Subgroups — C₉×H(3,𝔽₃)×C₇₃ #31801182

Order 17739.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(3,𝔽₃)×C₇₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉×C₇₃	C₃	1	4	yes	maximal, complemented
3	C₇₃×C₃²⋊C₉	C₃	1	8	yes	maximal, complemented
4	C₃³×C₇₃	C₃²	1	4	yes	—
5	H(3,𝔽₃)×C₇₃	C₉	1	9	yes	direct product factor
6	C₃×C₉×C₇₃	—	3	12	no	—
7	C₃×C₉×C₇₃	C₃²	1	8	yes	—
8	C₃×C₉×C₇₃	C₃²	1	1	yes	characteristic
9	C₃²×C₇₃	—	3	4	no	—
10	C₃²×C₇₃	—	3	8	no	—
11	C₃²×C₇₃	C₃×C₉	1	12	yes	complemented
12	C₃²×C₇₃	C₃³	1	1	yes	characteristic
13	C₉×C₇₃	—	3	8	no	—
14	C₉×C₇₃	H(3,𝔽₃)	1	3	yes	direct product factor
15	C₉×H(3,𝔽₃)	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₃×C₇₃	—	3	12	no	—
17	C₃×C₇₃	C₃²×C₉	1	1	yes	characteristic
18	C₃×C₇₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×C₇₃	#737	1	2	yes	—
20	C₃×H(3,𝔽₃)	C₃×C₇₃	1	1	yes	characteristic
21	C₃²×C₉	C₃×C₇₃	1	4	yes	complemented
22	C₃²⋊C₉	C₃×C₇₃	1	8	yes	complemented
23	C₇₃	C₉×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
24	C₃³	C₃²×C₇₃	1	4	yes	—
25	H(3,𝔽₃)	C₉×C₇₃	1	9	yes	direct product factor
26	C₃×C₉	—	3	12	no	—
27	C₃×C₉	C₃²×C₇₃	1	8	yes	—
28	C₃×C₉	C₃²×C₇₃	1	1	yes	characteristic
29	C₃²	—	3	4	no	—
30	C₃²	—	3	8	no	—
31	C₃²	C₃×C₉×C₇₃	1	12	yes	complemented
32	C₃²	C₃³×C₇₃	1	1	yes	characteristic
33	C₉	—	3	8	no	—
34	C₉	H(3,𝔽₃)×C₇₃	1	3	yes	direct product factor
35	C₃	—	3	12	no	—
36	C₃	C₃²×C₉×C₇₃	1	1	yes	characteristic
37	C₃	C₃×H(3,𝔽₃)×C₇₃	1	1	yes	characteristic
38	C₃	C₇₃×#737	1	2	yes	—

	subgroup	G/H
center	C₃×C₉×C₇₃	C₃²
commutator	C₃	C₃²×C₉×C₇₃
Frattini, cosocle	C₃²	C₃³×C₇₃
socle	C₃²×C₇₃	C₃³
prime radical	C₃×H(3,𝔽₃)×C₇₃	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₉×C₇₃	C₃	semidirect
3. C₇₃×C₃²⋊C₉	C₃	semidirect
5. H(3,𝔽₃)×C₇₃	C₉	direct (N × H)
11. C₃²×C₇₃	C₃×C₉	semidirect
14. C₉×C₇₃	H(3,𝔽₃)	direct (N × H)
15. C₉×H(3,𝔽₃)	C₇₃	direct (N × H)
21. C₃²×C₉	C₃×C₇₃	semidirect
22. C₃²⋊C₉	C₃×C₇₃	semidirect
23. C₇₃	C₉×H(3,𝔽₃)	direct (N × H)
25. H(3,𝔽₃)	C₉×C₇₃	direct (N × H)
31. C₃²	C₃×C₉×C₇₃	semidirect
34. C₉	H(3,𝔽₃)×C₇₃	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×H(3,𝔽₃)×C₇₃	3
C₃²×C₉×C₇₃	3
C₇₃×C₃²⋊C₉	3
C₉×H(3,𝔽₃)	73

Subgroups — C9×H(3,𝔽3)×C73 #31801182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×H(3,𝔽₃)×C₇₃ #31801182