Subgroups — C₃₇×#10624470 #41004259

Order 26973.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#6887	C₃	1	3	yes	maximal
2	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
4	C₃₇×C₉⋊C₉	—	3	3	no	—
5	C₃₇×C₉⋊C₉	—	3	2	no	—
6	C₃₇×C₉⋊C₉	C₉	1	9	yes	complemented
7	C₃₇×C₃²⋊C₉	—	3	3	no	—
8	C₃₇×C₃²⋊C₉	—	3	1	no	—
9	C₉²×C₃₇	—	3	3	no	—
10	C₉²×C₃₇	C₉	1	3	yes	complemented
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃×C₉×C₃₇	—	3	9	no	—
13	C₃×C₉×C₃₇	—	3	3	no	—
14	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
15	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
17	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
18	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
19	#10624470	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃²×C₃₇	—	9	1	no	—
21	C₃²×C₃₇	C₃²⋊C₉	1	3	yes	—
22	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
23	C₉×C₃₇	—	9	9	no	—
24	C₉×C₃₇	—	9	3	no	—
25	C₉×C₃₇	C₉²	1	1	yes	characteristic
26	C₉×C₃₇	C₉⋊C₉	1	6	yes	—
27	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
28	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
29	#6887	C₃×C₃₇	1	3	yes	—
30	#6887	C₃×C₃₇	1	1	yes	characteristic
31	C₃×C₃₇	—	3	3	no	—
32	C₃×C₃₇	—	3	1	no	—
33	C₃×C₃₇	#6855	1	1	yes	characteristic
34	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
35	C₉⋊C₉	—	3	3	no	—
36	C₉⋊C₉	—	3	2	no	—
37	C₉⋊C₉	C₉×C₃₇	1	9	yes	complemented
38	C₃²⋊C₉	—	3	3	no	—
39	C₃²⋊C₉	—	3	1	no	—
40	C₉²	—	3	3	no	—
41	C₉²	C₉×C₃₇	1	3	yes	complemented
42	C₃₇	#10624470	1	1	yes	characteristic, direct product factor, Sylow
43	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
44	C₃×C₉	—	3	9	no	—
45	C₃×C₉	—	3	3	no	—
46	C₃×C₉	C₃×C₉×C₃₇	1	1	yes	characteristic
47	C₃×C₉	C₃×C₉×C₃₇	1	3	yes	—
48	C₃×C₉	3₋¹⁺²×C₃₇	1	3	yes	—
49	C₃×C₉	3₋¹⁺²×C₃₇	1	3	yes	—
50	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
51	C₃²	—	9	1	no	—
52	C₃²	C₃₇×C₃²⋊C₉	1	3	yes	—
53	C₃²	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
54	C₉	—	9	9	no	—
55	C₉	—	9	3	no	—
56	C₉	C₉²×C₃₇	1	1	yes	characteristic
57	C₉	C₃₇×C₉⋊C₉	1	1	yes	characteristic
58	C₉	C₃₇×C₉⋊C₉	1	6	yes	—
59	C₉	C₃₇×C₉⋊C₉	1	1	yes	characteristic
60	C₃	—	3	3	no	—
61	C₃	—	3	1	no	—
62	C₃	C₃₇×#6855	1	1	yes	characteristic

	subgroup	G/H
center	C₉×C₃₇	C₉⋊C₉
commutator	C₉	C₉²×C₃₇
Frattini, cosocle	C₃²×C₉	C₃²×C₃₇
socle	C₃×C₃₇	#6855
prime radical	C₃³×C₃₇	H(3,𝔽₃)
LCS	G ▷ C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₃₇×C₉⋊C₉	C₉	semidirect
10. C₉²×C₃₇	C₉	semidirect
19. #10624470	C₃₇	direct (N × H)
37. C₉⋊C₉	C₉×C₃₇	semidirect
41. C₉²	C₉×C₃₇	semidirect
42. C₃₇	#10624470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#6887	3
#10624470	37

Subgroups — C37×#10624470 #41004259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624470 #41004259