Subgroups — C₃₇⋊C₃×H(5,𝔽₃) #41008252

Order 26973.

6776 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇⋊H(5,𝔽₃)	C₃	1	80	yes	maximal, complemented
2	C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	1	40	yes	maximal, complemented
3	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	C₃²	1	40	yes	complemented
5	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	320	yes	complemented
6	H(3,𝔽₃)×C₃₇⋊C₃	C₃²	1	90	yes	complemented
7	H(3,𝔽₃)×C₃₇⋊C₃	C₃²	1	720	yes	complemented
8	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	40	yes	complemented
9	C₃³×C₃₇	C₃³	1	40	yes	complemented
10	C₃²×C₃₇⋊C₃	—	9	120	no	—
11	C₃²×C₃₇⋊C₃	C₃³	1	40	yes	—
12	C₃²×C₃₇⋊C₃	C₃³	1	320	yes	complemented
13	H(3,𝔽₃)×C₃₇	C₃³	1	90	yes	complemented
14	C₃²⋊(C₃₇⋊C₃)	C₃³	1	720	yes	complemented
15	C₃×H(5,𝔽₃)	—	37	1	no	maximal, Sylow
16	C₃²×C₃₇	—	9	40	no	—
17	C₃²×C₃₇	C₃⁴	1	40	yes	—
18	C₃×C₃₇⋊C₃	—	3	120	no	—
19	C₃×C₃₇⋊C₃	—	9	320	no	—
20	C₃×C₃₇⋊C₃	C₃⁴	1	80	yes	—
21	C₃×C₃₇⋊C₃	C₃⁴	1	1	yes	characteristic
22	H(5,𝔽₃)	—	37	80	no	—
23	H(5,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
24	C₃²×H(3,𝔽₃)	—	37	40	no	—
25	C₃×C₃₇	—	3	40	no	—
26	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
27	C₃₇⋊C₃	—	3	80	no	—
28	C₃₇⋊C₃	H(5,𝔽₃)	1	3	yes	direct product factor
29	C₃⁴	—	37	40	no	—
30	C₃×H(3,𝔽₃)	—	37	320	no	—
31	C₃×H(3,𝔽₃)	—	37	90	no	—
32	C₃×H(3,𝔽₃)	—	37	720	no	—
33	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	40	yes	complemented
34	C₃₇	C₃×H(5,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
35	C₃³	—	37	320	no	—
36	C₃³	—	37	40	no	—
37	C₃³	—	333	120	no	—
38	C₃³	C₃²×C₃₇⋊C₃	1	40	yes	complemented
39	H(3,𝔽₃)	—	37	720	no	—
40	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	90	yes	complemented
41	C₃²	—	9	40	no	—
42	C₃²	—	37	1	no	—
43	C₃²	—	37	80	no	—
44	C₃²	—	333	320	no	—
45	C₃²	—	111	120	no	—
46	C₃²	C₃³×C₃₇⋊C₃	1	40	yes	—
47	C₃	—	3	40	no	—
48	C₃	—	37	3	no	—
49	C₃	—	111	80	no	—
50	C₃	C₃⁴×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃⁴×C₃₇⋊C₃
commutator, socle, cosocle	C₃×C₃₇	C₃⁵
Fitting	C₃₇×H(5,𝔽₃)	C₃
hypercenter	H(5,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	C₃×H(5,𝔽₃)
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃₇⋊H(5,𝔽₃)	C₃	semidirect
2. C₃×H(3,𝔽₃)×C₃₇⋊C₃	C₃	semidirect
3. C₃₇×H(5,𝔽₃)	C₃	semidirect
4. C₃³×C₃₇⋊C₃	C₃²	semidirect
5. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
6. H(3,𝔽₃)×C₃₇⋊C₃	C₃²	semidirect
7. H(3,𝔽₃)×C₃₇⋊C₃	C₃²	semidirect
8. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
9. C₃³×C₃₇	C₃³	semidirect
12. C₃²×C₃₇⋊C₃	C₃³	semidirect
13. H(3,𝔽₃)×C₃₇	C₃³	semidirect
14. C₃²⋊(C₃₇⋊C₃)	C₃³	semidirect
23. H(5,𝔽₃)	C₃₇⋊C₃	direct (N × H)
28. C₃₇⋊C₃	H(5,𝔽₃)	direct (N × H)
33. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
34. C₃₇	C₃×H(5,𝔽₃)	semidirect
38. C₃³	C₃²×C₃₇⋊C₃	semidirect
40. H(3,𝔽₃)	C₃²×C₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃₇⋊H(5,𝔽₃)	3
C₃×H(3,𝔽₃)×C₃₇⋊C₃	3
C₃₇×H(5,𝔽₃)	3
C₃×H(5,𝔽₃)	37

Subgroups — C37⋊C3×H(5,𝔽3) #41008252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊C₃×H(5,𝔽₃) #41008252