

—

C₂×C₃×C₄

C₃×C₄×D₃

yes

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

C₂×Dic₃

C₂×C₄×C₉

yes

—

C₂×C₃²

C₂×C₃×C₄²

yes

—

C₂×C₃²

C₂×C₃×C₄²

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₄²×D₃

yes

—

C₂×C₉

C₂×C₄×Dic₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃×C₉

yes

—

C₄²

—

C₄²

C₉×Dic₃

yes

direct product factor

C₂²×C₃

—

C₂²×C₃

—

100

C₂²×C₃

C₂²×C₄×C₉

yes

—

101

C₂²×C₃

C₂²×C₄×C₉

yes

characteristic

102

C₂²×C₃

C₂×C₃×C₄×D₃

yes

—

103

C₂²×C₃

C₂²×C₃×Dic₃

yes

characteristic

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

C₄²×C₉

yes

complemented

107

C₃×C₄

C₃×C₄×Dic₃

yes

—

108

Dic₃

C₄²×C₉

yes

direct product factor

109

C₃²

C₃×C₄³

yes

characteristic

110

C₉

—

111

C₉

C₄²×Dic₃

yes

characteristic, direct product factor

112

C₂³

C₂²×D₃×C₉

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

C₄×D₃×C₉

yes

—

115

C₂×C₄

C₂×C₉×Dic₃

yes

—

116

C₂×C₃

—

117

C₂×C₃

—

118

C₂×C₃

C₂×C₄²×C₉

yes

—

119

C₂×C₃

C₂×C₄²×C₉

yes

—

120

C₂×C₃

C₃×C₄²×D₃

yes

—

121

C₂×C₃

C₂×C₃×C₄×Dic₃

yes

—

122

C₂²

C₂×C₄×D₃×C₉

yes

—

123

C₂²

C₂²×C₉×Dic₃

yes

characteristic

124

C₄

—

125

C₄

C₄×C₉×Dic₃

yes

direct product factor

126

C₃

—

127

C₃

C₄³×C₉

yes

characteristic, complemented

128

C₃

C₃×C₄²×Dic₃

yes

characteristic

129

C₂

C₄²×D₃×C₉

yes

—

130

C₂

C₂×C₄×C₉×Dic₃

yes

—

	subgroup	G/H
center, hypercenter	C₂×C₄²×C₉	D₃
commutator, hypocenter	C₃	C₄³×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₄²×C₉

subgroup

G/H

center, hypercenter

C₂×C₄²×C₉

D₃

commutator, hypocenter

C₃

C₄³×C₉

Frattini

C₂³×C₃

C₂²×C₃×D₃

Fitting

C₂×C₃×C₄²×C₉

C₂

socle, prime radical, cosocle

C₂³×C₃²

C₂³×C₃

LCS

G ▷ C₃

UCS

C₁ ◁ C₂×C₄²×C₉

N	H ≅ G/N	type
7. C₃×C₄²×C₉	C₄	semidirect
8. C₄×C₉×Dic₃	C₄	direct (N × H)
19. C₄²×Dic₃	C₉	direct (N × H)
29. C₄²×C₉	Dic₃	direct (N × H)
32. C₃×C₄×C₉	C₄²	semidirect
33. C₉×Dic₃	C₄²	direct (N × H)
60. C₃×C₄²	C₄×C₉	semidirect
62. C₄×Dic₃	C₄×C₉	direct (N × H)
73. C₄×C₉	C₄×Dic₃	direct (N × H)
76. C₃×C₉	C₄³	semidirect
97. C₄²	C₉×Dic₃	direct (N × H)
106. C₃×C₄	C₄²×C₉	semidirect
108. Dic₃	C₄²×C₉	direct (N × H)
111. C₉	C₄²×Dic₃	direct (N × H)
125. C₄	C₄×C₉×Dic₃	direct (N × H)
127. C₃	C₄³×C₉	semidirect

H ≅ G/N

type

7. C₃×C₄²×C₉

C₄

semidirect

8. C₄×C₉×Dic₃

C₄

direct (N × H)

19. C₄²×Dic₃

C₉

direct (N × H)

29. C₄²×C₉

Dic₃

direct (N × H)

32. C₃×C₄×C₉

C₄²

semidirect

33. C₉×Dic₃

C₄²

direct (N × H)

60. C₃×C₄²

C₄×C₉

semidirect

62. C₄×Dic₃

C₄×C₉

direct (N × H)

73. C₄×C₉

C₄×Dic₃

direct (N × H)

76. C₃×C₉

C₄³

semidirect

97. C₄²

C₉×Dic₃

direct (N × H)

106. C₃×C₄

C₄²×C₉

semidirect

108. Dic₃

C₄²×C₉

direct (N × H)

111. C₉

C₄²×Dic₃

direct (N × H)

125. C₄

C₄×C₉×Dic₃

direct (N × H)

127. C₃

C₄³×C₉

semidirect

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₄×C₉×Dic₃	2
C₃×C₄²×Dic₃	3
C₄³×C₉	3

subgroup

index

C₂×C₃×C₄²×C₉

C₂×C₄×C₉×Dic₃

C₃×C₄²×Dic₃

C₄³×C₉

Subgroups — C₄²×C₉×Dic₃ #13142390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C42×C9×Dic3 #13142390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₉×Dic₃ #13142390