Subgroups — C₃₇×#10624503 #41004292

Order 26973.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#6903	C₃	1	1	yes	maximal, characteristic
2	C₃₇×C₈₁⋊C₃	C₃	1	2	yes	maximal, complemented
3	C₃×C₃₇×C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#737	C₉	1	1	yes	characteristic
5	C₃₇×C₂₇⋊C₃	—	3	2	no	—
6	C₃₇×C₂₇⋊C₃	C₉	1	2	yes	—
7	C₃×C₂₇×C₃₇	—	3	1	no	—
8	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
9	C₃₇×C₈₁	—	3	1	no	—
10	C₃₇×C₈₁	—	3	2	no	—
11	3₋¹⁺²×C₃₇	—	3	2	no	—
12	3₋¹⁺²×C₃₇	C₂₇	1	2	yes	—
13	H(3,𝔽₃)×C₃₇	C₂₇	1	1	yes	characteristic
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₂₇×C₃₇	—	3	2	no	—
17	C₂₇×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
18	C₂₇×C₃₇	3₋¹⁺²	1	2	yes	—
19	#10624503	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃²×C₃₇	—	3	1	no	—
21	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
22	C₉×C₃₇	—	3	2	no	—
23	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
24	C₉×C₃₇	C₂₇⋊C₃	1	2	yes	—
25	#6903	C₃×C₃₇	1	1	yes	characteristic
26	C₈₁⋊C₃	C₃×C₃₇	1	2	yes	complemented
27	C₃×C₈₁	C₃×C₃₇	1	1	yes	characteristic, complemented
28	C₃×C₃₇	—	3	1	no	—
29	C₃×C₃₇	—	9	1	no	—
30	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
31	#737	C₉×C₃₇	1	1	yes	characteristic
32	C₂₇⋊C₃	—	3	2	no	—
33	C₂₇⋊C₃	C₉×C₃₇	1	2	yes	—
34	C₃×C₂₇	—	3	1	no	—
35	C₃×C₂₇	C₃²×C₃₇	1	1	yes	characteristic
36	C₈₁	—	3	1	no	—
37	C₈₁	—	3	2	no	—
38	C₃₇	#10624503	1	1	yes	characteristic, direct product factor, Sylow
39	3₋¹⁺²	—	3	2	no	—
40	3₋¹⁺²	C₂₇×C₃₇	1	2	yes	—
41	H(3,𝔽₃)	C₂₇×C₃₇	1	1	yes	characteristic
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	C₃×C₉×C₃₇	1	1	yes	characteristic
44	C₂₇	—	3	2	no	—
45	C₂₇	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
46	C₂₇	3₋¹⁺²×C₃₇	1	2	yes	—
47	C₃²	—	3	1	no	—
48	C₃²	C₃×C₂₇×C₃₇	1	1	yes	characteristic
49	C₉	—	3	2	no	—
50	C₉	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
51	C₉	C₃₇×C₂₇⋊C₃	1	2	yes	—
52	C₃	—	3	1	no	—
53	C₃	—	9	1	no	—
54	C₃	C₃₇×C₃²⋊C₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₇×C₈₁⋊C₃	C₃	semidirect
3. C₃×C₃₇×C₈₁	C₃	semidirect
19. #10624503	C₃₇	direct (N × H)
26. C₈₁⋊C₃	C₃×C₃₇	semidirect
27. C₃×C₈₁	C₃×C₃₇	semidirect
38. C₃₇	#10624503	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×#6903	3
C₃₇×C₈₁⋊C₃	3
C₃×C₃₇×C₈₁	3
#10624503	37

Subgroups — C37×#10624503 #41004292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624503 #41004292