Subgroups — #41004744

Order 26973.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²⋊#11758601	C₃	1	3	yes	maximal
2	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃×#11758601	—	3	3	no	—
4	C₃²×C₉×C₃₇	C₉	1	2	yes	—
5	C₃²×C₉×C₃₇	C₉	1	1	yes	characteristic
6	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
7	C₉²×C₃₇	—	3	1	no	—
8	C₉²×C₃₇	—	3	2	no	—
9	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
10	C₃×C₉×C₃₇	—	3	1	no	—
11	C₃×C₉×C₃₇	—	3	2	no	—
12	C₃×C₉×C₃₇	—	3	2	no	—
13	C₃×C₉×C₃₇	—	3	1	no	—
14	C₃×C₉×C₃₇	—	3	2	no	—
15	C₃×C₉×C₃₇	—	3	6	no	—
16	C₃×C₉×C₃₇	C₂₇	1	3	yes	complemented
17	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
19	#11758601	—	9	3	no	—
20	#10624446	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	—	3	1	no	—
22	C₃²×C₃₇	—	3	2	no	—
23	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
24	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
25	C₃²×C₃₇	C₂₇⋊C₃	1	2	yes	—
26	C₉×C₃₇	—	3	3	no	—
27	C₉×C₃₇	—	3	2	no	—
28	C₉×C₃₇	—	3	6	no	—
29	C₉×C₃₇	#732	1	1	yes	characteristic
30	C₉×C₃₇	#733	1	2	yes	—
31	C₃×C₉²	C₃₇⋊C₃	1	1	yes	characteristic
32	C₃²⋊C₂₇	—	37	3	no	—
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	—	3	2	no	—
35	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
36	C₃×C₃₇	#6868	1	1	yes	characteristic
37	C₃×C₃₇	#6869	1	2	yes	—
38	C₃²×C₉	#66524	1	2	yes	—
39	C₃²×C₉	#66524	1	1	yes	characteristic
40	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
41	C₃×C₂₇	—	111	3	no	—
42	C₉²	—	3	2	no	—
43	C₉²	—	3	1	no	—
44	C₃₇	#10624446	1	1	yes	characteristic, complemented, Sylow
45	C₃³	C₃×#66524	1	1	yes	characteristic
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	3	2	no	—
48	C₃×C₉	—	3	2	no	—
49	C₃×C₉	—	3	1	no	—
50	C₃×C₉	—	3	2	no	—
51	C₃×C₉	—	3	6	no	—
52	C₃×C₉	#11758601	1	3	yes	complemented
53	C₃×C₉	#11758608	1	2	yes	—
54	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
55	C₂₇	—	333	3	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	3	2	no	—
58	C₃²	C₃²⋊#66524	1	1	yes	characteristic
59	C₃²	C₃×#11758601	1	1	yes	characteristic
60	C₃²	#15688511	1	2	yes	—
61	C₉	—	3	3	no	—
62	C₉	—	3	2	no	—
63	C₉	—	3	6	no	—
64	C₉	#15688519	1	1	yes	characteristic
65	C₉	#15688521	1	2	yes	—
66	C₃	—	3	1	no	—
67	C₃	—	3	2	no	—
68	C₃	C₃²⋊#11758601	1	1	yes	characteristic
69	C₃	#22582472	1	1	yes	characteristic
70	C₃	#22582474	1	2	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃×C₂₇
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃×C₉²×C₃₇	C₃
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×C₉²	C₃₇⋊C₃
hypocenter	C₃₇	#10624446
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
16. C₃×C₉×C₃₇	C₂₇	semidirect
44. C₃₇	#10624446	semidirect
52. C₃×C₉	#11758601	semidirect

Maximal subgroups

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

subgroup	index
C₃²⋊#11758601	3
C₃×C₉²×C₃₇	3
#10624446	37