Subgroups — C₉×C₇⋊H(3,𝔽₃) #13133532

Order 1701.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₇⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, complemented
4	C₃²⋊(C₇⋊C₉)	C₃	1	2	yes	maximal, complemented
5	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₉×H(3,𝔽₃)	—	7	1	no	maximal, Sylow
7	C₃³×C₇	C₃²	1	1	yes	characteristic
8	C₃²×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
9	C₃×C₇×C₉	—	3	3	no	—
10	C₃×C₇×C₉	C₃²	1	2	yes	—
11	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
12	C₇⋊H(3,𝔽₃)	C₉	1	9	yes	direct product factor
13	C₃×C₇⋊C₉	C₃²	1	6	yes	—
14	C₉×ΓL(1,𝔽₈)	—	3	9	no	—
15	C₃×H(3,𝔽₃)	—	7	1	no	—
16	C₃²×C₉	—	7	3	no	—
17	C₃²×C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
18	C₃²⋊C₉	—	7	2	no	—
19	C₃²⋊C₉	—	7	6	no	—
20	C₃²×C₇	—	3	1	no	—
21	C₃²×C₇	—	3	2	no	—
22	C₃²×C₇	C₃×C₉	1	3	yes	complemented
23	C₃²×C₇	C₃³	1	1	yes	characteristic
24	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
25	C₃×ΓL(1,𝔽₈)	—	3	6	no	—
26	C₃×ΓL(1,𝔽₈)	C₃×C₉	1	9	yes	complemented
27	C₇×C₉	—	3	2	no	—
28	C₇×C₉	H(3,𝔽₃)	1	3	yes	complemented
29	C₇⋊C₉	—	3	6	no	—
30	C₃³	—	7	3	no	—
31	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
32	H(3,𝔽₃)	—	7	9	no	—
33	C₃×C₉	—	3	3	no	—
34	C₃×C₉	—	7	6	no	—
35	C₃×C₉	—	21	9	no	—
36	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
37	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
38	C₃×C₇	—	3	3	no	—
39	C₃×C₇	C₃²×C₉	1	1	yes	characteristic
40	C₃×C₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃×C₇	#737	1	2	yes	—
42	ΓL(1,𝔽₈)	—	3	9	no	—
43	C₃²	—	3	2	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	7	9	no	—
46	C₃²	—	21	6	no	—
47	C₃²	—	21	3	no	—
48	C₃²	C₉×ΓL(1,𝔽₈)	1	3	yes	complemented
49	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
50	C₉	—	3	2	no	—
51	C₉	—	21	6	no	—
52	C₉	C₇⋊H(3,𝔽₃)	1	3	yes	direct product factor
53	C₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
54	C₃	—	3	3	no	—
55	C₃	—	21	9	no	—
56	C₃	C₃×C₉×ΓL(1,𝔽₈)	1	1	yes	characteristic
57	C₃	#10588061	1	2	yes	—
58	C₃	C₃×C₇⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃×ΓL(1,𝔽₈)
commutator	C₃×C₇	C₃²×C₉
Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
Fitting	C₃²×C₇×C₉	C₃
socle, cosocle	C₃²×C₇	C₃³
prime radical	C₃×C₇⋊H(3,𝔽₃)	C₃
hypercenter	C₃²×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	C₉×H(3,𝔽₃)
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
1. C₃²×C₇×C₉	C₃	semidirect
3. C₃×C₉×ΓL(1,𝔽₈)	C₃	semidirect
4. C₃²⋊(C₇⋊C₉)	C₃	semidirect
5. (C₃×ΓL(1,𝔽₈))⋊C₉	C₃	semidirect
12. C₇⋊H(3,𝔽₃)	C₉	direct (N × H)
17. C₃²×C₉	ΓL(1,𝔽₈)	semidirect
22. C₃²×C₇	C₃×C₉	semidirect
26. C₃×ΓL(1,𝔽₈)	C₃×C₉	semidirect
28. C₇×C₉	H(3,𝔽₃)	semidirect
48. C₃²	C₉×ΓL(1,𝔽₈)	semidirect
52. C₉	C₇⋊H(3,𝔽₃)	direct (N × H)
53. C₇	C₉×H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₇×C₉	3
C₃×C₇⋊H(3,𝔽₃)	3
C₃×C₉×ΓL(1,𝔽₈)	3
C₃²⋊(C₇⋊C₉)	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
C₉×H(3,𝔽₃)	7

Subgroups — C9×C7⋊H(3,𝔽3) #13133532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×C₇⋊H(3,𝔽₃) #13133532