Group #13152244 — subgroups

Subgroups — C₉⋊(#543⋊C₃) #13152244

Order 1728.

254 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 40 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×#543⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₉×#543	C₃	1	1	yes	maximal, characteristic, complemented
3	#543⋊C₉	C₃	1	2	yes	maximal, complemented
4	C₂²×C₉⋊A₄	—	4	1	no	maximal
5	C₂×C₉⋊SL(2,𝔽₃)	—	4	4	no	maximal
6	C₉×C₂²⋊Q₈	—	3	4	no	—
7	C₂×C₉×C₂²⋊C₄	—	3	1	no	—
8	C₂×C₉⋊A₄	—	4	2	no	—
9	C₂×C₉⋊A₄	—	4	1	no	—
10	C₉⋊SL(2,𝔽₃)	—	4	4	no	—
11	C₃×#543	C₃²	1	1	yes	characteristic
12	#543⋊C₃	—	3	1	no	—
13	C₂²×C₃×A₄	—	4	1	no	—
14	C₂×C₃×SL(2,𝔽₃)	—	4	4	no	—
15	C₂⁴×C₉	A₄	1	1	yes	characteristic
16	C₂²×C₂²⋊C₉	—	4	2	no	—
17	C₂×Q₈×C₉	A₄	1	4	yes	complemented
18	C₂²×C₄×C₉	—	3	2	no	—
19	C₂×Q₈⋊C₉	—	4	8	no	—
20	C₉×C₄⋊C₄	—	3	4	no	—
21	C₉×C₂²⋊C₄	—	3	4	no	—
22	C₂²×3₋¹⁺²	—	16	1	no	—
23	C₉⋊A₄	—	4	1	no	—
24	C₃×C₂²⋊Q₈	—	3	4	no	—
25	C₂×C₃×C₂²⋊C₄	—	3	1	no	—
26	C₂×C₃×A₄	—	4	2	no	—
27	C₂×C₃×A₄	—	4	1	no	—
28	C₃×SL(2,𝔽₃)	—	4	4	no	—
29	C₂³×C₉	—	3	1	no	—
30	C₂³×C₉	—	3	1	no	—
31	C₂³×C₉	—	4	2	no	—
32	C₂³×C₉	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₉	—	4	4	no	—
34	C₂×C₂²⋊C₉	—	4	2	no	—
35	Q₈×C₉	—	4	4	no	—
36	C₂×C₄×C₉	—	3	4	no	—
37	C₂×C₄×C₉	—	12	2	no	—
38	Q₈⋊C₉	—	4	8	no	—
39	#543	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
40	C₂×3₋¹⁺²	—	16	1	no	—
41	C₂×3₋¹⁺²	—	16	2	no	—
42	C₂⁴×C₃	C₃×A₄	1	1	yes	characteristic
43	C₂²×A₄	—	12	1	no	—
44	C₂×C₃×Q₈	C₃×A₄	1	4	yes	—
45	C₂²×C₃×C₄	—	3	2	no	—
46	C₂×SL(2,𝔽₃)	—	12	4	no	—
47	C₃×C₄⋊C₄	—	3	4	no	—
48	C₃×C₂²⋊C₄	—	3	4	no	—
49	C₂²×C₃²	—	16	1	no	—
50	C₃×A₄	—	4	1	no	—
51	C₂²×C₉	—	3	1	no	—
52	C₂²×C₉	—	3	1	no	—
53	C₂²×C₉	—	4	1	no	—
54	C₂²×C₉	—	6	2	no	—
55	C₂²×C₉	—	12	1	no	—
56	C₂²×C₉	—	16	2	no	—
57	C₂²×C₉	C₂²⋊A₄	1	1	yes	characteristic
58	C₂²⋊C₉	—	4	2	no	—
59	C₄×C₉	—	6	4	no	—
60	C₂²⋊Q₈	—	3	4	no	—
61	C₂×C₂²⋊C₄	—	3	1	no	—
62	3₋¹⁺²	—	16	1	no	Sylow
63	C₂³×C₃	—	3	1	no	—
64	C₂³×C₃	—	3	1	no	—
65	C₂³×C₃	—	4	2	no	—
66	C₂³×C₃	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
67	C₂×A₄	—	12	2	no	—
68	C₂×A₄	—	12	1	no	—
69	C₃×Q₈	—	4	4	no	—
70	C₂×C₃×C₄	—	3	4	no	—
71	C₂×C₃×C₄	—	12	2	no	—
72	SL(2,𝔽₃)	—	12	4	no	—
73	C₂×C₃²	—	16	1	no	—
74	C₂×C₃²	—	16	2	no	—
75	C₂×C₉	—	6	1	no	—
76	C₂×C₉	—	6	1	no	—
77	C₂×C₉	—	16	2	no	—
78	C₂×C₉	—	16	4	no	—
79	C₂×C₉	C₂²⋊SL(2,𝔽₃)	1	1	yes	characteristic
80	C₂×C₉	#997	1	2	yes	—
81	C₂⁴	C₉⋊A₄	1	1	yes	characteristic
82	C₂×Q₈	C₉⋊A₄	1	4	yes	complemented
83	C₂²×C₄	—	3	2	no	—
84	C₄⋊C₄	—	3	4	no	—
85	C₂²⋊C₄	—	3	4	no	—
86	C₂²×C₃	—	3	1	no	—
87	C₂²×C₃	—	3	1	no	—
88	C₂²×C₃	—	4	1	no	—
89	C₂²×C₃	—	6	2	no	—
90	C₂²×C₃	—	12	1	no	—
91	C₂²×C₃	—	48	1	no	—
92	C₂²×C₃	C₃×C₂²⋊A₄	1	1	yes	characteristic
93	A₄	—	12	1	no	—
94	C₃×C₄	—	6	4	no	—
95	C₃²	—	16	1	no	—
96	C₉	—	16	2	no	—
97	C₉	#543⋊C₃	1	1	yes	characteristic, complemented
98	C₂³	—	3	1	no	—
99	C₂³	—	3	1	no	—
100	C₂³	—	4	2	no	—
101	C₂³	C₉⋊SL(2,𝔽₃)	1	1	yes	characteristic, complemented
102	Q₈	—	4	4	no	—
103	C₂×C₄	—	3	4	no	—
104	C₂×C₄	—	12	2	no	—
105	C₂×C₃	—	6	1	no	—
106	C₂×C₃	—	6	1	no	—
107	C₂×C₃	—	48	1	no	—
108	C₂×C₃	—	48	2	no	—
109	C₂×C₃	C₃×C₂²⋊SL(2,𝔽₃)	1	1	yes	characteristic
110	C₂×C₃	C₃×#997	1	2	yes	—
111	C₂²	—	3	1	no	—
112	C₂²	—	3	1	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	6	2	no	—
115	C₂²	—	12	1	no	—
116	C₂²	C₉⋊(C₂²⋊A₄)	1	1	yes	characteristic
117	C₄	—	6	4	no	—
118	C₃	—	48	1	no	—
119	C₃	C₃×#543⋊C₃	1	1	yes	characteristic
120	C₂	—	6	1	no	—
121	C₂	—	6	1	no	—
122	C₂	C₉⋊(C₂²⋊SL(2,𝔽₃))	1	1	yes	characteristic
123	C₂	C₉⋊#997	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#543⋊C₃

