Group #14518867 — subgroups

Subgroups — SL(2,𝔽₃)×#737 #14518867

Order 1944.

753 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 753 conjugacy classes of subgroups — too many to draw. Only its 274 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	Q₈×#737	C₃	1	1	yes	maximal, characteristic, complemented
2	#10619916	C₃	1	16	yes	maximal, complemented
3	SL(2,𝔽₃)×3₋¹⁺²	C₃	1	8	yes	maximal, complemented
4	SL(2,𝔽₃)×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#10619913	C₃	1	2	yes	maximal, complemented
6	#10619912	C₃	1	8	yes	maximal, complemented
7	C₃×C₉×SL(2,𝔽₃)	C₃	1	4	yes	maximal, complemented
8	C₂×C₃×#737	—	4	1	no	maximal
9	C₄×#737	—	3	1	no	—
10	C₃×#737	—	4	1	no	Sylow
11	C₃²×SL(2,𝔽₃)	C₃²	1	4	yes	—
12	Q₈×3₋¹⁺²	C₃²	1	8	yes	complemented
13	Q₈×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃×Q₈×C₉	C₃²	1	4	yes	complemented
15	Q₈⋊H(3,𝔽₃)	C₃²	1	8	yes	—
16	#6204	C₃²	1	16	yes	complemented
17	C₃×Q₈⋊C₉	C₃²	1	8	yes	complemented
18	C₉⋊SL(2,𝔽₃)	C₃²	1	48	yes	complemented
19	C₉×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
20	C₉×SL(2,𝔽₃)	C₃²	1	8	yes	—
21	C₉×SL(2,𝔽₃)	C₃²	1	8	yes	complemented
22	C₉×SL(2,𝔽₃)	C₃²	1	16	yes	complemented
23	C₂×#737	—	4	16	no	—
24	C₂×#737	—	4	2	no	—
25	C₂×#737	—	4	8	no	—
26	C₂×#737	A₄	1	1	yes	characteristic
27	C₂×C₃×3₋¹⁺²	—	4	8	no	—
28	C₂×C₃×H(3,𝔽₃)	—	4	1	no	—
29	C₂×C₃²×C₉	—	4	4	no	—
30	C₄×3₋¹⁺²	—	3	8	no	—
31	C₄×H(3,𝔽₃)	—	3	1	no	—
32	C₃×C₄×C₉	—	3	4	no	—
33	#737	—	4	16	no	—
34	#737	—	4	2	no	—
35	#737	—	4	8	no	—
36	#737	SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
37	C₃×3₋¹⁺²	—	4	8	no	—
38	C₃×H(3,𝔽₃)	—	4	1	no	—
39	C₃²×C₉	—	4	4	no	—
40	C₃²×Q₈	C₃³	1	4	yes	—
41	C₃×SL(2,𝔽₃)	—	3	12	no	—
42	C₃×SL(2,𝔽₃)	C₃³	1	8	yes	—
43	C₃×SL(2,𝔽₃)	C₃³	1	1	yes	characteristic
44	Q₈×C₉	C₃³	1	1	yes	characteristic
45	Q₈×C₉	C₃³	1	8	yes	—
46	Q₈⋊C₉	C₃³	1	2	yes	—
47	Q₈⋊C₉	C₃³	1	16	yes	—
48	C₂×C₃³	—	4	4	no	—
49	C₂×3₋¹⁺²	—	4	16	no	—
50	C₂×3₋¹⁺²	—	4	48	no	—
51	C₂×3₋¹⁺²	C₃×A₄	1	8	yes	—
52	C₂×H(3,𝔽₃)	—	4	8	no	—
53	C₂×H(3,𝔽₃)	C₃×A₄	1	1	yes	characteristic
54	C₂×C₃×C₉	—	4	8	no	—
55	C₂×C₃×C₉	—	4	8	no	—
56	C₂×C₃×C₉	—	4	16	no	—
57	C₂×C₃×C₉	—	4	1	no	—
58	C₂×C₃×C₉	—	4	8	no	—
59	C₂×C₃×C₉	C₃×A₄	1	4	yes	—
60	C₃²×C₄	—	3	4	no	—
61	C₄×C₉	—	3	8	no	—
62	C₄×C₉	—	3	1	no	—
63	C₃³	—	4	4	no	—
64	3₋¹⁺²	—	4	16	no	—
65	3₋¹⁺²	—	4	48	no	—
66	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	8	yes	complemented
67	H(3,𝔽₃)	—	4	8	no	—
68	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
69	C₃×C₉	—	4	8	no	—
70	C₃×C₉	—	4	1	no	—
71	C₃×C₉	—	4	8	no	—
72	C₃×C₉	—	4	16	no	—
73	C₃×C₉	—	4	8	no	—
74	C₃×C₉	C₃×SL(2,𝔽₃)	1	4	yes	complemented
75	C₃×Q₈	—	3	4	no	—
76	C₃×Q₈	C₃⁴	1	1	yes	characteristic
77	SL(2,𝔽₃)	—	3	8	no	—
78	SL(2,𝔽₃)	#737	1	3	yes	direct product factor
79	C₂×C₃²	—	4	1	no	—
80	C₂×C₃²	—	4	8	no	—
81	C₂×C₃²	—	12	12	no	—
82	C₂×C₃²	C₃²×A₄	1	4	yes	—
83	C₂×C₉	—	4	2	no	—
84	C₂×C₉	—	4	16	no	—
85	C₂×C₉	C₃²×A₄	1	8	yes	—
86	C₂×C₉	C₃²×A₄	1	1	yes	characteristic
87	C₃×C₄	—	3	1	no	—
88	C₃×C₄	—	9	4	no	—
89	C₃²	—	4	1	no	—
90	C₃²	—	4	8	no	—
91	C₃²	—	12	12	no	—
92	C₃²	C₃²×SL(2,𝔽₃)	1	4	yes	—
93	C₉	—	4	2	no	—
94	C₉	—	4	16	no	—
95	C₉	C₃²×SL(2,𝔽₃)	1	8	yes	—
96	C₉	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
97	Q₈	C₃×#737	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₃	—	3	4	no	—
99	C₂×C₃	—	4	3	no	—
100	C₂×C₃	—	12	8	no	—
101	C₂×C₃	C₃³×A₄	1	1	yes	characteristic
102	C₄	—	3	1	no	—
103	C₃	—	3	4	no	—
104	C₃	—	4	3	no	—
105	C₃	—	12	8	no	—
106	C₃	C₃³×SL(2,𝔽₃)	1	1	yes	characteristic
107	C₂	A₄×#737	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#737

