Subgroups — #36379708

Order 22599.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419226	C₃	1	3	yes	maximal
2	C₃²×3₋¹⁺²×C₃₁	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₃₁⋊C₉)	—	3	3	no	—
4	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₃₁	—	3	1	no	—
6	C₃×3₋¹⁺²×C₃₁	—	3	2	no	—
7	C₃×3₋¹⁺²×C₃₁	—	3	9	no	—
8	C₃²×C₉×C₃₁	C₉	1	3	yes	complemented
9	C₃³×C₃₁	—	3	3	no	—
10	C₃³×C₃₁	—	9	3	no	—
11	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
12	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃³×C₃₁	3₋¹⁺²	1	2	yes	—
14	3₋¹⁺²×C₃₁	—	3	18	no	—
15	3₋¹⁺²×C₃₁	—	3	9	no	—
16	C₃×C₉×C₃₁	—	3	3	no	—
17	C₃×C₉×C₃₁	—	3	9	no	—
18	C₃×C₃₁⋊C₉	—	9	3	no	—
19	#10624506	—	31	1	no	maximal, Sylow
20	C₃²×C₃₁	—	3	3	no	—
21	C₃²×C₃₁	—	3	1	no	—
22	C₃²×C₃₁	—	3	2	no	—
23	C₃²×C₃₁	—	9	1	no	—
24	C₃²×C₃₁	—	9	9	no	—
25	C₃²×C₃₁	—	9	2	no	—
26	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
27	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
28	C₉×C₃₁	—	3	9	no	—
29	C₃₁⋊C₉	—	27	3	no	—
30	C₃²×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic
31	#6866	—	31	3	no	—
32	C₃×C₃₁	—	3	3	no	—
33	C₃×C₃₁	—	3	3	no	—
34	C₃×C₃₁	—	3	1	no	—
35	C₃×C₃₁	—	9	2	no	—
36	C₃×C₃₁	#6866	1	1	yes	characteristic
37	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	1	no	—
40	C₃×3₋¹⁺²	—	3	9	no	—
41	C₃²×C₉	C₃₁⋊C₉	1	3	yes	complemented
42	C₃²⋊C₉	—	93	3	no	—
43	C₃₁	#10624506	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	3	3	no	—
45	C₃³	—	9	3	no	—
46	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
47	C₃³	#11719689	1	2	yes	—
48	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
49	3₋¹⁺²	—	3	18	no	—
50	3₋¹⁺²	—	3	9	no	—
51	C₃×C₉	—	3	3	no	—
52	C₃×C₉	—	3	9	no	—
53	C₃×C₉	—	279	3	no	—
54	C₃²	—	3	3	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	3	2	no	—
57	C₃²	—	9	1	no	—
58	C₃²	—	9	9	no	—
59	C₃²	—	9	2	no	—
60	C₃²	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
61	C₃²	#14578894	1	3	yes	—
62	C₉	—	3	9	no	—
63	C₉	—	837	3	no	—
64	C₃	—	3	3	no	—
65	C₃	—	3	1	no	—
66	C₃	—	3	3	no	—
67	C₃	—	9	2	no	—
68	C₃	#21419226	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419226
commutator	C₃³×C₃₁	C₃×C₉
Frattini	C₃⁴	C₃×C₃₁⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₁	C₃
socle	C₃×C₃₁	#6866
prime radical, cosocle	C₃⁴×C₃₁	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₁⋊C₃
hypocenter	C₃₁	#10624506
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
8. C₃²×C₉×C₃₁	C₉	semidirect
41. C₃²×C₉	C₃₁⋊C₉	semidirect
43. C₃₁	#10624506	semidirect

Maximal subgroups

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

subgroup	index
#21419226	3
C₃²×3₋¹⁺²×C₃₁	3
#10624506	31