C₃

yes

maximal, characteristic

C₉×#543

C₃

yes

maximal, characteristic, complemented

#543⋊C₉

C₃

yes

maximal, complemented

C₂²×C₉⋊A₄

—

maximal

C₂×C₉⋊SL(2,𝔽₃)

—

maximal

C₉×C₂²⋊Q₈

—

C₂×C₉×C₂²⋊C₄

—

C₂×C₉⋊A₄

—

C₂×C₉⋊A₄

—

C₉⋊SL(2,𝔽₃)

—

C₃×#543

C₃²

yes

characteristic

#543⋊C₃

—

C₂²×C₃×A₄

—

C₂×C₃×SL(2,𝔽₃)

—

C₂⁴×C₉

A₄

yes

characteristic

C₂²×C₂²⋊C₉

—

C₂×Q₈×C₉

A₄

yes

complemented

C₂²×C₄×C₉

—

C₂×Q₈⋊C₉

—

C₉×C₄⋊C₄

—

C₉×C₂²⋊C₄

—

C₂²×3₋¹⁺²

—

C₉⋊A₄

—

C₃×C₂²⋊Q₈

—

C₂×C₃×C₂²⋊C₄

—

C₂×C₃×A₄

—

C₂×C₃×A₄

—

C₃×SL(2,𝔽₃)

