Subgroups — C₃×H(3,𝔽₃)×#66524 #41007501

Order 26973.

1688 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#66524	C₃	1	4	yes	maximal, complemented
2	H(3,𝔽₃)×#66524	C₃	1	27	yes	maximal, direct product factor
3	C₃×C₃²⋊#66524	C₃	1	8	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃²×#66524	—	3	36	no	—
6	C₃²×#66524	C₃²	1	8	yes	—
7	C₃²×#66524	C₃²	1	1	yes	characteristic
8	C₃²×#66524	C₃²	1	36	yes	complemented
9	C₃²⋊#66524	C₃²	1	72	yes	complemented
10	C₃⁴×C₃₇	C₃²	1	4	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₉	1	27	yes	complemented
12	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
13	C₃³×C₃₇	—	3	12	no	—
14	C₃³×C₃₇	—	3	24	no	—
15	C₃³×C₃₇	C₃×C₉	1	36	yes	complemented
16	C₃³×C₃₇	C₃³	1	12	yes	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	H(3,𝔽₃)×C₃₇	C₃×C₉	1	81	yes	complemented
19	C₃×#66524	—	3	24	no	—
20	C₃×#66524	—	3	108	no	—
21	C₃×#66524	H(3,𝔽₃)	1	9	yes	direct product factor
22	C₃×#66524	C₃³	1	24	yes	—
23	C₃×#66524	C₃³	1	3	yes	—
24	C₃×C₉×H(3,𝔽₃)	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	12	no	—
26	C₃²×C₃₇	—	3	24	no	—
27	C₃²×C₃₇	—	3	108	no	—
28	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
29	C₃²×C₃₇	C₃²×C₉	1	36	yes	complemented
30	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
31	C₃²×C₃₇	#737	1	6	yes	—
32	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
33	#66524	—	3	24	no	—
34	#66524	C₃×H(3,𝔽₃)	1	9	yes	direct product factor
35	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
36	C₃³×C₉	—	37	4	no	—
37	C₉×H(3,𝔽₃)	—	37	27	no	—
38	C₃×C₃²⋊C₉	—	37	8	no	—
39	C₃×C₃₇	—	3	36	no	—
40	C₃×C₃₇	C₉×H(3,𝔽₃)	1	9	yes	complemented
41	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
42	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
43	C₃×C₃₇	C₃×#737	1	2	yes	—
44	C₃⁴	C₃×C₃₇⋊C₃	1	4	yes	—
45	C₃×H(3,𝔽₃)	#66524	1	27	yes	direct product factor
46	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	9	yes	—
47	C₃²×C₉	—	37	1	no	—
48	C₃²×C₉	—	37	36	no	—
49	C₃²×C₉	—	37	8	no	—
50	C₃²×C₉	—	111	36	no	—
51	C₃²⋊C₉	—	37	72	no	—
52	C₃₇	C₃×C₉×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
53	C₃³	—	3	12	no	—
54	C₃³	—	3	24	no	—
55	C₃³	C₃×#66524	1	36	yes	complemented
56	C₃³	C₃²×C₃₇⋊C₃	1	12	yes	—
57	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
58	H(3,𝔽₃)	C₃×#66524	1	81	yes	direct product factor
59	C₃×C₉	—	37	9	no	—
60	C₃×C₉	—	37	3	no	—
61	C₃×C₉	—	37	24	no	—
62	C₃×C₉	—	111	108	no	—
63	C₃×C₉	—	111	24	no	—
64	C₃²	—	3	108	no	—
65	C₃²	—	3	12	no	—
66	C₃²	—	3	24	no	—
67	C₃²	C₃²×#66524	1	3	yes	—
68	C₃²	C₃²×#66524	1	36	yes	complemented
69	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
70	C₃²	#15688530	1	6	yes	—
71	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
72	C₉	—	37	9	no	—
73	C₉	—	111	24	no	—
74	C₃	—	3	36	no	—
75	C₃	H(3,𝔽₃)×#66524	1	9	yes	direct product factor
76	C₃	C₃³×#66524	1	1	yes	characteristic
77	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
78	C₃	C₃×#15688530	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃³×#66524	C₃	semidirect
2. H(3,𝔽₃)×#66524	C₃	direct (N × H)
3. C₃×C₃²⋊#66524	C₃	semidirect
8. C₃²×#66524	C₃²	semidirect
9. C₃²⋊#66524	C₃²	semidirect
11. C₃×H(3,𝔽₃)×C₃₇	C₉	semidirect
15. C₃³×C₃₇	C₃×C₉	semidirect
18. H(3,𝔽₃)×C₃₇	C₃×C₉	semidirect
21. C₃×#66524	H(3,𝔽₃)	direct (N × H)
29. C₃²×C₃₇	C₃²×C₉	semidirect
34. #66524	C₃×H(3,𝔽₃)	direct (N × H)
40. C₃×C₃₇	C₉×H(3,𝔽₃)	semidirect
45. C₃×H(3,𝔽₃)	#66524	direct (N × H)
52. C₃₇	C₃×C₉×H(3,𝔽₃)	semidirect
55. C₃³	C₃×#66524	semidirect
58. H(3,𝔽₃)	C₃×#66524	direct (N × H)
68. C₃²	C₃²×#66524	semidirect
75. C₃	H(3,𝔽₃)×#66524	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#66524	3
H(3,𝔽₃)×#66524	3
C₃×C₃²⋊#66524	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃×C₉×H(3,𝔽₃)	37

Subgroups — C3×H(3,𝔽3)×#66524 #41007501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×H(3,𝔽₃)×#66524 #41007501