Group #11899789 — subgroups

Subgroups — C₂²×C₄²×D₉ #11899789

Order 1152.

4500 conjugacy classes of subgroups; only the 1998 normal subgroups are shown.

normal subgroups only

This group has 4500 conjugacy classes of subgroups — too many to draw. Only its 1998 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	C₂³×C₄×D₉	C₂	1	3	yes	maximal
2	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×Dic₉	C₂	1	3	yes	maximal, complemented
4	C₂×C₄²×D₉	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄²×D₃	—	3	1	no	maximal
6	C₂⁴×D₉	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₉	C₂²	1	3	yes	—
8	C₂³×Dic₉	C₂²	1	3	yes	—
9	C₂²×C₄×D₉	C₄	1	48	yes	direct product factor
10	C₂²×C₄×D₉	C₂²	1	36	yes	—
11	C₂×C₄²×C₉	C₂²	1	12	yes	complemented
12	C₂×C₄×Dic₉	C₂²	1	36	yes	complemented
13	C₄²×D₉	C₂²	1	64	yes	direct product factor
14	C₂³×C₄×D₃	—	3	3	no	—
15	C₂²×C₃×C₄²	D₃	1	1	yes	characteristic
16	C₂²×C₄×Dic₃	—	3	3	no	—
17	C₂×C₄²×D₃	—	3	24	no	—
18	C₂⁴×C₉	C₂³	1	1	yes	characteristic
19	C₂³×D₉	C₂×C₄	1	24	yes	—
20	C₂³×D₉	C₂³	1	6	yes	—
21	C₂²×C₄×C₉	C₂×C₄	1	24	yes	complemented
22	C₂²×C₄×C₉	C₂³	1	18	yes	—
23	C₂²×Dic₉	C₂×C₄	1	24	yes	complemented
24	C₂²×Dic₉	C₂³	1	18	yes	—
25	C₂×C₄×D₉	C₂×C₄	1	288	yes	direct product factor
26	C₂×C₄×D₉	C₂³	1	48	yes	—
27	C₄²×C₉	C₂³	1	16	yes	complemented
28	C₄×Dic₉	C₂³	1	48	yes	complemented
29	C₂³×C₄²	—	9	1	no	Sylow
30	C₂⁴×D₃	—	3	1	no	—
31	C₂³×C₃×C₄	C₂×D₃	1	3	yes	—
32	C₂³×Dic₃	—	3	3	no	—
33	C₂²×C₄×D₃	—	3	48	no	—
34	C₂²×C₄×D₃	—	3	36	no	—
35	C₂×C₃×C₄²	C₂×D₃	1	12	yes	—
36	C₂×C₄×Dic₃	—	3	36	no	—
37	C₄²×D₃	—	3	64	no	—
38	C₂³×C₉	C₂²×C₄	1	12	yes	—
39	C₂³×C₉	C₂⁴	1	3	yes	—
40	C₂²×D₉	C₄²	1	64	yes	direct product factor
41	C₂²×D₉	C₂²×C₄	1	72	yes	—
42	C₂²×D₉	C₂⁴	1	4	yes	—
43	C₂×C₄×C₉	C₂²×C₄	1	72	yes	complemented
44	C₂×C₄×C₉	C₂⁴	1	12	yes	—
45	C₂×Dic₉	C₂²×C₄	1	72	yes	complemented
46	C₂×Dic₉	C₂⁴	1	12	yes	—
47	C₄×D₉	C₂²×C₄	1	192	yes	direct product factor
48	C₂⁴×C₄	—	9	3	no	—
49	C₂²×C₄²	—	9	3	no	—
50	C₂²×C₄²	—	9	24	no	—
51	C₂²×C₄²	D₉	1	1	yes	characteristic, direct product factor
52	C₂⁴×C₃	C₂²×D₃	1	1	yes	characteristic
53	C₂³×D₃	—	3	24	no	—
54	C₂³×D₃	—	3	6	no	—
55	C₂²×C₃×C₄	C₄×D₃	1	24	yes	—
56	C₂²×C₃×C₄	C₂²×D₃	1	18	yes	—
57	C₂²×Dic₃	—	3	24	no	—
58	C₂²×Dic₃	—	3	18	no	—
