Subgroups — #36379728

Order 22599.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419296	C₃	1	1	yes	maximal, characteristic
2	#21419228	C₃	1	2	yes	maximal
3	C₃₁×#6867	C₃	1	1	yes	maximal, characteristic
4	#14578887	—	3	1	no	—
5	#14578886	—	3	1	no	—
6	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
7	C₃²⋊(C₃₁⋊C₉)	—	3	2	no	—
8	C₉×C₃₁⋊C₉	C₉	1	3	yes	complemented
9	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
10	C₃₁×C₃²⋊C₉	—	3	1	no	—
11	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃×C₉×C₃₁	—	3	1	no	—
13	C₃×C₉×C₃₁	—	3	2	no	—
14	C₃×C₉×C₃₁	—	9	1	no	—
15	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
17	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
18	C₃×C₃₁⋊C₉	—	3	3	no	—
19	C₃×C₃₁⋊C₉	—	9	2	no	—
20	#10624513	—	31	1	no	maximal, Sylow
21	C₃²×C₃₁	—	9	1	no	—
22	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
23	C₃²×C₃₁	#731	1	1	yes	characteristic
24	C₃²×C₃₁	#731	1	2	yes	—
25	C₉×C₃₁	—	3	1	no	—
26	C₉×C₃₁	—	3	1	no	—
27	C₉×C₃₁	—	3	1	no	—
28	C₉×C₃₁	—	27	1	no	—
29	C₃₁⋊C₉	—	9	3	no	—
30	C₃₁⋊C₉	—	27	2	no	—
31	#6887	—	31	1	no	—
32	#6867	—	31	2	no	—
33	#6867	C₃₁⋊C₃	1	1	yes	characteristic
34	C₃×C₃₁	—	3	1	no	—
35	C₃×C₃₁	—	3	2	no	—
36	C₃×C₃₁	—	3	1	no	—
37	C₃×C₃₁	#6867	1	1	yes	characteristic
38	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
39	C₉⋊C₉	—	93	1	no	—
40	C₉⋊C₉	—	93	1	no	—
41	C₃²⋊C₉	—	3	1	no	—
42	C₃²⋊C₉	—	93	2	no	—
43	C₃²⋊C₉	—	93	1	no	—
44	C₉²	—	31	3	no	—
45	C₃₁	#10624513	1	1	yes	characteristic, complemented, Sylow
46	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	3	2	no	—
49	C₃×C₉	—	9	1	no	—
50	C₃×C₉	—	279	2	no	—
51	C₃×C₉	—	93	3	no	—
52	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
53	C₃×C₉	#11719686	1	1	yes	characteristic
54	C₃×C₉	#11719687	1	1	yes	characteristic
55	C₃²	—	9	1	no	—
56	C₃²	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
57	C₃²	#14578896	1	1	yes	characteristic
58	C₃²	#14578898	1	2	yes	—
59	C₉	—	3	1	no	—
60	C₉	—	3	1	no	—
61	C₉	—	3	1	no	—
62	C₉	—	279	3	no	—
63	C₉	—	27	1	no	—
64	C₉	—	837	2	no	—
65	C₃	—	3	1	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	2	no	—
68	C₃	#21419227	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419227
commutator	C₃×C₉×C₃₁	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6867	C₃
socle	C₃×C₃₁	#6867
prime radical	C₃³×C₃₁	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	#6867	C₃₁⋊C₃
hypocenter	C₃₁	#10624513
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6867

Complements

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

N	H ≅ G/N	type
8. C₉×C₃₁⋊C₉	C₉	semidirect
45. C₃₁	#10624513	semidirect

Maximal subgroups

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

subgroup	index
#21419296	3
#21419228	3
C₃₁×#6867	3
#10624513	31