Subgroups — #36379718

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	1	no	—
7	C₃×3₋¹⁺²×C₃₁	—	3	9	no	—
8	C₃×3₋¹⁺²×C₃₁	C₉	1	3	yes	complemented
9	C₃²×C₉×C₃₁	—	3	1	no	—
10	C₃³×C₃₁	—	3	3	no	—
11	C₃³×C₃₁	—	9	3	no	—
12	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
13	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
15	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
16	3₋¹⁺²×C₃₁	—	3	9	no	—
17	3₋¹⁺²×C₃₁	—	3	9	no	—
18	3₋¹⁺²×C₃₁	—	3	9	no	—
19	C₃×C₉×C₃₁	—	3	9	no	—
20	C₃×C₉×C₃₁	—	3	3	no	—
21	C₃×C₃₁⋊C₉	—	9	3	no	—
22	#10624509	—	31	1	no	maximal, Sylow
23	C₃²×C₃₁	—	3	3	no	—
24	C₃²×C₃₁	—	3	1	no	—
25	C₃²×C₃₁	—	3	1	no	—
26	C₃²×C₃₁	—	3	1	no	—
27	C₃²×C₃₁	—	9	1	no	—
28	C₃²×C₃₁	—	9	9	no	—
29	C₃²×C₃₁	—	9	1	no	—
30	C₃²×C₃₁	—	9	1	no	—
31	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
32	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
33	C₉×C₃₁	—	3	9	no	—
34	C₃₁⋊C₉	—	27	3	no	—
35	C₃²×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic
36	#6866	—	31	3	no	—
37	C₃×C₃₁	—	3	3	no	—
38	C₃×C₃₁	—	3	3	no	—
39	C₃×C₃₁	—	3	1	no	—
40	C₃×C₃₁	—	9	1	no	—
41	C₃×C₃₁	—	9	1	no	—
42	C₃×C₃₁	#6866	1	1	yes	characteristic
43	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
44	C₃×3₋¹⁺²	—	3	1	no	—
45	C₃×3₋¹⁺²	—	3	1	no	—
46	C₃×3₋¹⁺²	—	3	9	no	—
47	C₃×3₋¹⁺²	C₃₁⋊C₉	1	3	yes	complemented
48	C₃²×C₉	—	3	1	no	—
49	C₃²⋊C₉	—	93	3	no	—
50	C₃₁	#10624509	1	1	yes	characteristic, complemented, Sylow
51	C₃³	—	3	3	no	—
52	C₃³	—	9	3	no	—
53	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
54	C₃³	#11719689	1	1	yes	characteristic
55	C₃³	#11719689	1	1	yes	characteristic
56	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
57	3₋¹⁺²	—	3	9	no	—
58	3₋¹⁺²	—	3	9	no	—
59	3₋¹⁺²	—	3	9	no	—
60	C₃×C₉	—	3	9	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	279	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	9	1	no	—
68	C₃²	—	9	9	no	—
69	C₃²	—	9	1	no	—
70	C₃²	—	9	1	no	—
71	C₃²	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
72	C₃²	#14578894	1	3	yes	—
73	C₉	—	3	9	no	—
74	C₉	—	837	3	no	—
75	C₃	—	3	3	no	—
76	C₃	—	3	1	no	—
77	C₃	—	3	3	no	—
78	C₃	—	9	1	no	—
79	C₃	—	9	1	no	—
80	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	#10624509
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. C₃×3₋¹⁺²×C₃₁	C₉	semidirect
47. C₃×3₋¹⁺²	C₃₁⋊C₉	semidirect
50. C₃₁	#10624509	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
#10624509	31