Subgroups — C₃×C₃₇×#6875 #41004497

Order 26973.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×C₂₇⋊C₃	C₃	1	2	yes	maximal
2	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
4	C₃₇×#6875	C₃	1	9	yes	maximal, direct product factor
5	C₃²×C₉×C₃₇	—	3	1	no	—
6	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
7	C₃₇×C₂₇⋊C₃	—	3	4	no	—
8	C₃₇×C₂₇⋊C₃	C₉	1	6	yes	complemented
9	C₃₇×C₂₇⋊C₃	C₃²	1	6	yes	—
10	C₃₇×C₂₇⋊C₃	C₃²	1	3	yes	—
11	C₃×C₂₇×C₃₇	—	3	2	no	—
12	C₃×C₂₇×C₃₇	C₉	1	3	yes	complemented
13	C₃₇×C₉⋊C₉	—	3	2	no	—
14	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
15	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	3	no	—
17	C₃×C₉×C₃₇	—	9	3	no	—
18	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
21	C₃×C₉×C₃₇	3₋¹⁺²	1	4	yes	—
22	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
23	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
24	C₂₇×C₃₇	—	3	6	no	—
25	C₂₇×C₃₇	C₃×C₉	1	9	yes	complemented
26	C₃×#6875	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	C₉⋊C₉	1	2	yes	—
29	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
30	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
31	C₉×C₃₇	—	9	3	no	—
32	C₉×C₃₇	C₉⋊C₉	1	4	yes	—
33	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
34	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
35	C₉×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
36	C₃×C₂₇⋊C₃	C₃×C₃₇	1	2	yes	—
37	C₃×C₂₇⋊C₃	C₃×C₃₇	1	1	yes	characteristic
38	C₃×C₉⋊C₉	C₃×C₃₇	1	1	yes	characteristic
39	#6875	C₃×C₃₇	1	9	yes	direct product factor
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	—	3	2	no	—
42	C₃×C₃₇	#6875	1	3	yes	direct product factor
43	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
44	C₃²×C₉	—	3	1	no	—
45	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
46	C₂₇⋊C₃	—	3	4	no	—
47	C₂₇⋊C₃	C₉×C₃₇	1	6	yes	complemented
48	C₂₇⋊C₃	C₃²×C₃₇	1	6	yes	—
49	C₂₇⋊C₃	C₃²×C₃₇	1	3	yes	—
50	C₃×C₂₇	—	3	2	no	—
51	C₃×C₂₇	C₉×C₃₇	1	3	yes	complemented
52	C₉⋊C₉	—	3	2	no	—
53	C₉⋊C₉	C₃²×C₃₇	1	3	yes	—
54	C₃₇	C₃×#6875	1	1	yes	characteristic, direct product factor, Sylow
55	C₃³	3₋¹⁺²×C₃₇	1	1	yes	characteristic
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	9	3	no	—
58	C₃×C₉	C₃×C₉×C₃₇	1	1	yes	characteristic
59	C₃×C₉	C₃×C₉×C₃₇	1	2	yes	—
60	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
61	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
62	C₃×C₉	3₋¹⁺²×C₃₇	1	4	yes	—
63	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
64	C₂₇	—	3	6	no	—
65	C₂₇	C₃×C₉×C₃₇	1	9	yes	complemented
66	C₃²	—	3	3	no	—
67	C₃²	C₃₇×C₉⋊C₉	1	2	yes	—
68	C₃²	C₃₇×C₉⋊C₉	1	1	yes	characteristic
69	C₃²	C₃×3₋¹⁺²×C₃₇	1	1	yes	characteristic
70	C₉	—	9	3	no	—
71	C₉	C₃₇×C₉⋊C₉	1	2	yes	—
72	C₉	C₃₇×C₉⋊C₉	1	4	yes	—
73	C₉	C₃²×C₉×C₃₇	1	1	yes	characteristic
74	C₉	C₃×3₋¹⁺²×C₃₇	1	2	yes	—
75	C₃	—	3	1	no	—
76	C₃	—	3	2	no	—
77	C₃	C₃₇×#6875	1	3	yes	direct product factor
78	C₃	C₃×C₃₇×C₉⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₃₇	C₉⋊C₉
commutator	C₉	C₃²×C₉×C₃₇
Frattini, cosocle	C₃×C₉	C₃³×C₃₇
prime radical	C₃³×C₃₇	3₋¹⁺²
LCS	G ▷ C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₇×#6875	C₃	direct (N × H)
8. C₃₇×C₂₇⋊C₃	C₉	semidirect
12. C₃×C₂₇×C₃₇	C₉	semidirect
25. C₂₇×C₃₇	C₃×C₉	semidirect
26. C₃×#6875	C₃₇	direct (N × H)
39. #6875	C₃×C₃₇	direct (N × H)
42. C₃×C₃₇	#6875	direct (N × H)
47. C₂₇⋊C₃	C₉×C₃₇	semidirect
51. C₃×C₂₇	C₉×C₃₇	semidirect
54. C₃₇	C₃×#6875	direct (N × H)
65. C₂₇	C₃×C₉×C₃₇	semidirect
77. C₃	C₃₇×#6875	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×C₂₇⋊C₃	3
C₃×C₃₇×C₉⋊C₉	3
C₃₇×#6875	3
C₃×#6875	37

Subgroups — C3×C37×#6875 #41004497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6875 #41004497