Subgroups — C₂₇×3₋¹⁺²×C₃₇ #41004515

Order 26973.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃₇×C₉⋊C₂₇	C₃	1	6	yes	maximal, complemented
4	C₃₇×C₃²⋊C₂₇	C₃	1	2	yes	maximal
5	C₉×C₂₇×C₃₇	C₃	1	3	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×C₂₇×C₃₇	—	3	3	no	—
9	C₃×C₂₇×C₃₇	C₃²	1	6	yes	—
10	C₃×C₂₇×C₃₇	C₃²	1	2	yes	—
11	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₉⋊C₉	C₉	1	6	yes	—
13	C₃₇×C₃²⋊C₉	C₉	1	2	yes	—
14	C₉²×C₃₇	C₃²	1	3	yes	—
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	3₋¹⁺²×C₃₇	C₂₇	1	9	yes	direct product factor
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	3	2	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	6	yes	—
20	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
21	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
22	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
23	C₂₇×C₃₇	—	3	2	no	—
24	C₂₇×C₃₇	—	3	6	no	—
25	C₂₇×C₃₇	3₋¹⁺²	1	3	yes	direct product factor
26	C₂₇×3₋¹⁺²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	C₃×C₂₇	1	3	yes	—
30	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
31	C₉×C₃₇	—	3	2	no	—
32	C₉×C₃₇	—	3	6	no	—
33	C₉×C₃₇	C₃×C₂₇	1	9	yes	complemented
34	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
35	C₉×C₃₇	#737	1	2	yes	—
36	C₃²×C₂₇	C₃×C₃₇	1	1	yes	characteristic
37	C₉×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
38	C₉⋊C₂₇	C₃×C₃₇	1	6	yes	complemented
39	C₃²⋊C₂₇	C₃×C₃₇	1	2	yes	—
40	C₉×C₂₇	C₃×C₃₇	1	3	yes	complemented
41	C₃×C₃₇	—	3	3	no	—
42	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
43	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
44	C₃×C₃₇	#6903	1	2	yes	—
45	C₃×3₋¹⁺²	C₉×C₃₇	1	1	yes	characteristic
46	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
47	C₃×C₂₇	—	3	3	no	—
48	C₃×C₂₇	C₃²×C₃₇	1	6	yes	—
49	C₃×C₂₇	C₃²×C₃₇	1	2	yes	—
50	C₃×C₂₇	C₃²×C₃₇	1	1	yes	characteristic
51	C₉⋊C₉	C₉×C₃₇	1	6	yes	—
52	C₃²⋊C₉	C₉×C₃₇	1	2	yes	—
53	C₉²	C₃²×C₃₇	1	3	yes	—
54	C₃₇	C₂₇×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
55	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
56	3₋¹⁺²	C₂₇×C₃₇	1	9	yes	direct product factor
57	C₃×C₉	—	3	1	no	—
58	C₃×C₉	—	3	2	no	—
59	C₃×C₉	C₃×C₉×C₃₇	1	6	yes	—
60	C₃×C₉	C₃×C₉×C₃₇	1	2	yes	—
61	C₃×C₉	C₃×C₉×C₃₇	1	3	yes	—
62	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
63	C₂₇	—	3	2	no	—
64	C₂₇	—	3	6	no	—
65	C₂₇	3₋¹⁺²×C₃₇	1	3	yes	direct product factor
66	C₃²	—	3	1	no	—
67	C₃²	—	3	2	no	—
68	C₃²	C₃×C₂₇×C₃₇	1	3	yes	—
69	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
70	C₉	—	3	2	no	—
71	C₉	—	3	6	no	—
72	C₉	C₃×C₂₇×C₃₇	1	9	yes	complemented
73	C₉	C₃×3₋¹⁺²×C₃₇	1	1	yes	characteristic
74	C₉	C₃₇×#737	1	2	yes	—
75	C₃	—	3	3	no	—
76	C₃	C₉×3₋¹⁺²×C₃₇	1	1	yes	characteristic
77	C₃	C₃²×C₂₇×C₃₇	1	1	yes	characteristic
78	C₃	C₃₇×#6903	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₇×C₉⋊C₂₇	C₃	semidirect
5. C₉×C₂₇×C₃₇	C₃	semidirect
16. 3₋¹⁺²×C₃₇	C₂₇	direct (N × H)
25. C₂₇×C₃₇	3₋¹⁺²	direct (N × H)
26. C₂₇×3₋¹⁺²	C₃₇	direct (N × H)
33. C₉×C₃₇	C₃×C₂₇	semidirect
38. C₉⋊C₂₇	C₃×C₃₇	semidirect
40. C₉×C₂₇	C₃×C₃₇	semidirect
54. C₃₇	C₂₇×3₋¹⁺²	direct (N × H)
56. 3₋¹⁺²	C₂₇×C₃₇	direct (N × H)
65. C₂₇	3₋¹⁺²×C₃₇	direct (N × H)
72. C₉	C₃×C₂₇×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂₇×C₃₇	3
C₉×3₋¹⁺²×C₃₇	3
C₃₇×C₉⋊C₂₇	3
C₃₇×C₃²⋊C₂₇	3
C₉×C₂₇×C₃₇	3
C₂₇×3₋¹⁺²	37

Subgroups — C27×3−1+2×C37 #41004515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₇×3₋¹⁺²×C₃₇ #41004515