Subgroups — #36379714

Order 22599.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419226	C₃	1	3	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₃₁⋊C₉)	—	3	3	no	—
4	C₃⁴×C₃₁	—	3	1	no	—
5	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
6	C₃×H(3,𝔽₃)×C₃₁	—	3	1	no	—
7	C₃×H(3,𝔽₃)×C₃₁	—	3	1	no	—
8	C₃×H(3,𝔽₃)×C₃₁	—	3	9	no	—
9	C₃×H(3,𝔽₃)×C₃₁	C₉	1	3	yes	complemented
10	C₃³×C₃₁	—	3	9	no	—
11	C₃³×C₃₁	—	3	3	no	—
12	C₃³×C₃₁	—	3	3	no	—
13	C₃³×C₃₁	—	9	9	no	—
14	C₃³×C₃₁	—	9	3	no	—
15	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
16	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
18	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
19	H(3,𝔽₃)×C₃₁	—	3	9	no	—
20	H(3,𝔽₃)×C₃₁	—	3	9	no	—
21	H(3,𝔽₃)×C₃₁	—	3	9	no	—
22	C₃×C₃₁⋊C₉	—	9	3	no	—
23	#10624508	—	31	1	no	maximal, Sylow
24	C₃²×C₃₁	—	3	3	no	—
25	C₃²×C₃₁	—	3	1	no	—
26	C₃²×C₃₁	—	3	1	no	—
27	C₃²×C₃₁	—	3	1	no	—
28	C₃²×C₃₁	—	3	9	no	—
29	C₃²×C₃₁	—	9	1	no	—
30	C₃²×C₃₁	—	9	9	no	—
31	C₃²×C₃₁	—	9	1	no	—
32	C₃²×C₃₁	—	9	1	no	—
33	C₃²×C₃₁	—	9	27	no	—
34	C₃²×C₃₁	—	9	9	no	—
35	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
36	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
37	C₃₁⋊C₉	—	27	3	no	—
38	C₃²×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic
39	#6866	—	31	3	no	—
40	C₃×C₃₁	—	3	3	no	—
41	C₃×C₃₁	—	3	3	no	—
42	C₃×C₃₁	—	3	1	no	—
43	C₃×C₃₁	—	9	1	no	—
44	C₃×C₃₁	—	9	1	no	—
45	C₃×C₃₁	—	9	9	no	—
46	C₃×C₃₁	#6866	1	1	yes	characteristic
47	C₃⁴	—	3	1	no	—
48	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃×H(3,𝔽₃)	—	3	1	no	—
51	C₃×H(3,𝔽₃)	—	3	9	no	—
52	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	3	yes	complemented
53	C₃²⋊C₉	—	93	3	no	—
54	C₃₁	#10624508	1	1	yes	characteristic, complemented, Sylow
55	C₃³	—	3	9	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	3	3	no	—
58	C₃³	—	9	3	no	—
59	C₃³	—	9	9	no	—
60	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
61	C₃³	#11719689	1	1	yes	characteristic
62	C₃³	#11719689	1	1	yes	characteristic
63	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
64	H(3,𝔽₃)	—	3	9	no	—
65	H(3,𝔽₃)	—	3	9	no	—
66	H(3,𝔽₃)	—	3	9	no	—
67	C₃×C₉	—	279	3	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	1	no	—
70	C₃²	—	3	1	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	9	no	—
73	C₃²	—	9	1	no	—
74	C₃²	—	9	9	no	—
75	C₃²	—	9	1	no	—
76	C₃²	—	9	1	no	—
77	C₃²	—	9	9	no	—
78	C₃²	—	9	27	no	—
79	C₃²	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
80	C₃²	#14578894	1	3	yes	—
81	C₉	—	837	3	no	—
82	C₃	—	3	3	no	—
83	C₃	—	3	1	no	—
84	C₃	—	3	3	no	—
85	C₃	—	9	9	no	—
86	C₃	—	9	1	no	—
87	C₃	—	9	1	no	—
88	C₃	#21419226	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419226
commutator	C₃³×C₃₁	C₃×C₉
Frattini	C₃⁴	C₃×C₃₁⋊C₃
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₁	C₃
socle	C₃×C₃₁	#6866
cosocle	C₃⁴×C₃₁	C₃²
hypercenter	C₃²×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	#10624508
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #21419226	C₃	semidirect
9. C₃×H(3,𝔽₃)×C₃₁	C₉	semidirect
52. C₃×H(3,𝔽₃)	C₃₁⋊C₉	semidirect
54. C₃₁	#10624508	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×H(3,𝔽₃)×C₃₁	3
#10624508	31