Subgroups — C₃₇×#10624445 #41004234

Order 26973.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×C₃²⋊C₂₇	C₃	1	3	yes	maximal
3	C₃²×C₉×C₃₇	C₉	1	2	yes	—
4	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic
5	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×C₂₇×C₃₇	—	3	3	no	—
7	C₉²×C₃₇	—	3	1	no	—
8	C₉²×C₃₇	—	3	2	no	—
9	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
10	C₃×C₉×C₃₇	—	3	1	no	—
11	C₃×C₉×C₃₇	—	3	2	no	—
12	C₃×C₉×C₃₇	—	3	2	no	—
13	C₃×C₉×C₃₇	—	3	3	no	—
14	C₃×C₉×C₃₇	—	3	6	no	—
15	C₃×C₉×C₃₇	C₂₇	1	3	yes	complemented
16	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
18	C₂₇×C₃₇	—	9	3	no	—
19	#10624445	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃²×C₃₇	—	3	1	no	—
21	C₃²×C₃₇	—	3	2	no	—
22	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
23	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
24	C₃²×C₃₇	C₂₇⋊C₃	1	2	yes	—
25	C₉×C₃₇	—	3	3	no	—
26	C₉×C₃₇	—	3	2	no	—
27	C₉×C₃₇	—	3	6	no	—
28	C₉×C₃₇	#731	1	3	yes	—
29	C₃×C₉²	C₃×C₃₇	1	1	yes	characteristic
30	C₃²⋊C₂₇	C₃×C₃₇	1	3	yes	—
31	C₃×C₃₇	—	3	1	no	—
32	C₃×C₃₇	—	3	2	no	—
33	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
34	C₃×C₃₇	#6867	1	1	yes	characteristic
35	C₃×C₃₇	#6869	1	2	yes	—
36	C₃²×C₉	C₉×C₃₇	1	2	yes	—
37	C₃²×C₉	C₉×C₃₇	1	1	yes	characteristic
38	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
39	C₃×C₂₇	—	3	3	no	—
40	C₉²	—	3	1	no	—
41	C₉²	—	3	2	no	—
42	C₃₇	#10624445	1	1	yes	characteristic, direct product factor, Sylow
43	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	3	2	no	—
46	C₃×C₉	—	3	2	no	—
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	—	3	6	no	—
49	C₃×C₉	C₂₇×C₃₇	1	3	yes	complemented
50	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
51	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
52	C₂₇	—	9	3	no	—
53	C₃²	—	3	1	no	—
54	C₃²	—	3	2	no	—
55	C₃²	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
56	C₃²	C₃×C₂₇×C₃₇	1	1	yes	characteristic
57	C₃²	C₃₇×C₂₇⋊C₃	1	2	yes	—
58	C₉	—	3	3	no	—
59	C₉	—	3	2	no	—
60	C₉	—	3	6	no	—
61	C₉	C₃₇×#731	1	3	yes	—
62	C₃	—	3	1	no	—
63	C₃	—	3	2	no	—
64	C₃	C₃₇×C₃²⋊C₂₇	1	1	yes	characteristic
65	C₃	C₃₇×#6867	1	1	yes	characteristic
66	C₃	C₃₇×#6869	1	2	yes	—

	subgroup	G/H
center	C₃×C₉×C₃₇	H(3,𝔽₃)
commutator	C₃²	C₃×C₂₇×C₃₇
Frattini, cosocle	C₃²×C₉	C₃²×C₃₇
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇	C₃×C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃×C₉×C₃₇	C₂₇	semidirect
19. #10624445	C₃₇	direct (N × H)
42. C₃₇	#10624445	direct (N × H)
49. C₃×C₉	C₂₇×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉²×C₃₇	3
C₃₇×C₃²⋊C₂₇	3
#10624445	37

Subgroups — C37×#10624445 #41004234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624445 #41004234