Subgroups — #36379544

Order 22599.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578892	C₃	1	3	yes	maximal
2	C₃₁×#6887	C₃	1	1	yes	maximal, characteristic
3	#14578892	—	3	3	no	—
4	#14578892	—	3	3	no	—
5	#14578892	—	3	3	no	—
6	C₃×C₃₁⋊C₂₇	C₉	1	9	yes	complemented
7	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
8	C₃₁×C₉⋊C₉	—	3	1	no	—
9	C₃₁×C₉⋊C₉	—	3	1	no	—
10	C₃₁×C₃²⋊C₉	—	3	1	no	—
11	C₉²×C₃₁	C₉	1	3	yes	—
12	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
13	C₃×C₉×C₃₁	—	3	3	no	—
14	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
15	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃×C₉×C₃₁	3₋¹⁺²	1	3	yes	—
18	C₃×C₉×C₃₁	3₋¹⁺²	1	3	yes	—
19	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
20	C₃₁⋊C₂₇	—	3	9	no	—
21	#10624465	—	31	1	no	maximal, Sylow
22	C₃²×C₃₁	—	9	1	no	—
23	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
24	C₃²×C₃₁	C₉⋊C₉	1	3	yes	—
25	C₉×C₃₁	—	9	3	no	—
26	C₉×C₃₁	C₉²	1	1	yes	characteristic
27	C₉×C₃₁	C₃²⋊C₉	1	3	yes	—
28	C₉×C₃₁	C₉⋊C₉	1	3	yes	—
29	C₉×C₃₁	C₉⋊C₉	1	1	yes	characteristic
30	C₉×C₃₁	C₉⋊C₉	1	1	yes	characteristic
31	C₃×C₂₇⋊C₃	—	31	3	no	—
32	#6887	C₃₁⋊C₃	1	1	yes	characteristic
33	C₃×C₃₁	—	3	1	no	—
34	C₃×C₃₁	—	3	3	no	—
35	C₃×C₃₁	#6855	1	1	yes	characteristic
36	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
37	C₂₇⋊C₃	—	93	3	no	—
38	C₂₇⋊C₃	—	93	3	no	—
39	C₂₇⋊C₃	—	93	3	no	—
40	C₃×C₂₇	—	31	9	no	—
41	C₉⋊C₉	—	3	1	no	—
42	C₉⋊C₉	—	3	1	no	—
43	C₃²⋊C₉	—	3	1	no	—
44	C₉²	C₃₁⋊C₉	1	3	yes	—
45	C₃₁	#10624465	1	1	yes	characteristic, complemented, Sylow
46	C₃³	#11719689	1	1	yes	characteristic
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	C₉×C₃₁⋊C₃	1	3	yes	—
49	C₃×C₉	#11719686	1	3	yes	—
50	C₃×C₉	#11719687	1	3	yes	—
51	C₃×C₉	C₃×C₃₁⋊C₉	1	1	yes	characteristic
52	C₃×C₉	#11719689	1	1	yes	characteristic
53	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
54	C₂₇	—	93	9	no	—
55	C₃²	—	9	1	no	—
56	C₃²	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
57	C₃²	(C₃₁⋊C₉)⋊C₉	1	3	yes	—
58	C₉	—	9	3	no	—
59	C₉	C₉×C₃₁⋊C₉	1	1	yes	characteristic
60	C₉	(C₃×C₃₁⋊C₃)⋊C₉	1	3	yes	—
61	C₉	(C₃₁⋊C₉)⋊C₉	1	3	yes	—
62	C₉	#14578886	1	1	yes	characteristic
63	C₉	#14578887	1	1	yes	characteristic
64	C₃	—	3	1	no	—
65	C₃	—	3	3	no	—
66	C₃	C₃₁⋊#6855	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₉×C₃₁⋊C₉
commutator	C₉×C₃₁	C₉²
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6887	C₃
socle	C₃×C₃₁	#6855
prime radical	C₃³×C₃₁	3₋¹⁺²
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	#6887	C₃₁⋊C₃
hypocenter	C₃₁	#10624465
LCS	G ▷ C₉×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
6. C₃×C₃₁⋊C₂₇	C₉	semidirect
45. C₃₁	#10624465	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578892	3
C₃₁×#6887	3
#10624465	31