59	C₂×C₄×D₃	—	3	288	no	—
60	C₂×C₄×D₃	—	3	48	no	—
61	C₃×C₄²	C₂²×D₃	1	16	yes	—
62	C₄×Dic₃	—	3	48	no	—
63	C₂²×C₉	C₂×C₄²	1	16	yes	complemented
64	C₂²×C₉	C₂³×C₄	1	18	yes	—
65	C₂²×C₉	C₂⁵	1	1	yes	characteristic
66	C₂×D₉	C₂×C₄²	1	96	yes	direct product factor
67	C₂×D₉	C₂³×C₄	1	24	yes	—
68	C₄×C₉	C₂³×C₄	1	24	yes	complemented
69	Dic₉	C₂³×C₄	1	24	yes	complemented
70	C₂⁵	—	9	1	no	—
71	C₂³×C₄	—	9	3	no	—
72	C₂³×C₄	—	9	48	no	—
73	C₂³×C₄	—	9	36	no	—
74	C₂³×C₄	C₂×D₉	1	3	yes	—
75	C₂×C₄²	—	9	36	no	—
76	C₂×C₄²	—	9	64	no	—
77	C₂×C₄²	C₂×D₉	1	12	yes	direct product factor
78	C₂³×C₃	C₂×C₄×D₃	1	12	yes	—
79	C₂³×C₃	C₂³×D₃	1	3	yes	—
80	C₂²×D₃	—	3	64	no	—
81	C₂²×D₃	—	3	72	no	—
82	C₂²×D₃	—	3	4	no	—
83	C₂×C₃×C₄	C₂×C₄×D₃	1	72	yes	—
84	C₂×C₃×C₄	C₂³×D₃	1	12	yes	—
85	C₂×Dic₃	—	3	72	no	—
86	C₂×Dic₃	—	3	12	no	—
87	C₄×D₃	—	3	192	no	—
88	C₂×C₉	C₂²×C₄²	1	12	yes	complemented
89	C₂×C₉	C₂⁴×C₄	1	3	yes	—
90	D₉	C₂²×C₄²	1	16	yes	direct product factor
91	C₂⁴	—	9	24	no	—
92	C₂⁴	—	9	6	no	—
93	C₂⁴	C₂²×D₉	1	1	yes	characteristic
94	C₂²×C₄	—	9	24	no	—
95	C₂²×C₄	—	9	18	no	—
96	C₂²×C₄	—	9	288	no	—
97	C₂²×C₄	—	9	48	no	—
98	C₂²×C₄	C₄×D₉	1	24	yes	direct product factor
99	C₂²×C₄	C₂²×D₉	1	18	yes	—
100	C₄²	—	9	48	no	—
101	C₄²	C₂²×D₉	1	16	yes	direct product factor
102	C₂²×C₃	C₄²×D₃	1	16	yes	—
103	C₂²×C₃	C₂²×C₄×D₃	1	18	yes	—
104	C₂²×C₃	C₂⁴×D₃	1	1	yes	characteristic
105	C₂×D₃	—	3	24	no	—
106	C₂×D₃	—	3	96	no	—
107	C₃×C₄	C₂²×C₄×D₃	1	24	yes	—
108	Dic₃	—	3	24	no	—
109	C₉	C₂³×C₄²	1	1	yes	characteristic, complemented, Sylow
110	C₂³	—	9	4	no	—
111	C₂³	—	9	64	no	—
112	C₂³	—	9	72	no	—
113	C₂³	C₂×C₄×D₉	1	12	yes	—
114	C₂³	C₂³×D₉	1	3	yes	—
115	C₂×C₄	—	9	12	no	—
116	C₂×C₄	—	9	72	no	—
117	C₂×C₄	—	9	192	no	—
118	C₂×C₄	C₂×C₄×D₉	1	72	yes	direct product factor
119	C₂×C₄	C₂³×D₉	1	12	yes	—
120	C₂×C₃	C₂×C₄²×D₃	1	12	yes	—
121	C₂×C₃	C₂³×C₄×D₃	1	3	yes	—
122	D₃	—	3	16	no	—
123	C₂²	—	9	24	no	—
124	C₂²	—	9	96	no	—
125	C₂²	C₄²×D₉	1	16	yes	direct product factor
126	C₂²	C₂²×C₄×D₉	1	18	yes	—
127	C₂²	C₂⁴×D₉	1	1	yes	characteristic
128	C₄	—	9	24	no	—
129	C₄	C₂²×C₄×D₉	1	24	yes	direct product factor
130	C₃	C₂²×C₄²×D₃	1	1	yes	characteristic
131	C₂	—	9	16	no	—
132	C₂	C₂×C₄²×D₉	1	12	yes	direct product factor
133	C₂	C₂³×C₄×D₉	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄×D₉

