Group #13091648 — subgroups

Subgroups — #13091648

Order 1458.

366 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10624931	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	#92394	C₃	1	2	yes	maximal, complemented
3	#92392	C₃	1	1	yes	maximal, characteristic
4	C₃×#4208	C₃	1	1	yes	maximal, characteristic, complemented
5	D₃×#731	—	3	4	no	maximal
6	3₋¹⁺⁴	C₂×C₃	1	2	yes	complemented
7	H(5,𝔽₃)	C₂×C₃	1	1	yes	characteristic
8	C₃×#737	C₂×C₃	1	1	yes	characteristic, complemented
9	#6910	—	2	8	no	—
10	#6910	—	2	4	no	—
11	C₃×#731	—	2	4	no	—
12	C₃×#731	D₃	1	4	yes	complemented
13	#4208	—	3	1	no	—
14	#4208	—	3	2	no	—
15	C₃×#272	—	3	1	no	—
16	C₃×#272	C₃²	1	1	yes	characteristic
17	D₃×3₋¹⁺²	—	3	8	no	—
18	D₃×H(3,𝔽₃)	—	3	4	no	—
19	C₃×D₃×C₉	—	3	4	no	—
20	C₂×#731	—	9	1	no	—
21	#737	—	3	1	no	—
22	#737	—	3	2	no	—
23	#737	—	6	4	no	—
24	#737	—	6	8	no	—
25	C₃×3₋¹⁺²	—	2	4	no	—
26	C₃×3₋¹⁺²	—	2	8	no	—
27	C₃×3₋¹⁺²	C₃×D₃	1	8	yes	complemented
28	C₃×H(3,𝔽₃)	—	2	4	no	—
29	C₃×H(3,𝔽₃)	—	3	1	no	—
30	C₃×H(3,𝔽₃)	—	6	4	no	—
31	C₃×H(3,𝔽₃)	C₃×D₃	1	4	yes	—
32	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
33	C₃²×C₉	C₃×D₃	1	4	yes	complemented
34	#731	—	2	24	no	—
35	#731	—	2	8	no	—
36	#731	—	2	4	no	—
37	#731	—	2	4	no	—
38	#731	C₃⋊D₃	1	1	yes	characteristic
39	C₃²×D₃	—	3	4	no	—
40	C₃²×D₃	—	9	4	no	—
41	C₂×3₋¹⁺²	—	9	2	no	—
42	C₂×H(3,𝔽₃)	—	9	1	no	—
43	C₂×C₃×C₉	—	9	1	no	—
44	#272	H(3,𝔽₃)	1	1	yes	characteristic
45	D₃×C₉	—	9	4	no	—
46	D₃×C₉	—	9	8	no	—
47	C₃³	—	3	4	no	—
48	C₃³	—	6	4	no	—
49	C₃³	C₃²×D₃	1	4	yes	—
50	3₋¹⁺²	—	2	8	no	—
51	3₋¹⁺²	—	6	8	no	—
52	3₋¹⁺²	—	6	8	no	—
53	3₋¹⁺²	—	6	8	no	—
54	3₋¹⁺²	—	6	4	no	—
55	3₋¹⁺²	—	6	8	no	—
56	3₋¹⁺²	C₃×C₃⋊D₃	1	2	yes	—
57	H(3,𝔽₃)	—	2	4	no	—
58	H(3,𝔽₃)	—	2	4	no	—
59	H(3,𝔽₃)	—	6	4	no	—
60	H(3,𝔽₃)	—	6	4	no	—
61	H(3,𝔽₃)	—	6	4	no	—
62	H(3,𝔽₃)	C₂×H(3,𝔽₃)	1	1	yes	characteristic
63	H(3,𝔽₃)	C₃×C₃⋊D₃	1	1	yes	characteristic
64	C₃×C₉	—	2	4	no	—
65	C₃×C₉	—	3	4	no	—
66	C₃×C₉	—	3	8	no	—
67	C₃×C₉	—	6	4	no	—
68	C₃×C₉	—	6	8	no	—
69	C₃×C₉	C₃×C₃⋊D₃	1	1	yes	characteristic
70	C₂×C₃²	—	9	1	no	—
71	C₂×C₃²	—	27	1	no	—
72	C₃×D₃	—	3	4	no	—
73	C₃×D₃	—	9	4	no	—
74	C₃×D₃	—	27	4	no	—
75	C₂×C₉	—	27	1	no	—
76	C₂×C₉	—	27	2	no	—
77	C₃²	—	2	4	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	6	4	no	—
80	C₃²	—	9	4	no	—
81	C₃²	—	27	4	no	—
82	C₃²	—	54	4	no	—
83	C₃²	D₃×H(3,𝔽₃)	1	4	yes	—
84	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
85	C₉	—	3	1	no	—
86	C₉	—	3	2	no	—
87	C₉	—	6	4	no	—
88	C₉	—	6	8	no	—
89	C₂×C₃	—	9	1	no	—
90	C₂×C₃	—	27	1	no	—
91	C₂×C₃	—	81	1	no	—
92	D₃	—	3	4	no	—
93	C₃	—	3	4	no	—
94	C₃	—	3	1	no	—
95	C₃	—	6	4	no	—
96	C₃	—	9	1	no	—
97	C₃	—	18	4	no	—
98	C₃	C₃⋊D₃×H(3,𝔽₃)	1	1	yes	characteristic
99	C₂	—	9	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624931

