Subgroups — #41008126

Order 26973.

828 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582613	C₃	1	27	yes	maximal, complemented
2	C₃×#15688511	C₃	1	12	yes	maximal, complemented
3	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
4	#15688511	C₃²	1	9	yes	—
5	#15688511	C₃²	1	72	yes	complemented
6	C₃×#11758601	C₃²	1	36	yes	complemented
7	C₃₇×#737	C₉	1	18	yes	—
8	C₃₇×#737	C₃²	1	9	yes	—
9	C₃×3₋¹⁺²×C₃₇	C₉	1	8	yes	—
10	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
11	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
12	C₃³×C₃₇	C₃×C₉	1	4	yes	—
13	3₋¹⁺²×C₃₇	C₃×C₉	1	72	yes	—
14	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	—
15	C₃×C₉×C₃₇	C₃×C₉	1	8	yes	—
16	C₃×C₉×C₃₇	C₃×C₉	1	24	yes	—
17	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
18	C₃×C₉×C₃₇	C₃³	1	12	yes	—
19	#11758601	C₃³	1	27	yes	—
20	#10624918	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	—	9	4	no	—
22	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
23	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
24	C₉×C₃₇	C₃²×C₉	1	2	yes	—
25	C₉×C₃₇	C₃²×C₉	1	24	yes	—
26	C₉×C₃₇	C₃⁴	1	1	yes	characteristic
27	C₃×#737	C₃₇⋊C₃	1	1	yes	characteristic
28	#6903	—	37	27	no	—
29	C₃×C₂₇⋊C₃	—	37	12	no	—
30	C₃×C₃₇	—	3	1	no	—
31	C₃×C₃₇	—	3	12	no	—
32	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
33	#737	#66524	1	18	yes	—
34	#737	C₃×C₃₇⋊C₃	1	9	yes	—
35	C₃×3₋¹⁺²	#66524	1	8	yes	—
36	C₃×H(3,𝔽₃)	#66524	1	1	yes	characteristic
37	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
38	C₂₇⋊C₃	—	37	9	no	—
39	C₂₇⋊C₃	—	37	72	no	—
40	C₃×C₂₇	—	37	36	no	—
41	C₃₇	#10624918	1	1	yes	characteristic, complemented, Sylow
42	C₃³	C₃×#66524	1	4	yes	—
43	3₋¹⁺²	C₃×#66524	1	72	yes	—
44	H(3,𝔽₃)	C₃×#66524	1	9	yes	—
45	C₃×C₉	C₃×#66524	1	8	yes	—
46	C₃×C₉	C₃×#66524	1	24	yes	—
47	C₃×C₉	C₃²×C₃₇⋊C₃	1	12	yes	—
48	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
49	C₂₇	—	37	27	no	—
50	C₃²	—	9	4	no	—
51	C₃²	C₃²×#66524	1	1	yes	characteristic
52	C₃²	C₃²×#66524	1	12	yes	—
53	C₉	C₃²×#66524	1	2	yes	—
54	C₉	C₃²×#66524	1	24	yes	—
55	C₉	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
56	C₃	—	3	1	no	—
57	C₃	—	3	12	no	—
58	C₃	C₃³×#66524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₉	C₃³×C₃₇⋊C₃
commutator, socle	C₃×C₃₇	C₃³×C₉
Fitting	C₃×C₃₇×#737	C₃
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
cosocle	C₉×C₃₇	C₃⁴
hypercenter	C₃×#737	C₃₇⋊C₃
hypocenter	C₃₇	#10624918
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. #22582613	C₃	semidirect
2. C₃×#15688511	C₃	semidirect
5. #15688511	C₃²	semidirect
6. C₃×#11758601	C₃²	semidirect
41. C₃₇	#10624918	semidirect

Maximal subgroups

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

subgroup	index
#22582613	3
C₃×#15688511	3
C₃×C₃₇×#737	3
#10624918	37