C₂

yes

maximal

C₂²×C₄²×C₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×Dic₉

C₂

yes

maximal, complemented

C₂×C₄²×D₉

C₂

yes

maximal, direct product factor

C₂²×C₄²×D₃

—

maximal

C₂⁴×D₉

C₂²

yes

characteristic

C₂³×C₄×C₉

C₂²

yes

—

C₂³×Dic₉

C₂²

yes

—

C₂²×C₄×D₉

C₄

yes

direct product factor

C₂²×C₄×D₉

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

complemented

C₂×C₄×Dic₉

C₂²

yes

complemented

C₄²×D₉

C₂²

yes

direct product factor

C₂³×C₄×D₃

—

C₂²×C₃×C₄²

D₃

yes

characteristic

C₂²×C₄×Dic₃

—

C₂×C₄²×D₃

—

C₂⁴×C₉

C₂³

yes

characteristic

C₂³×D₉

C₂×C₄

yes

—

C₂³×D₉

C₂³

yes

—

C₂²×C₄×C₉

C₂×C₄

yes

complemented

C₂²×C₄×C₉

C₂³

yes

—

C₂²×Dic₉

C₂×C₄

yes

complemented

C₂²×Dic₉

C₂³

yes

—

C₂×C₄×D₉

C₂×C₄

288

yes

direct product factor

C₂×C₄×D₉

C₂³

yes

—

C₄²×C₉

C₂³

yes

complemented

C₄×Dic₉

C₂³

yes

complemented

C₂³×C₄²

—

Sylow

C₂⁴×D₃

—

C₂³×C₃×C₄

C₂×D₃

yes

—

C₂³×Dic₃

—

C₂²×C₄×D₃

—

C₂²×C₄×D₃

—

C₂×C₃×C₄²

C₂×D₃

yes

—

C₂×C₄×Dic₃

—

C₄²×D₃

—

C₂³×C₉

C₂²×C₄

yes

—

C₂³×C₉

C₂⁴

yes

—

C₂²×D₉

C₄²

yes

direct product factor

C₂²×D₉

C₂²×C₄

yes

—

C₂²×D₉

C₂⁴

yes

—

C₂×C₄×C₉

C₂²×C₄

yes

complemented

C₂×C₄×C₉

C₂⁴

yes

—

C₂×Dic₉

C₂²×C₄

yes

complemented

C₂×Dic₉

C₂⁴

yes

—

C₄×D₉

C₂²×C₄

192

yes

direct product factor

C₂⁴×C₄

—

C₂²×C₄²

—

C₂²×C₄²

—

C₂²×C₄²

D₉

yes

characteristic, direct product factor

C₂⁴×C₃

C₂²×D₃

yes

characteristic

C₂³×D₃

—

C₂³×D₃

—

C₂²×C₃×C₄

C₄×D₃

yes

—

C₂²×C₃×C₄

C₂²×D₃

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×C₄×D₃

—

288

—

C₂×C₄×D₃

—

C₃×C₄²

C₂²×D₃

yes

—

C₄×Dic₃

—

C₂²×C₉

C₂×C₄²

yes

complemented

C₂²×C₉

C₂³×C₄

yes

—

C₂²×C₉

C₂⁵

yes

characteristic

C₂×D₉

C₂×C₄²

yes

direct product factor

C₂×D₉

