Subgroups — #36379730

Order 22599.

86 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	#21419232	C₃	1	2	yes	maximal
3	C₃₁×#6869	C₃	1	1	yes	maximal, characteristic
4	#14578892	—	3	2	no	—
5	#14578887	—	3	1	no	—
6	#14578886	—	3	1	no	—
7	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
8	C₉×C₃₁⋊C₉	C₉	1	3	yes	—
9	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
10	C₃₁×C₂₇⋊C₃	—	3	1	no	—
11	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
12	C₃×C₉×C₃₁	—	3	1	no	—
13	C₃×C₉×C₃₁	—	3	2	no	—
14	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
15	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
17	C₃×C₃₁⋊C₉	—	3	3	no	—
18	C₂₇×C₃₁	—	9	1	no	—
19	C₃₁⋊C₂₇	—	9	2	no	—
20	#10624514	—	31	1	no	maximal, Sylow
21	C₃²×C₃₁	—	3	1	no	—
22	C₃²×C₃₁	—	9	1	no	—
23	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
24	C₉×C₃₁	—	3	1	no	—
25	C₉×C₃₁	—	3	1	no	—
26	C₉×C₃₁	#731	1	1	yes	characteristic
27	C₉×C₃₁	#731	1	2	yes	—
28	C₃₁⋊C₉	—	9	3	no	—
29	#6887	—	31	1	no	—
30	#6869	—	31	2	no	—
31	#6869	C₃₁⋊C₃	1	1	yes	characteristic
32	C₃×C₃₁	—	3	1	no	—
33	C₃×C₃₁	—	9	1	no	—
34	C₃×C₃₁	#6867	1	1	yes	characteristic
35	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
36	C₂₇⋊C₃	—	3	1	no	—
37	C₂₇⋊C₃	—	93	2	no	—
38	C₉⋊C₉	—	93	1	no	—
39	C₉⋊C₉	—	93	1	no	—
40	C₃²⋊C₉	—	93	1	no	—
41	C₉²	—	31	3	no	—
42	C₃₁	#10624514	1	1	yes	characteristic, complemented, Sylow
43	C₃³	#11719687	1	1	yes	characteristic
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	3	2	no	—
46	C₃×C₉	—	93	3	no	—
47	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
48	C₃×C₉	#11719686	1	1	yes	characteristic
49	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
50	C₂₇	—	9	1	no	—
51	C₂₇	—	279	2	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	9	1	no	—
54	C₃²	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
55	C₉	—	3	1	no	—
56	C₉	—	3	1	no	—
57	C₉	—	279	3	no	—
58	C₉	#14578896	1	1	yes	characteristic
59	C₉	#14578898	1	2	yes	—
60	C₃	—	3	1	no	—
61	C₃	—	9	1	no	—
62	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×#6869	C₃
socle	C₃×C₃₁	#6867
prime radical	C₃³×C₃₁	3₋¹⁺²
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	#6869	C₃₁⋊C₃
hypocenter	C₃₁	#10624514
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6869

Complements

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

N	H ≅ G/N	type
42. C₃₁	#10624514	semidirect

Maximal subgroups

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

subgroup	index
#21419296	3
#21419232	3
C₃₁×#6869	3
#10624514	31