Group #10589037 — subgroups

Subgroups — C₉×#483 #10589037

Order 576.

156 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₉×#106	C₂	1	2	yes	maximal, complemented
4	C₉×#103	C₂	1	2	yes	maximal, complemented
5	C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃×#483	C₃	1	1	yes	maximal, characteristic
7	C₂×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₉	—	2	1	no	—
9	C₂²×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₉	C₂²	1	2	yes	—
11	C₉×C₄⋊C₄	—	2	1	no	—
12	C₉×C₄⋊C₄	—	2	1	no	—
13	C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₉×C₂²⋊C₄	—	2	1	no	—
16	C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
18	C₃×#106	C₂×C₃	1	2	yes	—
19	C₃×#103	C₂×C₃	1	2	yes	—
20	C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
21	C₂³×C₉	D₄	1	1	yes	characteristic
22	Q₈×C₉	—	2	2	no	—
23	C₂×C₄×C₉	—	2	2	no	—
24	C₂×C₄×C₉	—	2	1	no	—
25	C₂×C₄×C₉	—	2	1	no	—
26	C₂×C₄×C₉	—	2	2	no	—
27	C₂×C₄×C₉	—	4	2	no	—
28	C₂×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₈×C₉	—	2	2	no	—
31	#483	C₉	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
33	C₂²×C₃×C₄	—	2	1	no	—
34	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₃×C₈	C₂²×C₃	1	2	yes	—
36	C₃×C₄⋊C₄	—	2	1	no	—
37	C₃×C₄⋊C₄	—	2	1	no	—
38	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
39	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
40	C₃×C₂²⋊C₄	—	2	1	no	—
41	C₂²×C₉	—	2	1	no	—
42	C₂²×C₉	—	2	1	no	—
43	C₂²×C₉	SD₁₆	1	2	yes	—
44	C₂²×C₉	C₂×D₄	1	1	yes	characteristic
45	C₄×C₉	—	2	1	no	—
46	C₄×C₉	—	4	1	no	—
47	C₄×C₉	—	4	2	no	—
48	C₄×C₉	—	4	1	no	—
49	C₄×C₉	#41	1	2	yes	—
50	C₂²⋊Q₈	C₂×C₉	1	1	yes	characteristic
51	C₂×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
52	#106	C₂×C₉	1	2	yes	complemented
53	#103	C₂×C₉	1	2	yes	complemented
54	C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic
55	C₂³×C₃	C₃×D₄	1	1	yes	characteristic
56	C₃×Q₈	—	2	2	no	—
57	C₂×C₃×C₄	—	2	2	no	—
58	C₂×C₃×C₄	—	2	1	no	—
59	C₂×C₃×C₄	—	2	1	no	—
60	C₂×C₃×C₄	—	2	2	no	—
61	C₂×C₃×C₄	—	4	2	no	—
62	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
64	C₃×C₈	—	2	2	no	—
65	C₂×C₉	—	2	2	no	—
66	C₂×C₉	#123	1	1	yes	characteristic
67	C₂×C₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₉	#137	1	1	yes	characteristic
69	C₂×Q₈	C₂²×C₉	1	1	yes	characteristic
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	C₂²×C₉	1	1	yes	characteristic
72	C₂×C₈	C₂²×C₉	1	2	yes	—
73	C₄⋊C₄	—	2	1	no	—
74	C₄⋊C₄	—	2	1	no	—
75	C₄⋊C₄	C₂²×C₉	1	2	yes	—
76	C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
77	C₂²⋊C₄	—	2	1	no	—
78	C₂²×C₃	—	2	1	no	—
79	C₂²×C₃	—	2	1	no	—
80	C₂²×C₃	C₃×SD₁₆	1	2	yes	—
81	C₂²×C₃	C₂×C₃×D₄	1	1	yes	characteristic
82	C₃×C₄	—	2	1	no	—
83	C₃×C₄	—	4	1	no	—
84	C₃×C₄	—	4	2	no	—
85	C₃×C₄	—	4	1	no	—
86	C₃×C₄	C₃×#41	1	2	yes	—
87	C₉	#483	1	1	yes	characteristic, direct product factor, Sylow
88	C₂³	D₄×C₉	1	1	yes	characteristic
89	Q₈	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	D₄×C₉	1	1	yes	characteristic
96	C₂×C₄	C₂³×C₉	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₂×C₃	—	2	2	no	—
99	C₂×C₃	C₃×#123	1	1	yes	characteristic
100	C₂×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
101	C₂×C₃	C₃×#137	1	1	yes	characteristic
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₉×SD₁₆	1	2	yes	—
105	C₂²	C₂×D₄×C₉	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	4	1	no	—
108	C₄	—	4	2	no	—
109	C₄	—	4	1	no	—
110	C₄	C₉×#41	1	2	yes	—
111	C₃	C₃×#483	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	C₉×#123	1	1	yes	characteristic
114	C₂	C₂×C₉×SD₁₆	1	1	yes	characteristic
115	C₂	C₉×#137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×C₂²⋊Q₈