C₃

yes

maximal, characteristic, complemented

#10619916

C₃

yes

maximal, complemented

SL(2,𝔽₃)×3₋¹⁺²

C₃

yes

maximal, complemented

SL(2,𝔽₃)×H(3,𝔽₃)

C₃

yes

maximal, characteristic

#10619913

C₃

yes

maximal, complemented

#10619912

C₃

yes

maximal, complemented

C₃×C₉×SL(2,𝔽₃)

C₃

yes

maximal, complemented

C₂×C₃×#737

—

—

—

C₃²×SL(2,𝔽₃)

C₃²

yes

—

Q₈×3₋¹⁺²

C₃²

yes

complemented

Q₈×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×Q₈×C₉

C₃²

yes

complemented

Q₈⋊H(3,𝔽₃)

C₃²

yes

—

#6204

C₃²

yes

complemented

C₃×Q₈⋊C₉

C₃²

yes

complemented

C₉⋊SL(2,𝔽₃)

C₃²

yes

complemented

C₉×SL(2,𝔽₃)

C₃²

yes

characteristic

C₉×SL(2,𝔽₃)

C₃²

yes

—

C₉×SL(2,𝔽₃)

C₃²

yes

complemented

C₉×SL(2,𝔽₃)

C₃²

yes

—

—

—

A₄

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×H(3,𝔽₃)

—

C₂×C₃²×C₉

—

C₄×3₋¹⁺²

—

C₄×H(3,𝔽₃)

—

C₃×C₄×C₉

—

—

—

—

yes

characteristic, direct product factor

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×Q₈

C₃³

yes

—

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

C₃³

yes

—

C₃×SL(2,𝔽₃)

