Subgroups — 3₋¹⁺²×C₇⋊C₉ #13133539

Order 1701.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₇⋊C₉)	C₃	1	2	yes	maximal
4	#10588036	C₃	1	3	yes	maximal, complemented
5	#10588035	C₃	1	3	yes	maximal, complemented
6	C₉×C₇⋊C₉	C₃	1	3	yes	maximal, complemented
7	C₉×3₋¹⁺²	—	7	1	no	maximal, Sylow
8	C₃³×C₇	C₃²	1	1	yes	characteristic
9	C₇×3₋¹⁺²	C₉	1	9	yes	complemented
10	C₃×C₇×C₉	C₃²	1	3	yes	—
11	C₃×C₇⋊C₉	—	3	3	no	—
12	C₃×C₇⋊C₉	C₃²	1	6	yes	—
13	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
15	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
16	C₃×3₋¹⁺²	ΓL(1,𝔽₈)	1	1	yes	characteristic
17	C₃²×C₉	—	7	1	no	—
18	C₉⋊C₉	—	7	3	no	—
19	C₉⋊C₉	—	7	3	no	—
20	C₃²⋊C₉	—	7	2	no	—
21	C₉²	—	7	3	no	—
22	C₃²×C₇	—	3	1	no	—
23	C₃²×C₇	—	3	2	no	—
24	C₃²×C₇	C₃×C₉	1	3	yes	—
25	C₃²×C₇	C₃³	1	1	yes	characteristic
26	C₇×C₉	C₃×C₉	1	9	yes	complemented
27	C₇⋊C₉	—	3	1	no	—
28	C₇⋊C₉	—	3	1	no	—
29	C₇⋊C₉	—	3	6	no	—
30	C₇⋊C₉	3₋¹⁺²	1	3	yes	direct product factor
31	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
32	3₋¹⁺²	C₇⋊C₉	1	9	yes	direct product factor
33	C₃×C₉	—	7	1	no	—
34	C₃×C₉	—	7	6	no	—
35	C₃×C₉	—	7	1	no	—
36	C₃×C₉	—	7	1	no	—
37	C₃×C₉	—	21	3	no	—
38	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
39	C₃×C₇	—	3	3	no	—
40	C₃×C₇	C₃²×C₉	1	1	yes	characteristic
41	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
42	C₃×C₇	#737	1	2	yes	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	2	no	—
45	C₃²	C₃×C₇⋊C₉	1	3	yes	—
46	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
47	C₉	—	7	3	no	—
48	C₉	—	21	1	no	—
49	C₉	—	21	1	no	—
50	C₉	—	21	6	no	—
51	C₉	C₃×C₇⋊C₉	1	9	yes	complemented
52	C₇	C₉×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
53	C₃	—	3	3	no	—
54	C₃	ΓL(1,𝔽₈)×3₋¹⁺²	1	1	yes	characteristic
55	C₃	#10588065	1	2	yes	—
56	C₃	C₃²×C₇⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator	C₃×C₇	C₃²×C₉
Fitting	C₃×C₇×3₋¹⁺²	C₃
socle, cosocle	C₃²×C₇	C₃³
prime radical	C₃³×C₇	C₃²
hypercenter	C₃×3₋¹⁺²	ΓL(1,𝔽₈)
hypocenter	C₇	C₉×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
4. #10588036	C₃	semidirect
5. #10588035	C₃	semidirect
6. C₉×C₇⋊C₉	C₃	semidirect
9. C₇×3₋¹⁺²	C₉	semidirect
26. C₇×C₉	C₃×C₉	semidirect
30. C₇⋊C₉	3₋¹⁺²	direct (N × H)
32. 3₋¹⁺²	C₇⋊C₉	direct (N × H)
51. C₉	C₃×C₇⋊C₉	semidirect
52. C₇	C₉×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×3₋¹⁺²	3
C₃²×C₇⋊C₉	3
C₃²⋊(C₇⋊C₉)	3
#10588036	3
#10588035	3
C₉×C₇⋊C₉	3
C₉×3₋¹⁺²	7

Subgroups — 3−1+2×C7⋊C9 #13133539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×C₇⋊C₉ #13133539