Subgroups — C₃×#22582507 #41007327

Order 26973.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688519	C₃	1	3	yes	maximal, complemented
2	#22582507	C₃	1	9	yes	maximal, direct product factor
3	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
5	#15688519	—	3	6	no	—
6	#15688519	C₃²	1	9	yes	complemented
7	C₃²×C₉×C₃₇	—	3	1	no	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₉²×C₃₇	—	3	2	no	—
10	C₉²×C₃₇	C₃²	1	3	yes	complemented
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃²×C₃₇⋊C₃	—	9	3	no	—
13	C₃×C₉×C₃₇	—	3	9	no	—
14	C₃×C₉×C₃₇	—	3	2	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	3	no	—
17	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
18	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
19	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
20	C₃²⋊(C₃₇⋊C₃)	—	9	6	no	—
21	C₃×#6879	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	3	no	—
23	C₃²×C₃₇	#732	1	2	yes	—
24	C₃²×C₃₇	#732	1	1	yes	characteristic
25	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×C₃₇⋊C₃	—	9	9	no	—
27	C₃×C₃₇⋊C₃	—	27	9	no	—
28	C₉×C₃₇	—	3	9	no	—
29	C₉×C₃₇	—	3	2	no	—
30	C₉×C₃₇	—	3	1	no	—
31	C₃×#732	—	37	3	no	—
32	C₃×C₉²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
33	#6879	—	37	9	no	—
34	C₃×C₃₇	—	3	1	no	—
35	C₃×C₃₇	—	3	2	no	—
36	C₃×C₃₇	#6879	1	3	yes	complemented
37	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
38	C₃₇⋊C₃	—	27	9	no	—
39	C₃×H(3,𝔽₃)	—	111	3	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
42	#732	—	37	9	no	—
43	#732	—	111	6	no	—
44	C₉²	—	3	2	no	—
45	C₉²	C₃×C₃₇⋊C₃	1	3	yes	complemented
46	C₃₇	C₃×#6879	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	333	3	no	—
48	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
49	H(3,𝔽₃)	—	333	6	no	—
50	H(3,𝔽₃)	—	111	9	no	—
51	C₃×C₉	—	3	9	no	—
52	C₃×C₉	—	3	2	no	—
53	C₃×C₉	—	3	1	no	—
54	C₃×C₉	—	3	3	no	—
55	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
56	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
57	C₃²	—	3	3	no	—
58	C₃²	—	333	9	no	—
59	C₃²	—	999	9	no	—
60	C₃²	#15688519	1	2	yes	—
61	C₃²	#15688519	1	1	yes	characteristic
62	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
63	C₉	—	3	9	no	—
64	C₉	—	3	2	no	—
65	C₉	—	3	1	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	2	no	—
68	C₃	—	999	9	no	—
69	C₃	#22582507	1	3	yes	direct product factor
70	C₃	C₃×#15688519	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688519
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃×C₉²×C₃₇	C₃
socle	C₃²×C₃₇	#732
hypercenter	C₃×C₉²	C₃₇⋊C₃
hypocenter	C₃₇	C₃×#6879
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×C₉²

Complements

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

N	H ≅ G/N	type
1. C₃×#15688519	C₃	semidirect
2. #22582507	C₃	direct (N × H)
3. C₃×C₉²×C₃₇	C₃	semidirect
6. #15688519	C₃²	semidirect
10. C₉²×C₃₇	C₃²	semidirect
32. C₃×C₉²	C₃₇⋊C₃	semidirect
36. C₃×C₃₇	#6879	semidirect
45. C₉²	C₃×C₃₇⋊C₃	semidirect
46. C₃₇	C₃×#6879	semidirect
69. C₃	#22582507	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×#15688519	3
#22582507	3
C₃×C₉²×C₃₇	3
C₃×#6879	37

Subgroups — C3×#22582507 #41007327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582507 #41007327