C₃³

yes

characteristic

Q₈×C₉

C₃³

yes

characteristic

Q₈×C₉

C₃³

yes

—

Q₈⋊C₉

C₃³

yes

—

Q₈⋊C₉

C₃³

yes

—

C₂×C₃³

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₃×A₄

yes

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₃×A₄

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×A₄

yes

—

C₃²×C₄

—

C₄×C₉

—

C₄×C₉

—

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×SL(2,𝔽₃)

yes

complemented

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×SL(2,𝔽₃)

yes

complemented

C₃×Q₈

—

C₃×Q₈

C₃⁴

yes

—

yes

direct product factor

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃²×A₄

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₃²×A₄

yes

—

C₂×C₉

C₃²×A₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×SL(2,𝔽₃)

yes

—

C₉

—

C₉

—

C₉

C₃²×SL(2,𝔽₃)

yes

—

C₉

C₃²×SL(2,𝔽₃)

yes

characteristic

Q₈

C₃×#737

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

100

C₂×C₃

—

101

C₂×C₃

C₃³×A₄

yes

characteristic

102

C₄

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

C₃³×SL(2,𝔽₃)

yes

characteristic

107

C₂

A₄×#737

yes

characteristic

	subgroup	G/H
center	C₂×C₉	C₃²×A₄
commutator, cosocle	C₃×Q₈	C₃⁴
Frattini, socle	C₂×C₃	C₃³×A₄
Fitting	Q₈×#737	C₃
prime radical	SL(2,𝔽₃)×H(3,𝔽₃)	C₃
hypercenter	C₂×#737	A₄
hypocenter	Q₈	C₃×#737
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₉ ◁ C₂×#737

subgroup

G/H

center

C₂×C₉

C₃²×A₄

commutator, cosocle

C₃×Q₈

C₃⁴

Frattini, socle

C₂×C₃

C₃³×A₄

C₃

C₃

A₄

Q₈

G ▷ C₃×Q₈ ▷ Q₈

C₁ ◁ C₂×C₉ ◁ C₂×#737

N	H ≅ G/N	type
1. Q₈×#737	C₃	semidirect
2. #10619916	C₃	semidirect
3. SL(2,𝔽₃)×3₋¹⁺²	C₃	semidirect
5. #10619913	C₃	semidirect
6. #10619912	C₃	semidirect
7. C₃×C₉×SL(2,𝔽₃)	C₃	semidirect
12. Q₈×3₋¹⁺²	C₃²	semidirect
14. C₃×Q₈×C₉	C₃²	semidirect
16. #6204	C₃²	semidirect
17. C₃×Q₈⋊C₉	C₃²	semidirect
18. C₉⋊SL(2,𝔽₃)	C₃²	semidirect
21. C₉×SL(2,𝔽₃)	C₃²	semidirect
22. C₉×SL(2,𝔽₃)	C₃²	semidirect
36. #737	SL(2,𝔽₃)	direct (N × H)
66. 3₋¹⁺²	C₃×SL(2,𝔽₃)	semidirect
74. C₃×C₉	C₃×SL(2,𝔽₃)	semidirect
78. SL(2,𝔽₃)	#737	direct (N × H)
97. Q₈	C₃×#737	semidirect

H ≅ G/N

type

1. Q₈×#737

C₃

semidirect

2. #10619916

C₃

semidirect

3. SL(2,𝔽₃)×3₋¹⁺²

C₃

semidirect

5. #10619913

C₃

semidirect

6. #10619912

C₃

semidirect

7. C₃×C₉×SL(2,𝔽₃)

C₃

semidirect

12. Q₈×3₋¹⁺²

C₃²

semidirect

14. C₃×Q₈×C₉

C₃²

semidirect

16. #6204

C₃²

semidirect

17. C₃×Q₈⋊C₉

C₃²

semidirect

18. C₉⋊SL(2,𝔽₃)

C₃²

semidirect

21. C₉×SL(2,𝔽₃)

C₃²

semidirect

22. C₉×SL(2,𝔽₃)

C₃²

semidirect

36. #737

SL(2,𝔽₃)

direct (N × H)

66. 3₋¹⁺²

C₃×SL(2,𝔽₃)

semidirect

74. C₃×C₉

C₃×SL(2,𝔽₃)

semidirect

78. SL(2,𝔽₃)

#737

direct (N × H)

97. Q₈

C₃×#737

semidirect

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

subgroup	index
Q₈×#737	3
#10619916	3
SL(2,𝔽₃)×3₋¹⁺²	3
SL(2,𝔽₃)×H(3,𝔽₃)	3
#10619913	3
#10619912	3
C₃×C₉×SL(2,𝔽₃)	3
C₂×C₃×#737	4

subgroup

index

SL(2,𝔽₃)×3₋¹⁺²

C₃×C₉×SL(2,𝔽₃)

C₂×C₃×#737

Subgroups — SL(2,𝔽₃)×#737 #14518867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SL(2,𝔽3)×#737 #14518867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SL(2,𝔽₃)×#737 #14518867