C₂³×C₄

yes

—

C₄×C₉

C₂³×C₄

yes

complemented

Dic₉

C₂³×C₄

yes

complemented

C₂⁵

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉

yes

direct product factor

C₂³×C₃

C₂×C₄×D₃

yes

—

C₂³×C₃

C₂³×D₃

yes

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

C₂×C₄×D₃

yes

—

C₂×C₃×C₄

C₂³×D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

192

—

C₂×C₉

C₂²×C₄²

yes

complemented

C₂×C₉

C₂⁴×C₄

yes

—

D₉

C₂²×C₄²

yes

direct product factor

C₂⁴

—

C₂⁴

—

C₂⁴

C₂²×D₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

288

—

C₂²×C₄

—

C₂²×C₄

C₄×D₉

yes

direct product factor

C₂²×C₄

C₂²×D₉

yes

—

100

C₄²

—

101

C₄²

C₂²×D₉

yes

direct product factor

102

C₂²×C₃

C₄²×D₃

yes

—

103

C₂²×C₃

C₂²×C₄×D₃

yes

—

104

C₂²×C₃

C₂⁴×D₃

yes

characteristic

105

C₂×D₃

—

106

C₂×D₃

—

107

C₃×C₄

C₂²×C₄×D₃

yes

—

108

Dic₃

—

109

C₉

C₂³×C₄²

yes

characteristic, complemented, Sylow

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₄×D₉

yes

—

114

C₂³

C₂³×D₉

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

192

—

118

C₂×C₄

C₂×C₄×D₉

yes

direct product factor

119

C₂×C₄

C₂³×D₉

yes

—

120

C₂×C₃

C₂×C₄²×D₃

yes

—

121

C₂×C₃

C₂³×C₄×D₃

yes

—

122

D₃

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₄²×D₉

yes

direct product factor

126

C₂²

C₂²×C₄×D₉

yes

—

127

C₂²

C₂⁴×D₉

yes

characteristic

128

C₄

—

129

C₄

C₂²×C₄×D₉

yes

direct product factor

130

C₃

C₂²×C₄²×D₃

yes

characteristic

131

C₂

—

132

C₂

C₂×C₄²×D₉

yes

direct product factor

133

C₂

C₂³×C₄×D₉

yes

—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₉
commutator, hypocenter	C₉	C₂³×C₄²
Frattini	C₂²×C₃	C₂⁴×D₃
Fitting	C₂²×C₄²×C₉	C₂
socle	C₂⁴×C₃	C₂²×D₃
prime radical	C₂⁴×D₉	C₂²
cosocle	C₂²×C₉	C₂⁵
LCS	G ▷ C₉
UCS	C₁ ◁ C₂²×C₄²

