Subgroups — H(3,𝔽₃)²×C₃₇ #41004682

Order 26973.

2102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	8	yes	maximal, complemented
2	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	32	yes	maximal, complemented
3	C₃⁴×C₃₇	C₃²	1	16	yes	complemented
4	C₃×H(3,𝔽₃)×C₃₇	—	3	72	no	—
5	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	2	yes	—
6	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	64	yes	complemented
7	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	48	yes	complemented
8	C₃³×C₃₇	—	3	96	no	—
9	C₃³×C₃₇	—	9	32	no	—
10	C₃³×C₃₇	C₃³	1	32	yes	—
11	C₃³×C₃₇	C₃³	1	8	yes	—
12	H(3,𝔽₃)×C₃₇	—	3	192	no	—
13	H(3,𝔽₃)×C₃₇	—	9	48	no	—
14	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	18	yes	direct product factor
15	H(3,𝔽₃)²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₃²×C₃₇	—	3	8	no	—
17	C₃²×C₃₇	—	3	16	no	—
18	C₃²×C₃₇	—	3	64	no	—
19	C₃²×C₃₇	—	3	64	no	—
20	C₃²×C₃₇	—	9	144	no	—
21	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	24	yes	complemented
22	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
23	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	8	yes	complemented
24	C₃²⋊H(3,𝔽₃)	C₃×C₃₇	1	32	yes	complemented
25	C₃×C₃₇	—	3	24	no	—
26	C₃×C₃₇	—	9	32	no	—
27	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	2	yes	—
28	C₃×C₃₇	H(5,𝔽₃)	1	2	yes	—
29	C₃⁴	C₃²×C₃₇	1	16	yes	complemented
30	C₃×H(3,𝔽₃)	—	3	72	no	—
31	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	2	yes	—
32	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	64	yes	complemented
33	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	48	yes	complemented
34	C₃₇	H(3,𝔽₃)²	1	1	yes	characteristic, direct product factor, Sylow
35	C₃³	—	3	96	no	—
36	C₃³	—	9	32	no	—
37	C₃³	C₃³×C₃₇	1	32	yes	—
38	C₃³	C₃³×C₃₇	1	8	yes	—
39	H(3,𝔽₃)	—	3	192	no	—
40	H(3,𝔽₃)	—	9	48	no	—
41	H(3,𝔽₃)	H(3,𝔽₃)×C₃₇	1	18	yes	direct product factor
42	C₃²	—	3	8	no	—
43	C₃²	—	3	16	no	—
44	C₃²	—	3	64	no	—
45	C₃²	—	3	64	no	—
46	C₃²	—	9	144	no	—
47	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	24	yes	complemented
48	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
49	C₃	—	3	24	no	—
50	C₃	—	9	32	no	—
51	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	2	yes	—
52	C₃	C₃₇×H(5,𝔽₃)	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
2. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
3. C₃⁴×C₃₇	C₃²	semidirect
6. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
7. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
14. H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	direct (N × H)
15. H(3,𝔽₃)²	C₃₇	direct (N × H)
21. C₃²×C₃₇	C₃×H(3,𝔽₃)	semidirect
23. C₃²×H(3,𝔽₃)	C₃×C₃₇	semidirect
24. C₃²⋊H(3,𝔽₃)	C₃×C₃₇	semidirect
29. C₃⁴	C₃²×C₃₇	semidirect
32. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
33. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
34. C₃₇	H(3,𝔽₃)²	direct (N × H)
41. H(3,𝔽₃)	H(3,𝔽₃)×C₃₇	direct (N × H)
47. C₃²	C₃×H(3,𝔽₃)×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×H(3,𝔽₃)×C₃₇	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
H(3,𝔽₃)²	37

Subgroups — H(3,𝔽3)2×C37 #41004682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)²×C₃₇ #41004682