Subgroups — C₃₇×#10624864 #41004653

Order 26973.

926 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
2	C₉×3₋¹⁺²×C₃₇	C₃	1	24	yes	maximal, complemented
3	C₉×H(3,𝔽₃)×C₃₇	C₃	1	3	yes	maximal
4	C₃₇×#6887	C₃	1	8	yes	maximal, complemented
5	C₃×C₃₇×C₉⋊C₉	C₃	1	4	yes	maximal, complemented
6	C₃₇×#737	C₉	1	27	yes	complemented
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	—
8	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₃₇	C₃²	1	12	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
11	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
12	C₃₇×C₉⋊C₉	C₃²	1	8	yes	—
13	C₃₇×C₉⋊C₉	C₃²	1	24	yes	—
14	C₃₇×C₉⋊C₉	C₃²	1	24	yes	—
15	C₃₇×C₃²⋊C₉	C₃²	1	24	yes	—
16	C₉²×C₃₇	C₃²	1	24	yes	—
17	C₃³×C₃₇	C₃³	1	4	yes	—
18	3₋¹⁺²×C₃₇	C₃×C₉	1	72	yes	complemented
19	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	—
20	C₃×C₉×C₃₇	—	9	12	no	—
21	C₃×C₉×C₃₇	C₃×C₉	1	36	yes	complemented
22	C₃×C₉×C₃₇	C₃³	1	24	yes	—
23	C₃×C₉×C₃₇	C₃³	1	3	yes	—
24	C₃×C₉×C₃₇	C₃³	1	8	yes	—
25	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
26	#10624864	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃²×C₃₇	—	3	12	no	—
28	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
29	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
30	C₉×C₃₇	—	3	3	no	—
31	C₉×C₃₇	—	3	24	no	—
32	C₉×C₃₇	C₃²×C₉	1	3	yes	—
33	C₉×C₃₇	C₃²×C₉	1	24	yes	—
34	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
35	C₉×3₋¹⁺²	C₃×C₃₇	1	24	yes	complemented
36	C₉×H(3,𝔽₃)	C₃×C₃₇	1	3	yes	—
37	#6887	C₃×C₃₇	1	8	yes	complemented
38	C₃×C₉⋊C₉	C₃×C₃₇	1	4	yes	complemented
39	C₃×C₃₇	—	3	12	no	—
40	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
41	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
42	#737	C₉×C₃₇	1	27	yes	complemented
43	C₃×3₋¹⁺²	C₃²×C₃₇	1	8	yes	—
44	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
45	C₃²×C₉	C₃²×C₃₇	1	12	yes	—
46	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
47	C₉⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
48	C₉⋊C₉	C₃²×C₃₇	1	8	yes	—
49	C₉⋊C₉	C₃²×C₃₇	1	24	yes	—
50	C₉⋊C₉	C₃²×C₃₇	1	24	yes	—
51	C₃²⋊C₉	C₃²×C₃₇	1	24	yes	—
52	C₉²	C₃²×C₃₇	1	24	yes	—
53	C₃₇	#10624864	1	1	yes	characteristic, direct product factor, Sylow
54	C₃³	C₃³×C₃₇	1	4	yes	—
55	3₋¹⁺²	C₃×C₉×C₃₇	1	72	yes	complemented
56	H(3,𝔽₃)	C₃×C₉×C₃₇	1	9	yes	—
57	C₃×C₉	—	9	12	no	—
58	C₃×C₉	C₃×C₉×C₃₇	1	36	yes	complemented
59	C₃×C₉	C₃³×C₃₇	1	24	yes	—
60	C₃×C₉	C₃³×C₃₇	1	3	yes	—
61	C₃×C₉	C₃³×C₃₇	1	8	yes	—
62	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
63	C₃²	—	3	12	no	—
64	C₃²	C₃²×C₉×C₃₇	1	12	yes	—
65	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
66	C₉	—	3	3	no	—
67	C₉	—	3	24	no	—
68	C₉	C₃²×C₉×C₃₇	1	3	yes	—
69	C₉	C₃²×C₉×C₃₇	1	24	yes	—
70	C₃	—	3	12	no	—
71	C₃	C₃³×C₉×C₃₇	1	1	yes	characteristic
72	C₃	C₃₇×3₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
4. C₃₇×#6887	C₃	semidirect
5. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
6. C₃₇×#737	C₉	semidirect
18. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
21. C₃×C₉×C₃₇	C₃×C₉	semidirect
26. #10624864	C₃₇	direct (N × H)
35. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
37. #6887	C₃×C₃₇	semidirect
38. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
42. #737	C₉×C₃₇	semidirect
53. C₃₇	#10624864	direct (N × H)
55. 3₋¹⁺²	C₃×C₉×C₃₇	semidirect
58. C₃×C₉	C₃×C₉×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃₇×#737	3
C₉×3₋¹⁺²×C₃₇	3
C₉×H(3,𝔽₃)×C₃₇	3
C₃₇×#6887	3
C₃×C₃₇×C₉⋊C₉	3
#10624864	37

Subgroups — C37×#10624864 #41004653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624864 #41004653