subgroup

G/H

center, hypercenter

C₂²×C₄²

D₉

commutator, hypocenter

C₉

C₂³×C₄²

Frattini

C₂²×C₃

C₂⁴×D₃

Fitting

C₂²×C₄²×C₉

C₂

socle

C₂⁴×C₃

C₂²×D₃

prime radical

C₂⁴×D₉

C₂²

cosocle

C₂²×C₉

C₂⁵

LCS

G ▷ C₉

UCS

C₁ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. C₂²×C₄²×C₉	C₂	semidirect
3. C₂²×C₄×Dic₉	C₂	semidirect
4. C₂×C₄²×D₉	C₂	direct (N × H)
9. C₂²×C₄×D₉	C₄	direct (N × H)
11. C₂×C₄²×C₉	C₂²	semidirect
12. C₂×C₄×Dic₉	C₂²	semidirect
13. C₄²×D₉	C₂²	direct (N × H)
21. C₂²×C₄×C₉	C₂×C₄	semidirect
23. C₂²×Dic₉	C₂×C₄	semidirect
25. C₂×C₄×D₉	C₂×C₄	direct (N × H)
27. C₄²×C₉	C₂³	semidirect
28. C₄×Dic₉	C₂³	semidirect
40. C₂²×D₉	C₄²	direct (N × H)
43. C₂×C₄×C₉	C₂²×C₄	semidirect
45. C₂×Dic₉	C₂²×C₄	semidirect
47. C₄×D₉	C₂²×C₄	direct (N × H)
51. C₂²×C₄²	D₉	direct (N × H)
63. C₂²×C₉	C₂×C₄²	semidirect
66. C₂×D₉	C₂×C₄²	direct (N × H)
68. C₄×C₉	C₂³×C₄	semidirect
69. Dic₉	C₂³×C₄	semidirect
77. C₂×C₄²	C₂×D₉	direct (N × H)
88. C₂×C₉	C₂²×C₄²	semidirect
90. D₉	C₂²×C₄²	direct (N × H)
98. C₂²×C₄	C₄×D₉	direct (N × H)
101. C₄²	C₂²×D₉	direct (N × H)
109. C₉	C₂³×C₄²	semidirect
118. C₂×C₄	C₂×C₄×D₉	direct (N × H)
125. C₂²	C₄²×D₉	direct (N × H)
129. C₄	C₂²×C₄×D₉	direct (N × H)
132. C₂	C₂×C₄²×D₉	direct (N × H)

H ≅ G/N

type

2. C₂²×C₄²×C₉

C₂

semidirect

3. C₂²×C₄×Dic₉

C₂

semidirect

4. C₂×C₄²×D₉

C₂

direct (N × H)

9. C₂²×C₄×D₉

C₄

direct (N × H)

11. C₂×C₄²×C₉

C₂²

semidirect

12. C₂×C₄×Dic₉

C₂²

semidirect

13. C₄²×D₉

C₂²

direct (N × H)

21. C₂²×C₄×C₉

C₂×C₄

semidirect

23. C₂²×Dic₉

C₂×C₄

semidirect

25. C₂×C₄×D₉

C₂×C₄

direct (N × H)

27. C₄²×C₉

C₂³

semidirect

28. C₄×Dic₉

C₂³

semidirect

40. C₂²×D₉

C₄²

direct (N × H)

43. C₂×C₄×C₉

C₂²×C₄

semidirect

45. C₂×Dic₉

C₂²×C₄

semidirect

47. C₄×D₉

C₂²×C₄

direct (N × H)

51. C₂²×C₄²

D₉

direct (N × H)

63. C₂²×C₉

C₂×C₄²

semidirect

66. C₂×D₉

C₂×C₄²

direct (N × H)

68. C₄×C₉

C₂³×C₄

semidirect

69. Dic₉

C₂³×C₄

semidirect

77. C₂×C₄²

C₂×D₉

direct (N × H)

88. C₂×C₉

C₂²×C₄²

semidirect

90. D₉

C₂²×C₄²

direct (N × H)

98. C₂²×C₄

C₄×D₉

direct (N × H)

101. C₄²

C₂²×D₉

direct (N × H)

109. C₉

C₂³×C₄²

semidirect

118. C₂×C₄

C₂×C₄×D₉

direct (N × H)

125. C₂²

C₄²×D₉

direct (N × H)

129. C₄

C₂²×C₄×D₉

direct (N × H)

132. C₂

C₂×C₄²×D₉

direct (N × H)

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

subgroup	index
C₂³×C₄×D₉	2
C₂²×C₄²×C₉	2
C₂²×C₄×Dic₉	2
C₂×C₄²×D₉	2
C₂²×C₄²×D₃	3

subgroup

index

C₂³×C₄×D₉

C₂²×C₄²×C₉

C₂²×C₄×Dic₉

C₂×C₄²×D₉

C₂²×C₄²×D₃

Subgroups — C₂²×C₄²×D₉ #11899789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C42×D9 #11899789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×D₉ #11899789