Subgroups — #13091717

Order 1458.

1662 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624943	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	#92394	C₃	1	8	yes	maximal, complemented
3	#92392	C₃	1	1	yes	maximal, characteristic
4	C₃×#4208	C₃	1	4	yes	maximal, complemented
5	D₃×#737	—	3	4	no	maximal
6	3₋¹⁺⁴	—	2	32	no	—
7	3₋¹⁺⁴	—	2	4	no	—
8	3₋¹⁺⁴	C₂×C₃	1	8	yes	complemented
9	H(5,𝔽₃)	C₂×C₃	1	1	yes	characteristic
10	C₃×#737	—	2	16	no	—
11	C₃×#737	D₃	1	4	yes	complemented
12	C₃×#737	C₂×C₃	1	4	yes	complemented
13	C₂×#737	—	9	1	no	—
14	#4208	C₃²	1	1	yes	characteristic, complemented
15	#4208	C₃²	1	8	yes	complemented
16	C₃×#272	C₃²	1	4	yes	—
17	D₃×3₋¹⁺²	—	3	32	no	—
18	D₃×H(3,𝔽₃)	—	3	4	no	—
19	C₃×D₃×C₉	—	3	16	no	—
20	#737	—	2	12	no	—
21	#737	—	2	96	no	—
22	#737	—	2	96	no	—
23	#737	—	2	32	no	—
24	#737	—	2	4	no	—
25	#737	—	2	16	no	—
26	#737	—	2	96	no	—
27	#737	—	2	32	no	—
28	#737	—	2	16	no	—
29	#737	C₃⋊D₃	1	1	yes	characteristic
30	#737	C₂×C₃²	1	1	yes	characteristic
31	#737	C₂×C₃²	1	8	yes	—
32	C₃×3₋¹⁺²	—	2	96	no	—
33	C₃×3₋¹⁺²	—	2	32	no	—
34	C₃×3₋¹⁺²	—	2	16	no	—
35	C₃×3₋¹⁺²	C₃×D₃	1	32	yes	complemented
36	C₃×H(3,𝔽₃)	—	2	16	no	—
37	C₃×H(3,𝔽₃)	C₃×D₃	1	4	yes	—
38	C₃×H(3,𝔽₃)	C₂×C₃²	1	4	yes	—
39	C₃²×C₉	—	2	12	no	—
40	C₃²×C₉	C₃×D₃	1	16	yes	complemented
41	C₃²×D₃	—	3	16	no	—
42	C₂×3₋¹⁺²	—	9	8	no	—
43	C₂×H(3,𝔽₃)	—	9	1	no	—
44	C₂×C₃×C₉	—	9	4	no	—
45	#272	C₃³	1	1	yes	characteristic
46	D₃×C₉	—	3	4	no	—
47	D₃×C₉	—	3	32	no	—
48	C₃³	—	2	12	no	—
49	C₃³	C₃²×D₃	1	16	yes	—
50	3₋¹⁺²	—	2	96	no	—
51	3₋¹⁺²	—	2	96	no	—
52	3₋¹⁺²	—	2	32	no	—
53	3₋¹⁺²	—	2	4	no	—
54	3₋¹⁺²	—	2	32	no	—
55	3₋¹⁺²	—	2	96	no	—
56	3₋¹⁺²	C₃×C₃⋊D₃	1	8	yes	—
57	H(3,𝔽₃)	—	2	12	no	—
58	H(3,𝔽₃)	—	2	16	no	—
59	H(3,𝔽₃)	—	2	16	no	—
60	H(3,𝔽₃)	C₃×C₃⋊D₃	1	1	yes	characteristic
61	H(3,𝔽₃)	C₂×C₃³	1	1	yes	characteristic
62	C₃×C₉	—	2	96	no	—
63	C₃×C₉	—	2	16	no	—
64	C₃×C₉	—	2	32	no	—
65	C₃×C₉	—	2	16	no	—
66	C₃×C₉	C₃²×D₃	1	32	yes	—
67	C₃×C₉	C₃²×D₃	1	4	yes	—
68	C₃×C₉	C₃×C₃⋊D₃	1	4	yes	—
69	C₂×C₃²	—	9	4	no	—
70	C₃×D₃	—	3	4	no	—
71	C₃×D₃	—	9	16	no	—
72	C₂×C₉	—	9	8	no	—
73	C₂×C₉	—	9	1	no	—
74	C₃²	—	2	16	no	—
75	C₃²	—	9	16	no	—
76	C₃²	—	18	12	no	—
77	C₃²	C₃³×D₃	1	4	yes	—
78	C₃²	C₃²×C₃⋊D₃	1	4	yes	—
79	C₉	—	2	32	no	—
80	C₉	—	2	4	no	—
81	C₉	C₃²×C₃⋊D₃	1	8	yes	—
82	C₉	C₃²×C₃⋊D₃	1	1	yes	characteristic
83	C₂×C₃	—	9	1	no	—
84	C₂×C₃	—	27	4	no	—
85	D₃	—	3	4	no	—
86	C₃	—	3	4	no	—
87	C₃	—	3	4	no	—
88	C₃	—	6	16	no	—
89	C₃	C₃³×C₃⋊D₃	1	1	yes	characteristic
90	C₂	—	9	1	no	Sylow

	subgroup	G/H
center	C₉	C₃²×C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂×C₃³
Frattini, socle	C₃	C₃³×C₃⋊D₃
Fitting	#10624943	C₂
prime radical	#92392	C₃
hypercenter	#737	C₃⋊D₃
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₉ ◁ #737

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #10624943	C₂	semidirect
2. #92394	C₃	semidirect
4. C₃×#4208	C₃	semidirect
8. 3₋¹⁺⁴	C₂×C₃	semidirect
11. C₃×#737	D₃	semidirect
12. C₃×#737	C₂×C₃	semidirect
14. #4208	C₃²	semidirect
15. #4208	C₃²	semidirect
35. C₃×3₋¹⁺²	C₃×D₃	semidirect
40. C₃²×C₉	C₃×D₃	semidirect

Maximal subgroups

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

subgroup	index
#10624943	2
#92394	3
#92392	3
C₃×#4208	3
D₃×#737	3