Subgroups — C₃²×#14578892 #36382868

Order 22599.

944 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578892	C₃	1	36	yes	maximal, direct product factor
2	C₃²×C₃₁⋊C₂₇	C₃	1	3	yes	maximal, complemented
3	C₃³×C₉×C₃₁	C₃	1	1	yes	maximal, characteristic
4	#14578892	C₃²	1	81	yes	direct product factor
5	C₃×C₃₁⋊C₂₇	C₃²	1	36	yes	complemented
6	C₃⁴×C₃₁	C₉	1	1	yes	characteristic
7	C₃²×C₉×C₃₁	C₉	1	24	yes	—
8	C₃²×C₉×C₃₁	C₉	1	2	yes	—
9	C₃²×C₉×C₃₁	C₃²	1	12	yes	—
10	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
11	C₃³×C₃₁	—	3	9	no	—
12	C₃³×C₃₁	C₃×C₉	1	12	yes	—
13	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
14	C₃×C₉×C₃₁	C₃×C₉	1	72	yes	—
15	C₃×C₉×C₃₁	C₃×C₉	1	24	yes	—
16	C₃×C₉×C₃₁	C₃×C₉	1	8	yes	—
17	C₃×C₉×C₃₁	C₃³	1	4	yes	—
18	C₃×C₉×C₃₁	C₃³	1	9	yes	—
19	C₃₁⋊C₂₇	C₃³	1	27	yes	complemented
20	C₃²×C₂₇⋊C₃	—	31	1	no	maximal, Sylow
21	C₃²×C₃₁	—	3	36	no	—
22	C₃²×C₃₁	C₂₇⋊C₃	1	9	yes	complemented
23	C₃²×C₃₁	C₃²×C₉	1	9	yes	—
24	C₃²×C₃₁	C₃²×C₉	1	4	yes	—
25	C₉×C₃₁	C₃²×C₉	1	8	yes	—
26	C₉×C₃₁	C₃²×C₉	1	18	yes	—
27	C₉×C₃₁	C₃⁴	1	1	yes	characteristic
28	C₃³×C₉	C₃₁⋊C₃	1	1	yes	characteristic
29	C₃×C₂₇⋊C₃	—	31	36	no	—
30	C₃²×C₂₇	—	31	3	no	—
31	C₃×C₃₁	—	3	9	no	—
32	C₃×C₃₁	C₃×C₂₇⋊C₃	1	12	yes	complemented
33	C₃×C₃₁	C₃³×C₉	1	1	yes	characteristic
34	C₃⁴	C₃₁⋊C₉	1	1	yes	characteristic
35	C₃²×C₉	C₃₁⋊C₉	1	2	yes	—
36	C₃²×C₉	C₃₁⋊C₉	1	24	yes	—
37	C₃²×C₉	C₃×C₃₁⋊C₃	1	12	yes	—
38	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
39	C₂₇⋊C₃	—	31	81	no	—
40	C₃×C₂₇	—	31	36	no	—
41	C₃₁	C₃²×C₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	3	9	no	—
43	C₃³	C₃×C₃₁⋊C₉	1	12	yes	—
44	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
45	C₃×C₉	C₃×C₃₁⋊C₉	1	72	yes	—
46	C₃×C₉	C₃×C₃₁⋊C₉	1	24	yes	—
47	C₃×C₉	C₃×C₃₁⋊C₉	1	8	yes	—
48	C₃×C₉	C₃²×C₃₁⋊C₃	1	4	yes	—
49	C₃×C₉	C₃²×C₃₁⋊C₃	1	9	yes	—
50	C₂₇	—	31	27	no	—
51	C₃²	—	3	36	no	—
52	C₃²	#14578892	1	9	yes	direct product factor
53	C₃²	C₃²×C₃₁⋊C₉	1	9	yes	—
54	C₃²	C₃²×C₃₁⋊C₉	1	4	yes	—
55	C₉	C₃²×C₃₁⋊C₉	1	18	yes	—
56	C₉	C₃²×C₃₁⋊C₉	1	8	yes	—
57	C₉	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
58	C₃	—	3	9	no	—
59	C₃	C₃×#14578892	1	12	yes	direct product factor
60	C₃	C₃³×C₃₁⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₉	C₃×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃³×C₉
Frattini	C₉	C₃³×C₃₁⋊C₃
Fitting	C₃³×C₉×C₃₁	C₃
socle	C₃³×C₃₁	C₃×C₉
prime radical	C₃⁴×C₃₁	C₉
cosocle	C₉×C₃₁	C₃⁴
hypercenter	C₃³×C₉	C₃₁⋊C₃
hypocenter	C₃₁	C₃²×C₂₇⋊C₃
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃²×C₉ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
1. C₃×#14578892	C₃	direct (N × H)
2. C₃²×C₃₁⋊C₂₇	C₃	semidirect
4. #14578892	C₃²	direct (N × H)
5. C₃×C₃₁⋊C₂₇	C₃²	semidirect
19. C₃₁⋊C₂₇	C₃³	semidirect
22. C₃²×C₃₁	C₂₇⋊C₃	semidirect
32. C₃×C₃₁	C₃×C₂₇⋊C₃	semidirect
41. C₃₁	C₃²×C₂₇⋊C₃	semidirect
52. C₃²	#14578892	direct (N × H)
59. C₃	C₃×#14578892	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#14578892	3
C₃²×C₃₁⋊C₂₇	3
C₃³×C₉×C₃₁	3
C₃²×C₂₇⋊C₃	31

Subgroups — C32×#14578892 #36382868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#14578892 #36382868