Group #72243151 — subgroups

Subgroups — 3₋¹⁺²×C₉⋊D₉ #72243151

Order 4374.

1108 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1108 conjugacy classes of subgroups — too many to draw. Only its 168 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	C₉²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	3₋¹⁺²×C₃⋊D₉	—	3	4	no	maximal
3	C₃²×C₉⋊D₉	C₃	1	1	yes	maximal, characteristic
4	C₉×C₉⋊D₉	C₃	1	3	yes	maximal, complemented
5	C₃×C₉×3₋¹⁺²	D₃	1	4	yes	—
6	C₃²×C₉²	C₂×C₃	1	1	yes	characteristic
7	C₉×C₉⋊C₉	—	2	12	no	—
8	C₉×C₃²⋊C₉	—	2	4	no	—
9	C₉³	C₂×C₃	1	3	yes	complemented
10	C₃⋊D₃×3₋¹⁺²	—	9	1	no	—
11	C₃²×C₃⋊D₉	—	3	4	no	—
12	C₃×C₉⋊D₉	—	3	1	no	—
13	C₃×C₉⋊D₉	C₃²	1	1	yes	characteristic
14	C₉×C₃⋊D₉	—	3	12	no	—
15	D₉×3₋¹⁺²	—	9	12	no	—
16	C₃²×3₋¹⁺²	C₃⋊D₃	1	1	yes	characteristic
17	C₃³×C₉	C₃×D₃	1	4	yes	—
18	C₉×3₋¹⁺²	—	2	12	no	—
19	C₉×3₋¹⁺²	—	2	36	no	—
20	C₉×3₋¹⁺²	D₉	1	12	yes	complemented
21	C₃×C₉⋊C₉	—	2	12	no	—
22	C₃×C₃²⋊C₉	—	2	4	no	—
23	C₃×C₉²	—	2	12	no	—
24	C₃×C₉²	—	2	4	no	—
25	C₃×C₉²	—	3	1	no	—
26	C₃×C₉²	—	6	4	no	—
27	C₃×C₉²	C₃×D₃	1	12	yes	—
28	C₃×C₉²	C₂×C₃²	1	1	yes	characteristic
29	#6855	—	2	24	no	—
30	C₃²×C₃⋊D₃	—	9	1	no	—
31	C₉×C₃⋊D₃	—	9	3	no	—
32	C₃×C₃⋊D₉	—	3	4	no	—
33	C₃×C₃⋊D₉	—	9	4	no	—
34	D₃×3₋¹⁺²	—	27	4	no	—
35	C₃²×D₉	—	9	12	no	—
36	C₉⋊D₉	3₋¹⁺²	1	1	yes	characteristic, direct product factor
37	C₉×D₉	—	9	36	no	—
38	C₃⁴	C₃×C₃⋊D₃	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	2	4	no	—
40	C₃×3₋¹⁺²	—	2	12	no	—
41	C₃×3₋¹⁺²	C₃⋊D₉	1	4	yes	—
42	C₃²×C₉	—	2	4	no	—
43	C₃²×C₉	—	2	12	no	—
44	C₃²×C₉	—	2	12	no	—
45	C₃²×C₉	—	3	4	no	—
46	C₃²×C₉	—	6	12	no	—
47	C₃²×C₉	—	6	4	no	—
48	C₃²×C₉	C₃×D₉	1	12	yes	—
49	C₃²×C₉	C₃²×D₃	1	4	yes	—
50	C₃²×C₉	C₃×C₃⋊D₃	1	3	yes	—
51	C₉⋊C₉	—	2	72	no	—
52	C₉⋊C₉	—	2	36	no	—
