Subgroups — C₃²×#15688515 #41008132

Order 26973.

1160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758608	C₃	1	2	yes	maximal, complemented
2	C₃³×#66524	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#15688515	C₃	1	36	yes	maximal, direct product factor
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	16	no	—
6	C₃×#11758608	C₃²	1	24	yes	complemented
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	complemented
11	#15688515	C₃²	1	81	yes	direct product factor
12	C₃⁴×C₃₇	—	3	1	no	—
13	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
14	C₃×H(3,𝔽₃)×C₃₇	—	3	8	no	—
15	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	—
16	C₃³×C₃₇	—	3	9	no	—
17	C₃³×C₃₇	—	3	12	no	—
18	C₃³×C₃₇	—	9	9	no	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	8	yes	—
20	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	H(3,𝔽₃)×C₃₇	—	3	24	no	—
23	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
24	#11758608	—	3	48	no	—
25	#11758608	C₃³	1	18	yes	complemented
26	C₃×#66524	—	3	24	no	—
27	C₃×#66524	—	3	12	no	—
28	C₃×#66524	—	3	24	no	—
29	C₃×#66524	C₃³	1	9	yes	complemented
30	C₃²×#731	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	—	3	24	no	—
32	C₃²×C₃₇	—	3	12	no	—
33	C₃²×C₃₇	—	3	9	no	—
34	C₃²×C₃₇	—	9	36	no	—
35	C₃²×C₃₇	#731	1	9	yes	complemented
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	8	yes	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
38	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
39	#66524	—	3	9	no	—
40	#66524	—	3	18	no	—
41	C₃²×3₋¹⁺²	—	37	2	no	—
42	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
43	C₃³×C₉	—	37	1	no	—
44	C₃×#731	—	37	36	no	—
45	C₃×C₃₇	—	3	8	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	—	9	9	no	—
48	C₃×C₃₇	C₃×#731	1	12	yes	complemented
49	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃⁴	—	3	1	no	—
51	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	C₃×3₋¹⁺²	—	37	24	no	—
53	C₃×3₋¹⁺²	—	111	16	no	—
54	C₃×H(3,𝔽₃)	—	3	8	no	—
55	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	12	yes	—
56	C₃²×C₉	—	37	12	no	—
57	C₃²×C₉	—	111	8	no	—
58	C₃²×C₉	—	111	1	no	—
59	C₃²×C₉	—	111	2	no	—
60	#731	—	37	81	no	—
61	C₃₇	C₃²×#731	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	3	9	no	—
63	C₃³	—	3	12	no	—
64	C₃³	—	9	9	no	—
65	C₃³	C₃²⋊(C₃₇⋊C₃)	1	8	yes	—
66	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
67	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
68	3₋¹⁺²	—	37	18	no	—
69	3₋¹⁺²	—	111	48	no	—
70	H(3,𝔽₃)	—	3	24	no	—
71	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	9	yes	—
72	C₃×C₉	—	37	9	no	—
73	C₃×C₉	—	111	24	no	—
74	C₃×C₉	—	111	12	no	—
75	C₃×C₉	—	111	24	no	—
76	C₃²	—	3	24	no	—
77	C₃²	—	3	12	no	—
78	C₃²	—	3	9	no	—
79	C₃²	—	9	36	no	—
80	C₃²	#15688515	1	9	yes	direct product factor
81	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	8	yes	—
82	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	4	yes	—
83	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
84	C₉	—	111	9	no	—
85	C₉	—	111	18	no	—
86	C₃	—	3	8	no	—
87	C₃	—	3	1	no	—
88	C₃	—	9	9	no	—
89	C₃	C₃×#15688515	1	12	yes	direct product factor
90	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, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃³×C₃₇	H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₃²×#731
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×#11758608	C₃	semidirect
2. C₃³×#66524	C₃	semidirect
3. C₃×#15688515	C₃	direct (N × H)
6. C₃×#11758608	C₃²	semidirect
10. C₃²×#66524	C₃²	semidirect
11. #15688515	C₃²	direct (N × H)
25. #11758608	C₃³	semidirect
29. C₃×#66524	C₃³	semidirect
35. C₃²×C₃₇	#731	semidirect
48. C₃×C₃₇	C₃×#731	semidirect
61. C₃₇	C₃²×#731	semidirect
80. C₃²	#15688515	direct (N × H)
89. C₃	C₃×#15688515	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×#15688515	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃²×#731	37

Subgroups — C32×#15688515 #41008132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#15688515 #41008132