C₂

yes

maximal, characteristic, complemented, Sylow

#92394

C₃

yes

maximal, complemented

#92392

C₃

yes

maximal, characteristic

C₃×#4208

C₃

yes

maximal, characteristic, complemented

D₃×#731

—

maximal

3₋¹⁺⁴

C₂×C₃

yes

complemented

H(5,𝔽₃)

C₂×C₃

yes

characteristic

C₃×#737

C₂×C₃

yes

characteristic, complemented

—

—

—

D₃

yes

—

—

—

C₃²

yes

characteristic

D₃×3₋¹⁺²

—

D₃×H(3,𝔽₃)

—

C₃×D₃×C₉

—

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×D₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×D₃

yes

—

C₃×H(3,𝔽₃)

C₂×C₃²

yes

characteristic

C₃²×C₉

C₃×D₃

yes

—

—

—

—

C₃⋊D₃

yes

characteristic

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

#272

H(3,𝔽₃)

yes

characteristic

D₃×C₉

—

D₃×C₉

—

C₃³

—

C₃³

—

C₃³

C₃²×D₃

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×C₃⋊D₃

yes

—

—

—

—

—

—

C₂×H(3,𝔽₃)

yes

characteristic

H(3,𝔽₃)

C₃×C₃⋊D₃

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₃⋊D₃

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

D₃×H(3,𝔽₃)

yes

—

C₃²

C₃²×C₃⋊D₃

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

D₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

—

C₃

C₃⋊D₃×H(3,𝔽₃)

yes

characteristic

C₂

—

Sylow

	subgroup	G/H
center, socle	C₃	C₃⋊D₃×H(3,𝔽₃)
commutator, cosocle	C₃×H(3,𝔽₃)	C₂×C₃²
Frattini	C₃²	C₃²×C₃⋊D₃
Fitting	#10624931	C₂
prime radical	#92392	C₃
hypercenter	#731	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×H(3,𝔽₃)
LCS	G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

subgroup

G/H

center, socle

C₃

C₃⋊D₃×H(3,𝔽₃)

commutator, cosocle

C₃×H(3,𝔽₃)

C₂×C₃²

Frattini

C₃²

C₃²×C₃⋊D₃

C₂

C₃

C₃⋊D₃

C₂×H(3,𝔽₃)

G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₃ ◁ C₃² ◁ #731

N	H ≅ G/N	type
1. #10624931	C₂	semidirect
2. #92394	C₃	semidirect
4. C₃×#4208	C₃	semidirect
6. 3₋¹⁺⁴	C₂×C₃	semidirect
8. C₃×#737	C₂×C₃	semidirect
12. C₃×#731	D₃	semidirect
27. C₃×3₋¹⁺²	C₃×D₃	semidirect
33. C₃²×C₉	C₃×D₃	semidirect

H ≅ G/N

type

1. #10624931

C₂

semidirect

2. #92394

C₃

semidirect

4. C₃×#4208

C₃

semidirect

6. 3₋¹⁺⁴

C₂×C₃

semidirect

8. C₃×#737

C₂×C₃

semidirect

12. C₃×#731

D₃

semidirect

27. C₃×3₋¹⁺²

C₃×D₃

semidirect

33. C₃²×C₉

C₃×D₃

semidirect

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

subgroup	index
#10624931	2
#92394	3
#92392	3
C₃×#4208	3
D₃×#731	3

subgroup

index

Subgroups — #13091648

Conjugacy classes of subgroups

Complements

Maximal subgroups