Subgroups — #41008653

Order 26973.

828 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582731	C₃	1	27	yes	maximal, complemented
2	C₃×#15688541	C₃	1	12	yes	maximal, complemented
3	C₃×#15688530	C₃	1	1	yes	maximal, characteristic
4	#15688541	C₃²	1	9	yes	—
5	#15688541	C₃²	1	72	yes	complemented
6	C₃×#11758603	C₃²	1	36	yes	complemented
7	#15688530	C₉	1	18	yes	—
8	#15688530	C₃²	1	9	yes	—
9	C₃×#11758608	C₉	1	8	yes	—
10	C₃²×#66524	C₃²	1	4	yes	—
11	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
12	C₃³×C₃₇	C₃×C₉	1	4	yes	—
13	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	—
14	#11758608	C₃×C₉	1	72	yes	—
15	C₃×#66524	C₃×C₉	1	8	yes	—
16	C₃×#66524	C₃×C₉	1	24	yes	—
17	C₃×#66524	C₃³	1	1	yes	characteristic
18	C₃×#66524	C₃³	1	12	yes	—
19	#11758603	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	#66524	C₃²×C₉	1	2	yes	—
25	#66524	C₃²×C₉	1	24	yes	—
26	#66524	C₃⁴	1	1	yes	characteristic
27	C₃×#737	—	37	1	no	—
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	—	37	18	no	—
34	#737	—	37	9	no	—
35	C₃×3₋¹⁺²	—	37	8	no	—
36	C₃×H(3,𝔽₃)	#66526	1	1	yes	characteristic
37	C₃²×C₉	—	37	4	no	—
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×#66526	1	4	yes	—
43	3₋¹⁺²	—	37	72	no	—
44	H(3,𝔽₃)	C₃×#66526	1	9	yes	—
45	C₃×C₉	—	37	8	no	—
46	C₃×C₉	—	37	24	no	—
47	C₃×C₉	—	37	12	no	—
48	C₃×C₉	—	37	1	no	—
49	C₂₇	—	37	27	no	—
50	C₃²	—	9	4	no	—
51	C₃²	C₃²×#66526	1	1	yes	characteristic
52	C₃²	C₃²×#66526	1	12	yes	—
53	C₉	—	37	1	no	—
54	C₉	—	37	2	no	—
55	C₉	—	37	24	no	—
56	C₃	—	3	1	no	—
57	C₃	—	3	12	no	—
58	C₃	C₃³×#66526	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×#66526
commutator, socle	C₃×C₃₇	C₃³×C₉
Fitting, prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
cosocle	#66524	C₃⁴
hypercenter	C₃×H(3,𝔽₃)	#66526
hypocenter	C₃₇	#10624918
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #22582731	C₃	semidirect
2. C₃×#15688541	C₃	semidirect
5. #15688541	C₃²	semidirect
6. C₃×#11758603	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
#22582731	3
C₃×#15688541	3
C₃×#15688530	3
#10624918	37