Subgroups — #41008468

Order 26973.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×(C₃₇⋊C₃)⋊C₂₇	C₃	1	3	yes	maximal
2	#22582688	C₃	1	9	yes	maximal, complemented
3	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	(C₃₇⋊C₃)⋊C₂₇	C₉	1	27	yes	complemented
5	C₃×#11758603	C₃²	1	9	yes	—
6	C₃×C₉×C₃₇⋊C₃	C₉	1	3	yes	—
7	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₉×#66524	C₉	1	2	yes	—
9	C₉×#66524	C₉	1	3	yes	—
10	C₉×#66524	C₃²	1	3	yes	—
11	C₉²×C₃₇	C₉	1	1	yes	characteristic
12	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
13	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
14	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃×#66524	C₃×C₉	1	6	yes	—
16	C₃×#66524	C₃×C₉	1	1	yes	characteristic
17	C₃×#66524	C₃³	1	1	yes	characteristic
18	C₉×C₃₇⋊C₃	—	3	9	no	—
19	C₉×C₃₇⋊C₃	C₃×C₉	1	9	yes	—
20	#11758603	C₃×C₉	1	27	yes	complemented
21	C₃×#6864	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
23	C₃×C₃₇⋊C₃	C₉²	1	3	yes	—
24	C₃×C₃₇⋊C₃	C₂₇⋊C₃	1	9	yes	—
25	C₉×C₃₇	—	3	3	no	—
26	C₉×C₃₇	C₉²	1	3	yes	—
27	#66524	—	3	6	no	—
28	#66524	C₉²	1	3	yes	—
29	#66524	C₃²×C₉	1	3	yes	—
30	C₃²×C₂₇	—	37	3	no	—
31	C₃×C₉²	—	37	1	no	—
32	#6864	—	37	9	no	—
33	C₃×C₃₇	C₃×C₉²	1	1	yes	characteristic
34	C₃×C₃₇	C₃×C₂₇⋊C₃	1	3	yes	—
35	C₃₇⋊C₃	#6864	1	9	yes	complemented
36	C₃²×C₉	—	37	1	no	—
37	C₃²×C₉	—	37	3	no	—
38	C₃×C₂₇	—	37	9	no	—
39	C₃×C₂₇	—	37	27	no	—
40	C₉²	—	37	3	no	—
41	C₉²	—	37	3	no	—
42	C₉²	—	37	2	no	—
43	C₉²	#66526	1	1	yes	characteristic
44	C₃₇	C₃×#6864	1	1	yes	characteristic, complemented, Sylow
45	C₃³	—	37	1	no	—
46	C₃×C₉	—	37	1	no	—
47	C₃×C₉	—	37	1	no	—
48	C₃×C₉	—	37	9	no	—
49	C₃×C₉	—	37	6	no	—
50	C₃×C₉	—	111	9	no	—
51	C₃×C₉	C₃×#66526	1	3	yes	—
52	C₃×C₉	C₃×#66526	1	1	yes	characteristic
53	C₂₇	—	37	27	no	—
54	C₃²	—	37	3	no	—
55	C₃²	—	37	9	no	—
56	C₃²	C₃²×#66526	1	1	yes	characteristic
57	C₉	—	3	3	no	—
58	C₉	—	37	3	no	—
59	C₉	—	37	3	no	—
60	C₉	—	111	6	no	—
61	C₉	C₉×#66526	1	3	yes	—
62	C₃	—	37	9	no	—
63	C₃	C₃×C₉×#66526	1	1	yes	characteristic
64	C₃	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×#66526
commutator	C₃×C₃₇	C₃×C₉²
Frattini	C₃²	C₃²×#66526
Fitting	C₉²×C₃₇	C₉
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₉²	#66526
hypocenter	C₃₇	C₃×#6864
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
2. #22582688	C₃	semidirect
4. (C₃₇⋊C₃)⋊C₂₇	C₉	semidirect
20. #11758603	C₃×C₉	semidirect
35. C₃₇⋊C₃	#6864	semidirect
44. C₃₇	C₃×#6864	semidirect

Maximal subgroups

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

subgroup	index
C₃×(C₃₇⋊C₃)⋊C₂₇	3
#22582688	3
C₉²×C₃₇⋊C₃	3
C₃×#6864	37