Subgroups — C₃₇⋊C₃×#6864 #41006192

Order 26973.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₇×C₃₇⋊C₃	C₃	1	3	yes	maximal
2	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#22582464	C₃	1	2	yes	maximal, complemented
4	#22582463	C₃	1	3	yes	maximal, complemented
5	#22582462	C₃	1	3	yes	maximal, complemented
6	C₃₇×#6864	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₉×C₃₇⋊C₃	C₉	1	3	yes	—
8	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758601	C₃²	1	6	yes	—
10	C₂₇×C₃₇⋊C₃	C₉	1	27	yes	complemented
11	C₉×#66524	C₉	1	6	yes	—
12	C₉×#66524	C₃²	1	1	yes	characteristic
13	C₉×#66524	C₃²	1	1	yes	characteristic
14	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
15	C₉²×C₃₇	C₃²	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
17	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
18	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
19	C₃×#66524	C₃×C₉	1	3	yes	—
20	C₃×#66524	C₃×C₉	1	3	yes	—
21	C₃×#66524	C₃×C₉	1	2	yes	—
22	C₉×C₃₇⋊C₃	—	3	9	no	—
23	C₉×C₃₇⋊C₃	C₃×C₉	1	9	yes	—
24	C₂₇×C₃₇	C₃×C₉	1	9	yes	complemented
25	#11758601	C₃×C₉	1	18	yes	complemented
26	C₃×#6864	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
28	C₃×C₃₇⋊C₃	C₉²	1	3	yes	—
29	C₃×C₃₇⋊C₃	C₂₇⋊C₃	1	9	yes	—
30	C₉×C₃₇	—	3	3	no	—
31	C₉×C₃₇	C₃²×C₉	1	3	yes	—
32	#66524	—	3	3	no	—
33	#66524	—	3	3	no	—
34	#66524	C₉²	1	6	yes	—
35	C₃²×C₂₇	—	37	3	no	—
36	C₃×C₉²	—	37	1	no	—
37	#6864	—	37	3	no	—
38	#6864	—	37	3	no	—
39	#6864	—	37	2	no	—
40	#6864	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
41	C₃×C₃₇	C₃×C₉²	1	1	yes	characteristic
42	C₃×C₃₇	C₃×C₂₇⋊C₃	1	3	yes	—
43	C₃₇⋊C₃	#6864	1	9	yes	direct product factor
44	C₃²×C₉	—	37	1	no	—
45	C₃²×C₉	—	37	3	no	—
46	C₃×C₂₇	—	37	6	no	—
47	C₃×C₂₇	—	37	27	no	—
48	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
49	C₉²	—	37	6	no	—
50	C₉²	—	37	1	no	—
51	C₉²	—	37	1	no	—
52	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃₇	C₃×#6864	1	1	yes	characteristic, complemented, Sylow
54	C₃³	—	37	1	no	—
55	C₃×C₉	—	37	2	no	—
56	C₃×C₉	—	37	9	no	—
57	C₃×C₉	—	37	3	no	—
58	C₃×C₉	—	37	3	no	—
59	C₃×C₉	—	111	9	no	—
60	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
61	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
62	C₂₇	—	37	18	no	—
63	C₂₇	C₉×C₃₇⋊C₃	1	9	yes	complemented
64	C₃²	—	37	9	no	—
65	C₃²	—	37	3	no	—
66	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
67	C₉	—	3	3	no	—
68	C₉	—	37	6	no	—
69	C₉	—	111	3	no	—
70	C₉	—	111	3	no	—
71	C₉	C₃×C₉×C₃₇⋊C₃	1	3	yes	—
72	C₃	—	37	9	no	—
73	C₃	C₉²×C₃₇⋊C₃	1	1	yes	characteristic
74	C₃	C₂₇⋊C₃×C₃₇⋊C₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃×C₉²
Fitting	C₃₇×#6864	C₃
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	#6864	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6864
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ #6864

Complements

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

N	H ≅ G/N	type
3. #22582464	C₃	semidirect
4. #22582463	C₃	semidirect
5. #22582462	C₃	semidirect
6. C₃₇×#6864	C₃	semidirect
10. C₂₇×C₃₇⋊C₃	C₉	semidirect
24. C₂₇×C₃₇	C₃×C₉	semidirect
25. #11758601	C₃×C₉	semidirect
40. #6864	C₃₇⋊C₃	direct (N × H)
43. C₃₇⋊C₃	#6864	direct (N × H)
53. C₃₇	C₃×#6864	semidirect
63. C₂₇	C₉×C₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂₇×C₃₇⋊C₃	3
C₉²×C₃₇⋊C₃	3
#22582464	3
#22582463	3
#22582462	3
C₃₇×#6864	3
C₃×#6864	37

Subgroups — C37⋊C3×#6864 #41006192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊C₃×#6864 #41006192