Subgroups — #13133565

Order 1701.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4430	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊(C₇⋊C₉)	C₃	1	1	yes	maximal, characteristic
5	#10588036	C₃	1	3	yes	maximal, complemented
6	(C₇⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
7	(C₇⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
8	#6894	—	7	1	no	maximal, Sylow
9	C₃³×C₇	C₃²	1	1	yes	characteristic
10	C₃×C₇×C₉	C₃²	1	3	yes	—
11	#4430	—	3	3	no	—
12	C₃×C₇⋊C₉	—	3	3	no	—
13	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₇⋊C₉	C₃²	1	3	yes	—
15	C₃×C₇⋊C₉	C₃²	1	3	yes	—
16	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
17	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
18	C₃×3₋¹⁺²	—	7	1	no	—
19	C₃²×C₉	—	7	1	no	—
20	C₉⋊C₉	—	7	3	no	—
21	C₉⋊C₉	—	7	3	no	—
22	C₉⋊C₉	—	7	3	no	—
23	C₃²⋊C₉	—	7	1	no	—
24	C₃²⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic
25	C₃²×C₇	—	3	1	no	—
26	C₃²×C₇	—	3	1	no	—
27	C₃²×C₇	—	3	1	no	—
28	C₃²×C₇	3₋¹⁺²	1	3	yes	—
29	C₃²×C₇	C₃³	1	1	yes	characteristic
30	C₇×C₉	—	3	3	no	—
31	C₇⋊C₉	—	3	1	no	—
32	C₇⋊C₉	—	3	1	no	—
33	C₇⋊C₉	—	3	3	no	—
34	C₇⋊C₉	—	3	3	no	—
35	C₇⋊C₉	3₋¹⁺²	1	3	yes	complemented
36	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	3₋¹⁺²	—	21	3	no	—
38	C₃×C₉	—	7	1	no	—
39	C₃×C₉	—	7	3	no	—
40	C₃×C₉	—	7	1	no	—
41	C₃×C₉	—	7	1	no	—
42	C₃×C₉	—	7	3	no	—
43	C₃×C₉	—	21	3	no	—
44	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
45	C₃×C₇	—	3	3	no	—
46	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
47	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
48	C₃×C₇	#737	1	1	yes	characteristic
49	C₃×C₇	#737	1	1	yes	characteristic
50	C₃²	—	3	1	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	3	1	no	—
53	C₃²	#4428	1	3	yes	—
54	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
55	C₉	—	3	3	no	—
56	C₉	—	7	3	no	—
57	C₉	—	21	3	no	—
58	C₉	—	21	1	no	—
59	C₉	—	21	1	no	—
60	C₉	—	21	3	no	—
61	C₇	#6894	1	1	yes	characteristic, complemented, Sylow
62	C₃	—	3	3	no	—
63	C₃	C₃×#4428	1	1	yes	characteristic
64	C₃	#10588062	1	1	yes	characteristic
65	C₃	ΓL(1,𝔽₈)×3₋¹⁺²	1	1	yes	characteristic
66	C₃	#10588065	1	1	yes	characteristic

	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	#6894
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
5. #10588036	C₃	semidirect
6. (C₇⋊C₉)⋊C₉	C₃	semidirect
7. (C₇⋊C₉)⋊C₉	C₃	semidirect
35. C₇⋊C₉	3₋¹⁺²	semidirect
61. C₇	#6894	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
C₃²×C₇⋊C₉	3
C₇×C₃²⋊C₉	3
C₃²⋊(C₇⋊C₉)	3
#10588036	3
(C₇⋊C₉)⋊C₉	3
#6894	7