Subgroups — #13133553

Order 1701.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#4427	C₃	1	3	yes	maximal, complemented
4	C₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²⋊(C₇⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
6	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	6	yes	maximal, complemented
7	C₉⋊H(3,𝔽₃)	—	7	1	no	maximal, Sylow
8	C₃³×C₇	C₃²	1	1	yes	characteristic
9	C₃²×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
10	C₃×C₇×C₉	C₃²	1	3	yes	—
11	C₇⋊H(3,𝔽₃)	—	3	3	no	—
12	C₃×C₇⋊C₉	—	3	3	no	—
13	C₃×C₇⋊C₉	C₃²	1	2	yes	—
14	C₃×C₇⋊C₉	C₃²	1	3	yes	—
15	C₃×C₇⋊C₉	C₃²	1	1	yes	characteristic
16	#4427	—	3	9	no	—
17	C₃×3₋¹⁺²	—	7	3	no	—
18	C₃×H(3,𝔽₃)	—	7	1	no	—
19	C₃²×C₉	—	7	1	no	—
20	C₃²⋊C₉	—	7	1	no	—
21	C₃²⋊C₉	—	7	6	no	—
22	C₃²⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₃²×C₇	—	3	1	no	—
24	C₃²×C₇	—	3	2	no	—
25	C₃²×C₇	3₋¹⁺²	1	3	yes	complemented
26	C₃²×C₇	C₃³	1	1	yes	characteristic
27	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
28	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
29	C₃×ΓL(1,𝔽₈)	—	3	6	no	—
30	C₇×C₉	—	3	3	no	—
31	C₇⋊C₉	—	3	3	no	—
32	C₇⋊C₉	—	3	2	no	—
33	C₇⋊C₉	H(3,𝔽₃)	1	3	yes	complemented
34	C₃³	—	7	3	no	—
35	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
36	3₋¹⁺²	—	21	9	no	—
37	H(3,𝔽₃)	—	21	3	no	—
38	C₃×C₉	—	7	1	no	—
39	C₃×C₉	—	7	2	no	—
40	C₃×C₉	—	7	3	no	—
41	C₃×C₉	—	21	3	no	—
42	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
43	C₃×C₇	—	3	3	no	—
44	C₃×C₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₇	C₃×3₋¹⁺²	1	1	yes	characteristic
46	C₃×C₇	#737	1	2	yes	—
47	ΓL(1,𝔽₈)	—	9	3	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	3	2	no	—
50	C₃²	—	21	3	no	—
51	C₃²	—	21	3	no	—
52	C₃²	—	21	6	no	—
53	C₃²	#4427	1	3	yes	complemented
54	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
55	C₉	—	3	3	no	—
56	C₉	—	7	3	no	—
57	C₉	—	21	2	no	—
58	C₉	—	21	3	no	—
59	C₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
60	C₃	—	3	3	no	—
61	C₃	—	63	3	no	—
62	C₃	C₃×#4427	1	1	yes	characteristic
63	C₃	ΓL(1,𝔽₈)×H(3,𝔽₃)	1	1	yes	characteristic
64	C₃	#10588065	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	C₃×C₇⋊H(3,𝔽₃)	C₃
hypercenter	C₃²⋊C₉	ΓL(1,𝔽₈)
hypocenter	C₇	C₉⋊H(3,𝔽₃)
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
2. C₃²×C₇⋊C₉	C₃	semidirect
3. C₃×#4427	C₃	semidirect
4. C₇×C₃²⋊C₉	C₃	semidirect
5. C₃²⋊(C₇⋊C₉)	C₃	semidirect
6. (C₃×ΓL(1,𝔽₈))⋊C₉	C₃	semidirect
22. C₃²⋊C₉	ΓL(1,𝔽₈)	semidirect
25. C₃²×C₇	3₋¹⁺²	semidirect
33. C₇⋊C₉	H(3,𝔽₃)	semidirect
53. C₃²	#4427	semidirect
59. 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
C₃×C₇⋊H(3,𝔽₃)	3
C₃²×C₇⋊C₉	3
C₃×#4427	3
C₇×C₃²⋊C₉	3
C₃²⋊(C₇⋊C₉)	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
C₉⋊H(3,𝔽₃)	7