Subgroups — C₃×#20238272 #31801397

Order 17739.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14520297	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊(C₇₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²×#10620321	C₃	1	1	yes	maximal, characteristic, complemented
4	#20238272	C₃	1	9	yes	maximal, direct product factor
5	C₃×3₋¹⁺²×C₇₃	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₇₃⋊C₃	—	3	1	no	—
7	#14520297	—	3	2	no	—
8	#14520297	C₃²	1	3	yes	complemented
9	C₃²⋊(C₇₃⋊C₃)	—	3	2	no	—
10	C₃²⋊(C₇₃⋊C₃)	C₃²	1	3	yes	—
11	C₃×#10620321	—	3	2	no	—
12	C₃×#10620321	—	3	1	no	—
13	C₃×#10620321	—	3	1	no	—
14	C₃×#10620321	C₃²	1	3	yes	complemented
15	C₃³×C₇₃	C₃²	1	1	yes	characteristic
16	3₋¹⁺²×C₇₃	—	3	2	no	—
17	3₋¹⁺²×C₇₃	C₃²	1	3	yes	complemented
18	C₃×C₉×C₇₃	—	3	1	no	—
19	C₃²×C₇₃	—	3	3	no	—
20	C₃²×C₇₃	H(3,𝔽₃)	1	2	yes	—
21	C₃²×C₇₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃²×C₇₃	C₃³	1	1	yes	characteristic
23	C₃×C₇₃⋊C₃	—	3	3	no	—
24	C₃×C₇₃⋊C₃	—	9	3	no	—
25	C₉×C₇₃	—	3	3	no	—
26	#10620321	—	3	3	no	—
27	#10620321	—	3	3	no	—
28	C₃×#731	—	73	1	no	maximal, Sylow
29	C₃×C₇₃	—	3	2	no	—
30	C₃×C₇₃	—	3	1	no	—
31	C₃×C₇₃	#731	1	3	yes	complemented
32	C₃×C₇₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₇₃⋊C₃	—	9	3	no	—
34	C₃×3₋¹⁺²	—	73	1	no	—
35	C₃×3₋¹⁺²	C₇₃⋊C₃	1	1	yes	characteristic, complemented
36	C₃×H(3,𝔽₃)	—	73	1	no	—
37	C₃²×C₉	—	73	1	no	—
38	#731	—	73	9	no	—
39	C₇₃	C₃×#731	1	1	yes	characteristic, complemented, Sylow
40	C₃³	—	219	1	no	—
41	C₃³	C₃×C₇₃⋊C₃	1	1	yes	characteristic
42	3₋¹⁺²	—	3	2	no	—
43	3₋¹⁺²	—	73	3	no	—
44	3₋¹⁺²	—	219	2	no	—
45	3₋¹⁺²	C₃×C₇₃⋊C₃	1	3	yes	complemented
46	H(3,𝔽₃)	—	73	3	no	—
47	H(3,𝔽₃)	—	219	2	no	—
48	C₃×C₉	—	3	1	no	—
49	C₃×C₉	—	73	3	no	—
50	C₃×C₉	—	219	2	no	—
51	C₃×C₉	—	219	1	no	—
52	C₃×C₉	—	219	1	no	—
53	C₃²	—	3	3	no	—
54	C₃²	—	657	3	no	—
55	C₃²	—	219	3	no	—
56	C₃²	C₃²⋊(C₇₃⋊C₃)	1	2	yes	—
57	C₃²	C₃²⋊(C₇₃⋊C₃)	1	1	yes	characteristic
58	C₃²	C₃²×C₇₃⋊C₃	1	1	yes	characteristic
59	C₉	—	3	3	no	—
60	C₉	—	219	3	no	—
61	C₉	—	219	3	no	—
62	C₃	—	3	2	no	—
63	C₃	—	3	1	no	—
64	C₃	—	657	3	no	—
65	C₃	#20238272	1	3	yes	direct product factor
66	C₃	C₃×C₃²⋊(C₇₃⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₇₃⋊C₃)
commutator, cosocle	C₃²×C₇₃	C₃³
Frattini	C₃²	C₃²×C₇₃⋊C₃
Fitting	C₃×3₋¹⁺²×C₇₃	C₃
socle	C₃²×C₇₃	H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₇₃⋊C₃)	C₃
hypercenter	C₃×3₋¹⁺²	C₇₃⋊C₃
hypocenter	C₇₃	C₃×#731
LCS	G ▷ C₃²×C₇₃ ▷ C₃×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×#14520297	C₃	semidirect
3. C₃²×#10620321	C₃	semidirect
4. #20238272	C₃	direct (N × H)
5. C₃×3₋¹⁺²×C₇₃	C₃	semidirect
8. #14520297	C₃²	semidirect
14. C₃×#10620321	C₃²	semidirect
17. 3₋¹⁺²×C₇₃	C₃²	semidirect
31. C₃×C₇₃	#731	semidirect
35. C₃×3₋¹⁺²	C₇₃⋊C₃	semidirect
39. C₇₃	C₃×#731	semidirect
45. 3₋¹⁺²	C₃×C₇₃⋊C₃	semidirect
65. C₃	#20238272	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#14520297	3
C₃×C₃²⋊(C₇₃⋊C₃)	3
C₃²×#10620321	3
#20238272	3
C₃×3₋¹⁺²×C₇₃	3
C₃×#731	73

Subgroups — C3×#20238272 #31801397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20238272 #31801397