53	C₃²⋊C₉	—	2	24	no	—
54	C₃²⋊C₉	—	2	12	no	—
55	C₉²	—	2	36	no	—
56	C₉²	—	2	4	no	—
57	C₉²	—	6	36	no	—
58	C₉²	—	6	8	no	—
59	C₉²	—	6	4	no	—
60	C₉²	C₃×D₉	1	36	yes	complemented
61	C₉²	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
62	C₃×C₃⋊D₃	—	9	1	no	—
63	C₃×C₃⋊D₃	—	27	1	no	—
64	C₃²×D₃	—	27	4	no	—
65	C₂×3₋¹⁺²	—	81	1	no	—
66	C₃⋊D₉	—	3	4	no	—
67	D₃×C₉	—	27	12	no	—
68	C₃×D₉	—	9	12	no	—
69	C₃×D₉	—	27	12	no	—
70	C₃³	—	2	4	no	—
71	C₃³	—	3	1	no	—
72	C₃³	—	6	4	no	—
73	C₃³	C₃×C₃⋊D₉	1	4	yes	—
74	C₃³	C₃²×C₃⋊D₃	1	1	yes	characteristic
75	3₋¹⁺²	—	2	24	no	—
76	3₋¹⁺²	—	2	4	no	—
77	3₋¹⁺²	—	2	12	no	—
78	3₋¹⁺²	C₉⋊D₉	1	1	yes	characteristic, direct product factor
79	C₃×C₉	—	2	12	no	—
80	C₃×C₉	—	2	4	no	—
81	C₃×C₉	—	2	12	no	—
82	C₃×C₉	—	2	36	no	—
83	C₃×C₉	—	2	12	no	—
84	C₃×C₉	—	3	12	no	—
85	C₃×C₉	—	6	24	no	—
86	C₃×C₉	—	6	12	no	—
87	C₃×C₉	—	6	36	no	—
88	C₃×C₉	—	6	8	no	—
89	C₃×C₉	—	6	4	no	—
90	C₃×C₉	—	6	12	no	—
91	C₃×C₉	C₃²×D₉	1	12	yes	—
92	C₃×C₉	D₃×3₋¹⁺²	1	4	yes	—
93	C₃×C₉	C₃×C₃⋊D₉	1	12	yes	—
94	C₂×C₃²	—	81	1	no	—
95	C₃⋊D₃	—	9	1	no	—
96	C₃×D₃	—	27	4	no	—
97	C₃×D₃	—	81	4	no	—
98	C₂×C₉	—	81	3	no	—
99	D₉	—	9	12	no	—
100	C₃²	—	2	4	no	—
101	C₃²	—	2	4	no	—
102	C₃²	—	3	4	no	—
103	C₃²	—	6	4	no	—
104	C₃²	—	6	8	no	—
105	C₃²	—	6	4	no	—
106	C₃²	C₃×C₉⋊D₉	1	1	yes	characteristic
107	C₃²	C₃²×C₃⋊D₉	1	4	yes	—
108	C₃²	C₃⋊D₃×3₋¹⁺²	1	1	yes	characteristic
109	C₉	—	2	12	no	—
110	C₉	—	2	12	no	—
111	C₉	—	6	36	no	—
112	C₉	—	6	12	no	—
113	C₉	D₉×3₋¹⁺²	1	12	yes	complemented
114	C₉	C₃×C₉⋊D₉	1	3	yes	complemented
115	C₂×C₃	—	81	1	no	—
116	C₂×C₃	—	243	1	no	—
117	D₃	—	27	4	no	—
118	C₃	—	2	4	no	—
119	C₃	—	3	1	no	—
120	C₃	—	6	4	no	—
121	C₃	C₃²×C₉⋊D₉	1	1	yes	characteristic
122	C₃	3₋¹⁺²×C₃⋊D₉	1	4	yes	—
123	C₂	—	81	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