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

SL(2,𝔽₃)

yes

characteristic, complemented

C₂×C₂²⋊C₉

—

C₂×C₂²⋊C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

Q₈⋊C₉

—

#543

3₋¹⁺²

yes

characteristic, complemented, Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂⁴×C₃

C₃×A₄

yes

characteristic

C₂²×A₄

—

C₂×C₃×Q₈

C₃×A₄

yes

—

C₂²×C₃×C₄

—

C₂×SL(2,𝔽₃)

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₂²×C₃²

—

C₃×A₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂²⋊A₄

yes

characteristic

C₂²⋊C₉

—

C₄×C₉

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

3₋¹⁺²

—

Sylow

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₃×SL(2,𝔽₃)

yes

characteristic

C₂×A₄

—

C₂×A₄

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

SL(2,𝔽₃)

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²⋊SL(2,𝔽₃)

yes

characteristic

C₂×C₉

#997

yes

—

C₂⁴

C₉⋊A₄

yes

characteristic

C₂×Q₈

C₉⋊A₄

yes

complemented

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₃×C₂²⋊A₄

yes

characteristic

A₄

—

C₃×C₄

—

C₃²

—

C₉

—

C₉

#543⋊C₃

yes

characteristic, complemented

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₉⋊SL(2,𝔽₃)

yes

characteristic, complemented

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₃

—

106

C₂×C₃

—

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

C₃×C₂²⋊SL(2,𝔽₃)

yes

characteristic

110

C₂×C₃

C₃×#997

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₉⋊(C₂²⋊A₄)

yes

characteristic

117

C₄

—

118

C₃

—

119

C₃

C₃×#543⋊C₃

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

C₉⋊(C₂²⋊SL(2,𝔽₃))

yes

characteristic

123

C₂

C₉⋊#997

yes

—

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₃×C₂²⋊A₄
commutator, cosocle	C₃×#543	C₃²
Fitting	C₉×#543	C₃
prime radical	C₃×#543⋊C₃	C₃
hypercenter	C₂²×C₉	C₂²⋊A₄
hypocenter	#543	3₋¹⁺²
LCS	G ▷ C₃×#543 ▷ #543
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₉

subgroup

G/H

center, Frattini, socle

C₂²×C₃

C₃×C₂²⋊A₄

C₃²

C₃

C₃×#543⋊C₃

C₃

hypercenter

C₂²×C₉

C₂²⋊A₄

3₋¹⁺²

G ▷ C₃×#543 ▷ #543

C₁ ◁ C₂²×C₃ ◁ C₂²×C₉

N	H ≅ G/N	type
2. C₉×#543	C₃	semidirect
3. #543⋊C₉	C₃	semidirect
17. C₂×Q₈×C₉	A₄	semidirect
32. C₂³×C₉	SL(2,𝔽₃)	semidirect
39. #543	3₋¹⁺²	semidirect
82. C₂×Q₈	C₉⋊A₄	semidirect
97. C₉	#543⋊C₃	semidirect
101. C₂³	C₉⋊SL(2,𝔽₃)	semidirect

H ≅ G/N

type

2. C₉×#543

C₃

semidirect

3. #543⋊C₉

C₃

semidirect

17. C₂×Q₈×C₉

A₄

semidirect

32. C₂³×C₉

SL(2,𝔽₃)

semidirect

39. #543

3₋¹⁺²

semidirect

82. C₂×Q₈

C₉⋊A₄

semidirect

97. C₉

#543⋊C₃

semidirect

101. C₂³

C₉⋊SL(2,𝔽₃)

semidirect

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

subgroup	index
C₃×#543⋊C₃	3
C₉×#543	3
#543⋊C₉	3
C₂²×C₉⋊A₄	4
C₂×C₉⋊SL(2,𝔽₃)	4

subgroup

index

C₃×#543⋊C₃

C₉×#543

#543⋊C₉

C₂²×C₉⋊A₄

C₂×C₉⋊SL(2,𝔽₃)

Subgroups — C₉⋊(#543⋊C₃) #13152244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9⋊(#543⋊C3) #13152244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉⋊(#543⋊C₃) #13152244