Subgroups — C₃×#22582717 #41008629

Order 26973.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582717	C₃	1	9	yes	maximal, direct product factor
2	C₃×#15688534	C₃	1	3	yes	maximal, complemented
3	C₃×C₂₇×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#15688534	C₃²	1	9	yes	complemented
5	C₃×C₉×C₃₇⋊C₃	C₉	1	1	yes	characteristic
6	C₃×#11758601	C₉	1	1	yes	characteristic
7	C₃×#11758601	C₃²	1	1	yes	characteristic
8	C₂₇×C₃₇⋊C₃	C₉	1	6	yes	—
9	C₂₇×C₃₇⋊C₃	C₃²	1	3	yes	—
10	C₃×C₂₇×C₃₇	C₉	1	1	yes	characteristic
11	C₃²×C₃₇⋊C₃	C₂₇	1	1	yes	characteristic
12	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃×#66524	C₂₇	1	2	yes	—
14	C₉×C₃₇⋊C₃	C₂₇	1	6	yes	—
15	C₉×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
16	C₂₇×C₃₇	C₃×C₉	1	3	yes	—
17	#11758601	C₃×C₉	1	2	yes	—
18	#11758601	C₃×C₉	1	3	yes	—
19	#11758601	C₃³	1	1	yes	characteristic
20	C₃×C₈₁⋊C₃	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
22	C₃×C₃₇⋊C₃	—	3	3	no	—
23	C₃×C₃₇⋊C₃	C₃×C₂₇	1	3	yes	—
24	C₉×C₃₇	C₃×C₂₇	1	2	yes	—
25	C₉×C₃₇	C₃²×C₉	1	1	yes	characteristic
26	#66524	C₃×C₂₇	1	6	yes	—
27	C₃²×C₂₇	—	37	1	no	—
28	C₈₁⋊C₃	—	37	9	no	—
29	C₃×C₈₁	—	37	3	no	—
30	C₃×C₃₇	C₈₁⋊C₃	1	3	yes	complemented
31	C₃×C₃₇	C₃²×C₂₇	1	1	yes	characteristic
32	C₃₇⋊C₃	—	3	3	no	—
33	C₃²×C₉	—	37	1	no	—
34	C₃×C₂₇	—	37	1	no	—
35	C₃×C₂₇	—	37	1	no	—
36	C₃×C₂₇	—	37	6	no	—
37	C₃×C₂₇	—	37	3	no	—
38	C₃×C₂₇	#66526	1	1	yes	characteristic
39	C₈₁	—	37	9	no	—
40	C₃₇	C₃×C₈₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
41	C₃³	—	37	1	no	—
42	C₃×C₉	—	37	2	no	—
43	C₃×C₉	—	37	6	no	—
44	C₃×C₉	—	37	3	no	—
45	C₃×C₉	C₃×#66526	1	1	yes	characteristic
46	C₂₇	—	37	2	no	—
47	C₂₇	—	37	3	no	—
48	C₂₇	—	37	1	no	—
49	C₂₇	C₃×#66526	1	3	yes	—
50	C₃²	—	37	3	no	—
51	C₃²	—	111	3	no	—
52	C₃²	(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic
53	C₉	—	37	6	no	—
54	C₉	(C₃₇⋊C₃)⋊C₂₇	1	2	yes	—
55	C₉	C₃²×#66526	1	1	yes	characteristic
56	C₃	—	111	3	no	—
57	C₃	#22582717	1	3	yes	direct product factor
58	C₃	C₃×(C₃₇⋊C₃)⋊C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃×#66526
commutator	C₃×C₃₇	C₃²×C₂₇
Frattini	C₉	C₃²×#66526
Fitting	C₃×C₂₇×C₃₇	C₉
socle	C₃²×C₃₇	C₃×C₂₇
prime radical	C₃²×C₃₇⋊C₃	C₂₇
cosocle	#11758601	C₃³
hypercenter	C₃×C₂₇	#66526
hypocenter	C₃₇	C₃×C₈₁⋊C₃
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₂₇

Complements

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

N	H ≅ G/N	type
1. #22582717	C₃	direct (N × H)
2. C₃×#15688534	C₃	semidirect
4. #15688534	C₃²	semidirect
30. C₃×C₃₇	C₈₁⋊C₃	semidirect
40. C₃₇	C₃×C₈₁⋊C₃	semidirect
57. C₃	#22582717	direct (N × H)

Maximal subgroups

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

subgroup	index
#22582717	3
C₃×#15688534	3
C₃×C₂₇×C₃₇⋊C₃	3
C₃×C₈₁⋊C₃	37

Subgroups — C3×#22582717 #41008629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#22582717 #41008629