Subgroups — C₃²×#15688521 #41008140

Order 26973.

1052 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758608	C₃	1	3	yes	maximal
2	C₃×#15688521	C₃	1	36	yes	maximal, direct product factor
3	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	—	3	24	no	—
5	C₃×#11758608	C₃²	1	36	yes	—
6	C₃²×#66524	—	3	3	no	—
7	#15688521	C₃²	1	81	yes	direct product factor
8	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₃₇	—	3	8	no	—
10	C₃²×C₉×C₃₇	—	3	1	no	—
11	C₃²×C₉×C₃₇	C₃²	1	12	yes	—
12	C₃³×C₃₇	—	3	9	no	—
13	C₃³×C₃₇	H(3,𝔽₃)	1	8	yes	—
14	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃³×C₃₇	C₃³	1	4	yes	—
16	C₃×C₉×C₃₇	—	3	24	no	—
17	C₃×C₉×C₃₇	—	3	12	no	—
18	C₃×C₉×C₃₇	C₃³	1	9	yes	—
19	#11758608	—	3	72	no	—
20	#11758608	C₃³	1	27	yes	—
21	C₃×#66524	—	3	36	no	—
22	C₃²×#733	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	24	no	—
24	C₃²×C₃₇	—	3	12	no	—
25	C₃²×C₃₇	#733	1	9	yes	complemented
26	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	8	yes	—
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
28	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
29	C₉×C₃₇	—	3	9	no	—
30	#66524	—	3	27	no	—
31	C₃²×3₋¹⁺²	—	37	3	no	—
32	C₃³×C₉	C₃₇⋊C₃	1	1	yes	characteristic
33	C₃×#733	—	37	36	no	—
34	C₃×C₃₇	—	3	8	no	—
35	C₃×C₃₇	—	3	1	no	—
36	C₃×C₃₇	C₃×#733	1	12	yes	complemented
37	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	37	36	no	—
40	C₃×3₋¹⁺²	—	111	24	no	—
41	C₃²×C₉	—	3	8	no	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	—	111	3	no	—
44	C₃²×C₉	C₃×C₃₇⋊C₃	1	12	yes	—
45	#733	—	37	81	no	—
46	C₃₇	C₃²×#733	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	3	9	no	—
48	C₃³	C₃²⋊(C₃₇⋊C₃)	1	8	yes	—
49	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
50	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
51	3₋¹⁺²	—	37	27	no	—
52	3₋¹⁺²	—	111	72	no	—
53	C₃×C₉	—	3	24	no	—
54	C₃×C₉	—	3	12	no	—
55	C₃×C₉	—	111	36	no	—
56	C₃×C₉	C₃²×C₃₇⋊C₃	1	9	yes	—
57	C₃²	—	3	24	no	—
58	C₃²	—	3	12	no	—
59	C₃²	#15688521	1	9	yes	direct product factor
60	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	8	yes	—
61	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	4	yes	—
62	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
63	C₉	—	3	9	no	—
64	C₉	—	111	27	no	—
65	C₃	—	3	8	no	—
66	C₃	—	3	1	no	—
67	C₃	C₃×#15688521	1	12	yes	direct product factor
68	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃³×C₉×C₃₇	C₃
socle	C₃³×C₃₇	H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃³×C₉	C₃₇⋊C₃
hypocenter	C₃₇	C₃²×#733
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
2. C₃×#15688521	C₃	direct (N × H)
7. #15688521	C₃²	direct (N × H)
25. C₃²×C₃₇	#733	semidirect
36. C₃×C₃₇	C₃×#733	semidirect
46. C₃₇	C₃²×#733	semidirect
59. C₃²	#15688521	direct (N × H)
67. C₃	C₃×#15688521	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#11758608	3
C₃×#15688521	3
C₃³×C₉×C₃₇	3
C₃²×#733	37

Subgroups — C32×#15688521 #41008140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688521 #41008140