C₂

yes

maximal, characteristic

C₂×C₉×C₄⋊C₄

C₂

yes

maximal, characteristic

C₉×#106

C₂

yes

maximal, complemented

C₉×#103

C₂

yes

maximal, complemented

C₉×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₃×#483

C₃

yes

maximal, characteristic

C₂×Q₈×C₉

C₂²

yes

characteristic

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²

yes

characteristic

C₂×C₈×C₉

C₂²

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂²

yes

—

C₉×C₄⋊C₄

C₂²

yes

characteristic

C₉×C₂²⋊C₄

—

C₃×C₂²⋊Q₈

C₂×C₃

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂×C₃

yes

characteristic

C₃×#106

C₂×C₃

yes

—

C₃×#103

C₂×C₃

yes

—

C₃×C₂²⋊C₈

C₂×C₃

yes

characteristic

C₂³×C₉

D₄

yes

characteristic

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

D₄

yes

characteristic

C₂×C₄×C₉

C₂³

yes

characteristic

C₈×C₉

—

#483

C₉

yes

characteristic, direct product factor, Sylow

C₂×C₃×Q₈

C₂²×C₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²×C₃

yes

characteristic

C₂×C₃×C₈

C₂²×C₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₃×C₄⋊C₄

C₂²×C₃

yes

characteristic

C₃×C₂²⋊C₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

SD₁₆

yes

—

C₂²×C₉

C₂×D₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#41

yes

—

C₂²⋊Q₈

C₂×C₉

yes

characteristic

C₂×C₄⋊C₄

C₂×C₉

yes

characteristic

#106

C₂×C₉

yes

complemented

#103

C₂×C₉

yes

complemented

C₂²⋊C₈

C₂×C₉

yes

characteristic

C₂³×C₃

C₃×D₄

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×D₄

yes

characteristic

C₂×C₃×C₄

C₂³×C₃

yes

characteristic

C₃×C₈

—

C₂×C₉

—

C₂×C₉

#123

yes

characteristic

C₂×C₉

C₂×SD₁₆

yes

characteristic

C₂×C₉

#137

yes

characteristic

C₂×Q₈

C₂²×C₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×C₉

yes

characteristic

C₂×C₈

C₂²×C₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₉

yes

—

C₄⋊C₄

C₂²×C₉

yes

characteristic

C₂²⋊C₄

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₃×SD₁₆

yes

—

C₂²×C₃

C₂×C₃×D₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₃×#41

yes

—

C₉

#483

yes

characteristic, direct product factor, Sylow

C₂³

D₄×C₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×C₉

yes

characteristic

C₂×C₄

C₂³×C₉

yes

characteristic

C₈

—

C₂×C₃

—

C₂×C₃

C₃×#123

yes

characteristic

100

C₂×C₃

C₂×C₃×SD₁₆

yes

characteristic

101

C₂×C₃

C₃×#137

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

C₉×SD₁₆

yes

—

105

C₂²

C₂×D₄×C₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

C₉×#41

yes

—

111

C₃

C₃×#483

yes

characteristic

112

C₂

—

113

C₂

C₉×#123

yes

characteristic

114

C₂

C₂×C₉×SD₁₆

yes

characteristic

115

C₂

C₉×#137

yes

characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×D₄
prime radical	C₂³×C₃	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

subgroup

G/H

center

C₂²×C₉

C₂×D₄

commutator

C₂×C₄

C₂³×C₉

Frattini, cosocle

C₂×C₃×C₄

C₂³×C₃

socle

C₂²×C₃

C₂×C₃×D₄

prime radical

C₂³×C₃

C₃×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

N	H ≅ G/N	type
3. C₉×#106	C₂	semidirect
4. C₉×#103	C₂	semidirect
31. #483	C₉	direct (N × H)
52. #106	C₂×C₉	semidirect
53. #103	C₂×C₉	semidirect
87. C₉	#483	direct (N × H)

H ≅ G/N

type

3. C₉×#106

C₂

semidirect

4. C₉×#103

C₂

semidirect

31. #483

C₉

direct (N × H)

52. #106

C₂×C₉

semidirect

53. #103

C₂×C₉

semidirect

87. C₉

#483

direct (N × H)

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

subgroup	index
C₉×C₂²⋊Q₈	2
C₂×C₉×C₄⋊C₄	2
C₉×#106	2
C₉×#103	2
C₉×C₂²⋊C₈	2
C₃×#483	3

subgroup

index

C₉×C₂²⋊Q₈

C₂×C₉×C₄⋊C₄

C₉×#106

C₉×#103

C₉×C₂²⋊C₈

C₃×#483

Subgroups — C₉×#483 #10589037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#483 #10589037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#483 #10589037