C₉²×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented, Sylow

3₋¹⁺²×C₃⋊D₉

—

maximal

C₃²×C₉⋊D₉

C₃

yes

maximal, characteristic

C₉×C₉⋊D₉

C₃

yes

maximal, complemented

C₃×C₉×3₋¹⁺²

D₃

yes

—

C₃²×C₉²

C₂×C₃

yes

characteristic

C₉×C₉⋊C₉

—

C₉×C₃²⋊C₉

—

C₉³

C₂×C₃

yes

complemented

C₃⋊D₃×3₋¹⁺²

—

C₃²×C₃⋊D₉

—

C₃×C₉⋊D₉

—

C₃×C₉⋊D₉

C₃²

yes

characteristic

C₉×C₃⋊D₉

—

D₉×3₋¹⁺²

—

C₃²×3₋¹⁺²

C₃⋊D₃

yes

characteristic

C₃³×C₉

C₃×D₃

yes

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

D₉

yes

complemented

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

C₃×C₉²

—

C₃×C₉²

—

C₃×C₉²

—

C₃×C₉²

—

C₃×C₉²

C₃×D₃

yes

—

C₃×C₉²

C₂×C₃²

yes

characteristic

#6855

—

C₃²×C₃⋊D₃

—

C₉×C₃⋊D₃

—

C₃×C₃⋊D₉

—

C₃×C₃⋊D₉

—

D₃×3₋¹⁺²

—

C₃²×D₉

—

C₉⋊D₉

3₋¹⁺²

yes

characteristic, direct product factor

C₉×D₉

—

C₃⁴

C₃×C₃⋊D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃⋊D₉

yes

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×D₉

yes

—

C₃²×C₉

C₃²×D₃

yes

—

C₃²×C₉

C₃×C₃⋊D₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

—

C₉²

—

C₉²

—

C₉²

—

C₉²

C₃×D₉

yes

complemented

C₉²

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₃⋊D₉

—

D₃×C₉

—

C₃×D₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₃⋊D₉

yes

—

C₃³

C₃²×C₃⋊D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₉⋊D₉

yes

characteristic, direct product factor

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²×D₉

yes

—

C₃×C₉

D₃×3₋¹⁺²

yes

—

C₃×C₉

C₃×C₃⋊D₉

yes

—

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

D₉

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

C₃×C₉⋊D₉

yes

characteristic

107

C₃²

C₃²×C₃⋊D₉

yes

—

108

C₃²

C₃⋊D₃×3₋¹⁺²

yes

characteristic

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

D₉×3₋¹⁺²

yes

complemented

114

C₉

C₃×C₉⋊D₉

yes

complemented

115

C₂×C₃

—

116

C₂×C₃

—

243

—

117

D₃

—

118

C₃

—

119

C₃

—

120

C₃

—

121

C₃

C₃²×C₉⋊D₉

yes

characteristic

122

C₃

3₋¹⁺²×C₃⋊D₉

yes

—

123

C₂

—

Sylow

	subgroup	G/H
center	C₃	C₃²×C₉⋊D₉
commutator, cosocle	C₃×C₉²	C₂×C₃²
Frattini, socle	C₃³	C₃²×C₃⋊D₃
Fitting	C₉²×3₋¹⁺²	C₂
prime radical	C₃²×C₉⋊D₉	C₃
hypercenter	3₋¹⁺²	C₉⋊D₉
hypocenter	C₉²	C₂×3₋¹⁺²
LCS	G ▷ C₃×C₉² ▷ C₉²
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

subgroup

G/H

center

C₃

C₃²×C₉⋊D₉

commutator, cosocle

C₃×C₉²

C₂×C₃²

Frattini, socle

C₃³

C₃²×C₃⋊D₃

Fitting

C₉²×3₋¹⁺²

C₂

prime radical

C₃²×C₉⋊D₉

C₃

hypercenter

3₋¹⁺²

C₉⋊D₉

hypocenter

C₉²

C₂×3₋¹⁺²

LCS

G ▷ C₃×C₉² ▷ C₉²

UCS

C₁ ◁ C₃ ◁ 3₋¹⁺²

N	H ≅ G/N	type
1. C₉²×3₋¹⁺²	C₂	semidirect
4. C₉×C₉⋊D₉	C₃	semidirect
9. C₉³	C₂×C₃	semidirect
20. C₉×3₋¹⁺²	D₉	semidirect
36. C₉⋊D₉	3₋¹⁺²	direct (N × H)
60. C₉²	C₃×D₉	semidirect
61. C₉²	C₂×3₋¹⁺²	semidirect
78. 3₋¹⁺²	C₉⋊D₉	direct (N × H)
113. C₉	D₉×3₋¹⁺²	semidirect
114. C₉	C₃×C₉⋊D₉	semidirect

H ≅ G/N

type

1. C₉²×3₋¹⁺²

C₂

semidirect

4. C₉×C₉⋊D₉

C₃

semidirect

9. C₉³

C₂×C₃

semidirect

20. C₉×3₋¹⁺²

D₉

semidirect

36. C₉⋊D₉

3₋¹⁺²

direct (N × H)

60. C₉²

C₃×D₉

semidirect

61. C₉²

C₂×3₋¹⁺²

semidirect

78. 3₋¹⁺²

C₉⋊D₉

direct (N × H)

113. C₉

D₉×3₋¹⁺²

semidirect

114. C₉

C₃×C₉⋊D₉

semidirect

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

subgroup	index
C₉²×3₋¹⁺²	2
3₋¹⁺²×C₃⋊D₉	3
C₃²×C₉⋊D₉	3
C₉×C₉⋊D₉	3

subgroup

index

C₉²×3₋¹⁺²

3₋¹⁺²×C₃⋊D₉

C₃²×C₉⋊D₉

C₉×C₉⋊D₉

Subgroups — 3₋¹⁺²×C₉⋊D₉ #72243151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×C9⋊D9 #72243151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×C₉⋊D₉ #72243151