Subgroups — #21419233

Order 7533.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊(C₃₁⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₃₁⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₁×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₁	C₃²	1	1	yes	characteristic
6	C₃²×C₃₁⋊C₃	—	3	1	no	—
7	C₃×C₉×C₃₁	—	3	1	no	—
8	C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
9	C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
10	C₃²⋊(C₃₁⋊C₃)	C₉	1	3	yes	complemented
11	C₃×C₃₁⋊C₉	—	3	1	no	—
12	C₃×C₃₁⋊C₉	—	3	3	no	—
13	C₃×C₃₁⋊C₉	—	3	1	no	—
14	C₃²×C₃₁	—	3	1	no	—
15	C₃²×C₃₁	—	3	1	no	—
16	C₃²×C₃₁	—	3	1	no	—
17	C₃²×C₃₁	C₃×C₉	1	1	yes	characteristic
18	C₃²×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₃₁	3₋¹⁺²	1	1	yes	characteristic
20	C₃²×C₃₁	3₋¹⁺²	1	1	yes	characteristic
21	C₃×C₃₁⋊C₃	—	3	3	no	—
22	C₃×C₃₁⋊C₃	—	9	1	no	—
23	C₃×C₃₁⋊C₃	—	9	1	no	—
24	C₃×C₃₁⋊C₃	—	9	1	no	—
25	C₉×C₃₁	—	9	1	no	—
26	C₃₁⋊C₉	—	3	3	no	—
27	C₃₁⋊C₉	—	9	1	no	—
28	H(3,𝔽₃)⋊C₉	—	31	1	no	maximal, Sylow
29	C₃×C₃₁	—	3	1	no	—
30	C₃×C₃₁	—	3	1	no	—
31	C₃×C₃₁	—	3	1	no	—
32	C₃×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
33	C₃×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
34	C₃×C₃₁	#731	1	1	yes	characteristic
35	C₃×C₃₁	#732	1	1	yes	characteristic
36	C₃₁⋊C₃	—	9	3	no	—
37	C₃×H(3,𝔽₃)	—	31	1	no	—
38	C₃²×C₉	—	31	1	no	—
39	C₃²⋊C₉	—	31	1	no	—
40	C₃²⋊C₉	C₃₁⋊C₃	1	1	yes	characteristic, complemented
41	C₃₁	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	93	1	no	—
43	C₃³	C₃×C₃₁⋊C₃	1	1	yes	characteristic
44	H(3,𝔽₃)	—	31	3	no	—
45	H(3,𝔽₃)	—	93	1	no	—
46	H(3,𝔽₃)	—	93	1	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	93	1	no	—
49	C₃×C₉	—	93	1	no	—
50	C₃×C₉	—	93	3	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	3	1	no	—
54	C₃²	—	279	1	no	—
55	C₃²	—	279	1	no	—
56	C₃²	—	279	1	no	—
57	C₃²	—	93	3	no	—
58	C₃²	C₉×C₃₁⋊C₃	1	1	yes	characteristic
59	C₃²	#11719686	1	1	yes	characteristic
60	C₃²	#11719687	1	1	yes	characteristic
61	C₃²	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
62	C₉	—	9	1	no	—
63	C₉	—	279	1	no	—
64	C₉	—	93	3	no	—
65	C₃	—	3	1	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	1	no	—
68	C₃	—	279	3	no	—
69	C₃	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
70	C₃	#14578895	1	1	yes	characteristic
71	C₃	#14578898	1	1	yes	characteristic
72	C₃	#14578899	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃²×C₃₁⋊C₉	C₃	semidirect
3. C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
4. C₃₁×C₃²⋊C₉	C₃	semidirect
10. C₃²⋊(C₃₁⋊C₃)	C₉	semidirect
40. C₃²⋊C₉	C₃₁⋊C₃	semidirect
41. C₃₁	H(3,𝔽₃)⋊C₉	semidirect