Subgroups — C₃×C₃₇×#6882 #41004632

Order 26973.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#733	C₃	1	3	yes	maximal
2	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃₇×#6882	C₃	1	9	yes	maximal, direct product factor
4	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
5	C₃²×C₉×C₃₇	—	3	1	no	—
6	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
7	C₃₇×#733	—	3	6	no	—
8	C₃₇×#733	C₃²	1	9	yes	—
9	C₃₇×C₉⋊C₉	—	3	2	no	—
10	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	3₋¹⁺²×C₃₇	—	3	9	no	—
13	3₋¹⁺²×C₃₇	—	9	6	no	—
14	C₃×C₉×C₃₇	—	3	2	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	3	no	—
17	C₃×C₉×C₃₇	—	9	3	no	—
18	C₃×C₉×C₃₇	—	9	3	no	—
19	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
20	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
21	C₃×#6882	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₃²×C₃₇	—	3	3	no	—
23	C₃²×C₃₇	#732	1	2	yes	—
24	C₃²×C₃₇	#732	1	1	yes	characteristic
25	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
26	C₉×C₃₇	—	3	2	no	—
27	C₉×C₃₇	—	3	1	no	—
28	C₉×C₃₇	—	9	3	no	—
29	C₉×C₃₇	—	9	9	no	—
30	C₃×#733	C₃×C₃₇	1	3	yes	—
31	C₃×C₉⋊C₉	C₃×C₃₇	1	1	yes	characteristic
32	#6882	C₃×C₃₇	1	9	yes	direct product factor
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	—	3	2	no	—
35	C₃×C₃₇	#6882	1	3	yes	direct product factor
36	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	3	no	—
38	C₃²×C₉	—	3	1	no	—
39	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
40	#733	—	3	6	no	—
41	#733	C₃²×C₃₇	1	9	yes	—
42	C₉⋊C₉	—	3	2	no	—
43	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
44	C₃₇	C₃×#6882	1	1	yes	characteristic, direct product factor, Sylow
45	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
46	3₋¹⁺²	—	3	9	no	—
47	3₋¹⁺²	—	9	6	no	—
48	C₃×C₉	—	3	2	no	—
49	C₃×C₉	—	3	1	no	—
50	C₃×C₉	—	3	3	no	—
51	C₃×C₉	—	9	3	no	—
52	C₃×C₉	—	9	3	no	—
53	C₃×C₉	H(3,𝔽₃)×C₃₇	1	2	yes	—
54	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
55	C₃²	—	3	3	no	—
56	C₃²	C₃₇×#732	1	2	yes	—
57	C₃²	C₃₇×#732	1	1	yes	characteristic
58	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
59	C₉	—	3	2	no	—
60	C₉	—	3	1	no	—
61	C₉	—	9	3	no	—
62	C₉	—	9	9	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	2	no	—
65	C₃	C₃₇×#6882	1	3	yes	direct product factor
66	C₃	C₃×C₃₇×#732	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₃₇	#732
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₃₇
prime radical	C₃³×C₃₇	H(3,𝔽₃)
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃³×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#6882	C₃	direct (N × H)
21. C₃×#6882	C₃₇	direct (N × H)
32. #6882	C₃×C₃₇	direct (N × H)
35. C₃×C₃₇	#6882	direct (N × H)
44. C₃₇	C₃×#6882	direct (N × H)
65. C₃	C₃₇×#6882	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₃₇×#733	3
C₃×C₃₇×C₉⋊C₉	3
C₃₇×#6882	3
C₃×#6882	37

Subgroups — C3×C37×#6882 #41004632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6882 #41004632