Subgroups — #13133571

Order 1701.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4430	C₃	1	2	yes	maximal
2	C₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₇⋊C₉)	C₃	1	1	yes	maximal, characteristic
4	#10588036	C₃	1	1	yes	maximal, characteristic, complemented
5	#10588035	C₃	1	1	yes	maximal, characteristic, complemented
6	(C₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
7	(C₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
8	(C₇⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
9	C₉×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
10	#6896	—	7	1	no	maximal, Sylow
11	C₃³×C₇	C₃²	1	1	yes	characteristic
12	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
13	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
14	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
15	#4430	—	3	6	no	—
16	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
17	C₃×C₇⋊C₉	C₃²	1	2	yes	—
18	C₃×C₇⋊C₉	C₃²	1	2	yes	—
19	C₃×C₇⋊C₉	C₃²	1	2	yes	—
20	C₃×C₇⋊C₉	C₃²	1	2	yes	—
21	C₃×3₋¹⁺²	—	7	2	no	—
22	C₉⋊C₉	—	7	2	no	—
23	C₉⋊C₉	—	7	2	no	—
24	C₉⋊C₉	—	7	2	no	—
25	C₉⋊C₉	—	7	1	no	—
26	C₉⋊C₉	—	7	1	no	—
27	C₃²⋊C₉	—	7	1	no	—
28	C₃²⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₉²	—	7	1	no	—
30	C₃²×C₇	—	3	2	no	—
31	C₃²×C₇	—	3	1	no	—
32	C₃²×C₇	—	3	1	no	—
33	C₃²×C₇	C₃³	1	1	yes	characteristic
34	C₇×C₉	—	3	1	no	—
35	C₇×C₉	—	3	1	no	—
36	C₇×C₉	—	3	1	no	—
37	C₇⋊C₉	—	3	2	no	—
38	C₇⋊C₉	—	3	2	no	—
39	C₇⋊C₉	—	3	1	no	—
40	C₇⋊C₉	—	3	2	no	—
41	C₇⋊C₉	3₋¹⁺²	1	6	yes	complemented
42	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
43	3₋¹⁺²	—	21	6	no	—
44	C₃×C₉	—	7	2	no	—
45	C₃×C₉	—	7	1	no	—
46	C₃×C₉	—	7	2	no	—
47	C₃×C₉	—	7	2	no	—
48	C₃×C₉	—	7	2	no	—
49	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
50	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
51	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
52	C₃×C₇	—	9	1	no	—
53	C₃×C₇	C₃×3₋¹⁺²	1	2	yes	—
54	C₃×C₇	#737	1	1	yes	characteristic
55	C₃×C₇	#737	1	1	yes	characteristic
56	C₃²	—	3	1	no	—
57	C₃²	—	3	1	no	—
58	C₃²	—	3	2	no	—
59	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
60	C₉	—	3	1	no	—
61	C₉	—	3	1	no	—
62	C₉	—	3	1	no	—
63	C₉	—	7	6	no	—
64	C₉	—	21	1	no	—
65	C₉	—	21	2	no	—
66	C₉	—	21	2	no	—
67	C₉	—	21	2	no	—
68	C₇	#6896	1	1	yes	characteristic, complemented, Sylow
69	C₃	—	9	1	no	—
70	C₃	#10588061	1	1	yes	characteristic
71	C₃	#10588062	1	1	yes	characteristic
72	C₃	ΓL(1,𝔽₈)×3₋¹⁺²	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₃²×C₇	C₃³
Fitting	C₇×C₃²⋊C₉	C₃
prime radical	C₃³×C₇	C₃²
hypercenter	C₃²⋊C₉	ΓL(1,𝔽₈)
hypocenter	C₇	#6896
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

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. (C₇⋊C₉)⋊C₉	C₃	semidirect
7. (C₇⋊C₉)⋊C₉	C₃	semidirect
8. (C₇⋊C₉)⋊C₉	C₃	semidirect
9. C₉×C₇⋊C₉	C₃	semidirect
41. C₇⋊C₉	3₋¹⁺²	semidirect
68. C₇	#6896	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4430	3
C₇×C₃²⋊C₉	3
C₃²⋊(C₇⋊C₉)	3
#10588036	3
#10588035	3
(C₇⋊C₉)⋊C₉	3
C₉×C₇⋊C₉	3
#6896	7