Subgroups — #21419319

Order 7533.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#11719686	C₃	1	3	yes	maximal, complemented
2	C₃×C₃²⋊(C₃₁⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²⋊(C₃₁⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₃×C₃₁⋊C₃)⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₃₁×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₃₁	C₃²	1	1	yes	characteristic
8	C₃²×C₃₁⋊C₃	C₃²	1	3	yes	—
9	C₃×C₉×C₃₁	C₃²	1	3	yes	—
10	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
11	C₃×C₃₁⋊C₉	—	3	3	no	—
12	C₃×C₃₁⋊C₉	C₃²	1	2	yes	—
13	C₃×C₃₁⋊C₉	C₃²	1	3	yes	—
14	C₃×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
15	#11719686	—	3	9	no	—
16	C₃²×C₃₁	—	3	1	no	—
17	C₃²×C₃₁	—	3	2	no	—
18	C₃²×C₃₁	3₋¹⁺²	1	3	yes	complemented
19	C₃²×C₃₁	C₃³	1	1	yes	characteristic
20	C₃×C₃₁⋊C₃	—	3	3	no	—
21	C₃×C₃₁⋊C₃	—	3	3	no	—
22	C₃×C₃₁⋊C₃	—	3	6	no	—
23	C₉×C₃₁	—	3	3	no	—
24	C₃₁⋊C₉	—	3	3	no	—
25	C₃₁⋊C₉	—	3	2	no	—
26	C₃₁⋊C₉	H(3,𝔽₃)	1	3	yes	complemented
27	C₉⋊H(3,𝔽₃)	—	31	1	no	maximal, Sylow
28	C₃×C₃₁	—	3	3	no	—
29	C₃×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₃×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
31	C₃×C₃₁	#737	1	2	yes	—
32	C₃₁⋊C₃	—	9	3	no	—
33	C₃×3₋¹⁺²	—	31	3	no	—
34	C₃×H(3,𝔽₃)	—	31	1	no	—
35	C₃²×C₉	—	31	1	no	—
36	C₃²⋊C₉	—	31	1	no	—
37	C₃²⋊C₉	—	31	6	no	—
38	C₃²⋊C₉	C₃₁⋊C₃	1	1	yes	characteristic, complemented
39	C₃₁	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
40	C₃³	—	31	3	no	—
41	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic
42	3₋¹⁺²	—	93	9	no	—
43	H(3,𝔽₃)	—	93	3	no	—
44	C₃×C₉	—	31	1	no	—
45	C₃×C₉	—	31	2	no	—
46	C₃×C₉	—	31	3	no	—
47	C₃×C₉	—	93	3	no	—
48	C₃×C₉	C₃×C₃₁⋊C₃	1	3	yes	—
49	C₃²	—	3	1	no	—
50	C₃²	—	3	2	no	—
51	C₃²	—	93	3	no	—
52	C₃²	—	93	3	no	—
53	C₃²	—	93	6	no	—
54	C₃²	#11719686	1	3	yes	complemented
55	C₃²	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
56	C₉	—	3	3	no	—
57	C₉	—	31	3	no	—
58	C₉	—	93	2	no	—
59	C₉	—	93	3	no	—
60	C₃	—	3	3	no	—
61	C₃	—	279	3	no	—
62	C₃	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
63	C₃	C₃×#11719686	1	1	yes	characteristic
64	C₃	#14578913	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃³
Fitting	C₃₁×C₃²⋊C₉	C₃
hypercenter	C₃²⋊C₉	C₃₁⋊C₃
hypocenter	C₃₁	C₉⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
1. C₃×#11719686	C₃	semidirect
3. C₃²×C₃₁⋊C₉	C₃	semidirect
4. C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
5. (C₃×C₃₁⋊C₃)⋊C₉	C₃	semidirect
6. C₃₁×C₃²⋊C₉	C₃	semidirect
18. C₃²×C₃₁	3₋¹⁺²	semidirect
26. C₃₁⋊C₉	H(3,𝔽₃)	semidirect
38. C₃²⋊C₉	C₃₁⋊C₃	semidirect
39. C₃₁	C₉⋊H(3,𝔽₃)	semidirect
54. C₃²	#11719686	semidirect