Subgroups — C₃₇⋊C₃×#6855 #41005084

Order 26973.

1052 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉×C₃₇⋊C₃	C₃	1	4	yes	maximal
2	#22582441	C₃	1	8	yes	maximal, complemented
3	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
5	C₃²×#66524	C₃²	1	8	yes	—
6	C₃×C₉×C₃₇⋊C₃	—	3	12	no	—
7	C₃×C₉×C₃₇⋊C₃	—	3	24	no	—
8	C₃×C₉×C₃₇⋊C₃	C₉	1	36	yes	complemented
9	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
10	C₃³×C₃₇	C₃³	1	1	yes	characteristic
11	C₃²×C₃₇⋊C₃	C₃×C₉	1	12	yes	—
12	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
13	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	24	yes	—
14	C₃×C₉×C₃₇	—	3	4	no	—
15	C₃×C₉×C₃₇	—	3	8	no	—
16	C₃×C₉×C₃₇	C₃×C₉	1	12	yes	complemented
17	C₃×#66524	—	3	8	no	—
18	C₃×#66524	—	3	16	no	—
19	C₃×#66524	C₃×C₉	1	24	yes	complemented
20	C₉×C₃₇⋊C₃	—	3	108	no	—
21	C₃×#6855	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	C₃²×C₉	1	4	yes	—
23	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
24	C₃²×C₃₇	C₃×3₋¹⁺²	1	8	yes	—
25	C₃×C₃₇⋊C₃	C₉²	1	9	yes	—
26	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	36	yes	—
27	C₃×C₃₇⋊C₃	C₉⋊C₉	1	72	yes	—
28	C₉×C₃₇	—	3	12	no	—
29	#66524	—	3	24	no	—
30	C₃³×C₉	—	37	4	no	—
31	#6855	—	37	8	no	—
32	#6855	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
33	C₃×C₃₇	C₃×C₉²	1	1	yes	characteristic
34	C₃×C₃₇	C₃×C₃²⋊C₉	1	4	yes	—
35	C₃×C₃₇	C₃×C₉⋊C₉	1	8	yes	—
36	C₃₇⋊C₃	#6855	1	27	yes	direct product factor
37	C₃⁴	—	37	1	no	—
38	C₃²×C₉	—	37	8	no	—
39	C₃²×C₉	—	37	36	no	—
40	C₃²×C₉	—	111	12	no	—
41	C₃²×C₉	—	111	24	no	—
42	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
43	C₃₇	C₃×#6855	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	37	12	no	—
45	C₃³	—	37	3	no	—
46	C₃³	—	37	24	no	—
47	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
48	C₃×C₉	—	3	4	no	—
49	C₃×C₉	—	3	8	no	—
50	C₃×C₉	—	37	24	no	—
51	C₃×C₉	—	111	108	no	—
52	C₃×C₉	—	111	8	no	—
53	C₃×C₉	—	111	16	no	—
54	C₃×C₉	C₉×C₃₇⋊C₃	1	12	yes	complemented
55	C₃²	—	37	36	no	—
56	C₃²	—	37	72	no	—
57	C₃²	—	37	9	no	—
58	C₃²	C₃×C₉×C₃₇⋊C₃	1	4	yes	—
59	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
60	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	8	yes	—
61	C₉	—	3	12	no	—
62	C₉	—	111	24	no	—
63	C₃	—	37	27	no	—
64	C₃	C₉²×C₃₇⋊C₃	1	1	yes	characteristic
65	C₃	C₃²⋊C₉×C₃₇⋊C₃	1	4	yes	—
66	C₃	C₉⋊C₉×C₃₇⋊C₃	1	8	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃×C₉²
Fitting	C₃₇×#6855	C₃
socle, cosocle	C₃³×C₃₇	C₃³
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6855
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

Complements

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

N	H ≅ G/N	type
2. #22582441	C₃	semidirect
3. C₃₇×#6855	C₃	semidirect
8. C₃×C₉×C₃₇⋊C₃	C₉	semidirect
16. C₃×C₉×C₃₇	C₃×C₉	semidirect
19. C₃×#66524	C₃×C₉	semidirect
32. #6855	C₃₇⋊C₃	direct (N × H)
36. C₃₇⋊C₃	#6855	direct (N × H)
43. C₃₇	C₃×#6855	semidirect
54. C₃×C₉	C₉×C₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₉×C₃₇⋊C₃	3
#22582441	3
C₃₇×#6855	3
C₃×#6855	37

Subgroups — C37⋊C3×#6855 #41005084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊C₃×#6855 #41005084