Subgroups — C₃²×#15688520 #41008139

Order 26973.

1052 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758608	C₃	1	2	yes	maximal
2	C₃³×#66524	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688520	C₃	1	36	yes	maximal, direct product factor
4	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	16	no	—
6	C₃×#11758608	C₃²	1	24	yes	—
7	C₃²×#66524	—	3	8	no	—
8	C₃²×#66524	—	3	1	no	—
9	C₃²×#66524	—	3	2	no	—
10	C₃²×#66524	C₃²	1	12	yes	—
11	#15688520	C₃²	1	81	yes	direct product factor
12	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
13	C₃×3₋¹⁺²×C₃₇	—	3	8	no	—
14	C₃×3₋¹⁺²×C₃₇	C₃²	1	12	yes	—
15	C₃²×C₉×C₃₇	—	3	1	no	—
16	C₃³×C₃₇	—	3	9	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	8	yes	—
18	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃³×C₃₇	C₃³	1	4	yes	—
20	3₋¹⁺²×C₃₇	—	3	24	no	—
21	3₋¹⁺²×C₃₇	C₃³	1	9	yes	—
22	C₃×C₉×C₃₇	—	3	12	no	—
23	#11758608	—	3	48	no	—
24	#11758608	C₃³	1	18	yes	—
25	C₃×#66524	—	3	24	no	—
26	C₃×#66524	—	3	12	no	—
27	C₃×#66524	—	3	24	no	—
28	C₃×#66524	C₃³	1	9	yes	—
29	C₃²×#733	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	24	no	—
31	C₃²×C₃₇	—	3	12	no	—
32	C₃²×C₃₇	#733	1	9	yes	complemented
33	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	8	yes	—
34	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
35	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
36	C₉×C₃₇	—	3	9	no	—
37	#66524	—	3	9	no	—
38	#66524	—	3	18	no	—
39	C₃²×3₋¹⁺²	—	37	2	no	—
40	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
41	C₃³×C₉	—	37	1	no	—
42	C₃×#733	—	37	36	no	—
43	C₃×C₃₇	—	3	8	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	C₃×#733	1	12	yes	complemented
46	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	3	8	no	—
49	C₃×3₋¹⁺²	—	37	24	no	—
50	C₃×3₋¹⁺²	—	111	16	no	—
51	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	12	yes	—
52	C₃²×C₉	—	3	1	no	—
53	C₃²×C₉	—	37	12	no	—
54	C₃²×C₉	—	111	8	no	—
55	C₃²×C₉	—	111	1	no	—
56	C₃²×C₉	—	111	2	no	—
57	#733	—	37	81	no	—
58	C₃₇	C₃²×#733	1	1	yes	characteristic, complemented, Sylow
59	C₃³	—	3	9	no	—
60	C₃³	C₃²⋊(C₃₇⋊C₃)	1	8	yes	—
61	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
62	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
63	3₋¹⁺²	—	3	24	no	—
64	3₋¹⁺²	—	37	18	no	—
65	3₋¹⁺²	—	111	48	no	—
66	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	9	yes	—
67	C₃×C₉	—	3	12	no	—
68	C₃×C₉	—	37	9	no	—
69	C₃×C₉	—	111	24	no	—
70	C₃×C₉	—	111	12	no	—
71	C₃×C₉	—	111	24	no	—
72	C₃²	—	3	24	no	—
73	C₃²	—	3	12	no	—
74	C₃²	#15688520	1	9	yes	direct product factor
75	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	8	yes	—
76	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	4	yes	—
77	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
78	C₉	—	3	9	no	—
79	C₉	—	111	9	no	—
80	C₉	—	111	18	no	—
81	C₃	—	3	8	no	—
82	C₃	—	3	1	no	—
83	C₃	C₃×#15688520	1	12	yes	direct product factor
84	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	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃³×C₃₇	H(3,𝔽₃)
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	C₃²×#733
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
3. C₃×#15688520	C₃	direct (N × H)
11. #15688520	C₃²	direct (N × H)
32. C₃²×C₃₇	#733	semidirect
45. C₃×C₃₇	C₃×#733	semidirect
58. C₃₇	C₃²×#733	semidirect
74. C₃²	#15688520	direct (N × H)
83. C₃	C₃×#15688520	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
C₃³×#66524	3
C₃×#15688520	3
C₃²×3₋¹⁺²×C₃₇	3
C₃²×#733	37

Subgroups — C32×#15688520 #41008139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688520 #41008139