Subgroups — C₃²×H(3,𝔽₃)×C₃₇⋊C₃ #41008242

Order 26973.

11744 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₃₇⋊C₃	C₃	1	4	yes	maximal, complemented
2	C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	1	8	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	108	yes	maximal, direct product factor
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₃₇⋊C₃	—	3	108	no	—
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃³×C₃₇⋊C₃	C₃²	1	144	yes	complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	8	yes	complemented
9	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	288	yes	complemented
10	H(3,𝔽₃)×C₃₇⋊C₃	C₃²	1	729	yes	direct product factor
11	C₃⁴×C₃₇	C₃²	1	4	yes	complemented
12	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	36	yes	complemented
13	C₃³×C₃₇	—	3	36	no	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	C₃³×C₃₇	C₃³	1	48	yes	complemented
16	C₃²×C₃₇⋊C₃	—	3	72	no	—
17	C₃²×C₃₇⋊C₃	—	3	1296	no	—
18	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	27	yes	direct product factor
19	C₃²×C₃₇⋊C₃	C₃³	1	12	yes	—
20	C₃²×C₃₇⋊C₃	C₃³	1	324	yes	complemented
21	C₃²×C₃₇⋊C₃	C₃³	1	96	yes	complemented
22	H(3,𝔽₃)×C₃₇	C₃³	1	81	yes	complemented
23	C₃²⋊(C₃₇⋊C₃)	C₃³	1	648	yes	complemented
24	C₃³×H(3,𝔽₃)	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	144	no	—
26	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	complemented
27	C₃²×C₃₇	C₃⁴	1	4	yes	—
28	C₃²×C₃₇	C₃⁴	1	36	yes	complemented
29	C₃×C₃₇⋊C₃	—	3	288	no	—
30	C₃×C₃₇⋊C₃	—	3	972	no	—
31	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	108	yes	direct product factor
32	C₃×C₃₇⋊C₃	C₃⁴	1	9	yes	—
33	C₃×C₃₇⋊C₃	C₃⁴	1	72	yes	complemented
34	C₃⁵	—	37	4	no	—
35	C₃²×H(3,𝔽₃)	—	37	8	no	—
36	C₃²×H(3,𝔽₃)	—	37	108	no	—
37	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
38	C₃×C₃₇	—	3	36	no	—
39	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	12	yes	complemented
40	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
41	C₃₇⋊C₃	—	3	72	no	—
42	C₃₇⋊C₃	C₃²×H(3,𝔽₃)	1	27	yes	direct product factor
43	C₃⁴	—	37	1	no	—
44	C₃⁴	—	37	8	no	—
45	C₃⁴	—	37	144	no	—
46	C₃⁴	—	111	108	no	—
47	C₃⁴	C₃×C₃₇⋊C₃	1	4	yes	complemented
48	C₃×H(3,𝔽₃)	—	37	288	no	—
49	C₃×H(3,𝔽₃)	—	37	729	no	—
50	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	36	yes	direct product factor
51	C₃₇	C₃³×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
52	C₃³	—	3	36	no	—
53	C₃³	—	37	27	no	—
54	C₃³	—	37	12	no	—
55	C₃³	—	37	96	no	—
56	C₃³	—	37	324	no	—
57	C₃³	—	111	1296	no	—
58	C₃³	—	111	72	no	—
59	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
60	C₃³	C₃²×C₃₇⋊C₃	1	48	yes	complemented
61	H(3,𝔽₃)	—	37	648	no	—
62	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	81	yes	direct product factor
63	C₃²	—	3	144	no	—
64	C₃²	—	37	108	no	—
65	C₃²	—	37	9	no	—
66	C₃²	—	37	72	no	—
67	C₃²	—	111	972	no	—
68	C₃²	—	111	288	no	—
69	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	9	yes	direct product factor
70	C₃²	C₃³×C₃₇⋊C₃	1	4	yes	—
71	C₃²	C₃³×C₃₇⋊C₃	1	36	yes	complemented
72	C₃	—	3	36	no	—
73	C₃	—	37	27	no	—
74	C₃	—	111	72	no	—
75	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	12	yes	direct product factor
76	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃²×C₃₇⋊C₃
commutator, cosocle	C₃×C₃₇	C₃⁵
Frattini	C₃	C₃⁴×C₃₇⋊C₃
Fitting	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃³×C₃₇	C₃³
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₃³×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. C₃⁴×C₃₇⋊C₃	C₃	semidirect
2. C₃²×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
3. C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	direct (N × H)
4. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
7. C₃³×C₃₇⋊C₃	C₃²	semidirect
8. C₃³×C₃₇⋊C₃	C₃²	semidirect
9. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
10. H(3,𝔽₃)×C₃₇⋊C₃	C₃²	direct (N × H)
11. C₃⁴×C₃₇	C₃²	semidirect
12. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
15. C₃³×C₃₇	C₃³	semidirect
18. C₃²×C₃₇⋊C₃	H(3,𝔽₃)	direct (N × H)
20. C₃²×C₃₇⋊C₃	C₃³	semidirect
21. C₃²×C₃₇⋊C₃	C₃³	semidirect
22. H(3,𝔽₃)×C₃₇	C₃³	semidirect
23. C₃²⋊(C₃₇⋊C₃)	C₃³	semidirect
26. C₃²×C₃₇	C₃×H(3,𝔽₃)	semidirect
28. C₃²×C₃₇	C₃⁴	semidirect
31. C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	direct (N × H)
33. C₃×C₃₇⋊C₃	C₃⁴	semidirect
37. C₃²×H(3,𝔽₃)	C₃₇⋊C₃	direct (N × H)
39. C₃×C₃₇	C₃²×H(3,𝔽₃)	semidirect
42. C₃₇⋊C₃	C₃²×H(3,𝔽₃)	direct (N × H)
47. C₃⁴	C₃×C₃₇⋊C₃	semidirect
50. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	direct (N × H)
51. C₃₇	C₃³×H(3,𝔽₃)	semidirect
60. C₃³	C₃²×C₃₇⋊C₃	semidirect
62. H(3,𝔽₃)	C₃²×C₃₇⋊C₃	direct (N × H)
69. C₃²	H(3,𝔽₃)×C₃₇⋊C₃	direct (N × H)
71. C₃²	C₃³×C₃₇⋊C₃	semidirect
75. C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₃₇⋊C₃	3
C₃²×C₃²⋊(C₃₇⋊C₃)	3
C₃×H(3,𝔽₃)×C₃₇⋊C₃	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃³×H(3,𝔽₃)	37

Subgroups — C32×H(3,𝔽3)×C37⋊C3 #41008242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×H(3,𝔽₃)×C₃₇⋊C₃ #41008242