Subgroups — #41007517

Order 26973.

1178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#66524	C₃	1	27	yes	maximal, complemented
2	C₃×C₃²⋊#66524	C₃	1	12	yes	maximal, complemented
3	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	C₃²	1	36	yes	complemented
5	C₃²⋊#66524	C₃²	1	9	yes	—
6	C₃²⋊#66524	C₃²	1	72	yes	complemented
7	C₃⁴×C₃₇	C₃²	1	4	yes	—
8	C₃×H(3,𝔽₃)×C₃₇	C₉	1	27	yes	complemented
9	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
10	C₃³×C₃₇	—	9	4	no	—
11	C₃³×C₃₇	—	9	8	no	—
12	C₃³×C₃₇	C₃×C₉	1	36	yes	complemented
13	C₃³×C₃₇	C₃³	1	12	yes	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	H(3,𝔽₃)×C₃₇	C₃×C₉	1	81	yes	complemented
16	C₃×#66524	—	9	36	no	—
17	C₃×#66524	C₃³	1	27	yes	—
18	#10624862	—	37	1	no	maximal, Sylow
19	C₃²×C₃₇	—	3	12	no	—
20	C₃²×C₃₇	—	3	1	no	—
21	C₃²×C₃₇	—	3	2	no	—
22	C₃²×C₃₇	—	3	24	no	—
23	C₃²×C₃₇	—	9	36	no	—
24	C₃²×C₃₇	C₃²×C₉	1	36	yes	—
25	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
26	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
27	#66524	—	3	27	no	—
28	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
29	C₉×H(3,𝔽₃)	—	37	27	no	—
30	C₃×C₃²⋊C₉	—	37	12	no	—
31	C₃×C₃₇	—	3	3	no	—
32	C₃×C₃₇	—	3	36	no	—
33	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
34	C₃×C₃₇	H(5,𝔽₃)	1	1	yes	characteristic
35	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
36	C₃⁴	C₃×C₃₇⋊C₃	1	4	yes	—
37	C₃×H(3,𝔽₃)	#66524	1	27	yes	complemented
38	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	9	yes	—
39	C₃²×C₉	—	37	36	no	—
40	C₃²⋊C₉	—	37	9	no	—
41	C₃²⋊C₉	—	37	72	no	—
42	C₃₇	#10624862	1	1	yes	characteristic, complemented, Sylow
43	C₃³	—	9	4	no	—
44	C₃³	—	9	8	no	—
45	C₃³	C₃×#66524	1	36	yes	complemented
46	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
47	C₃³	C₃²×C₃₇⋊C₃	1	12	yes	—
48	H(3,𝔽₃)	C₃×#66524	1	81	yes	complemented
49	C₃×C₉	—	37	27	no	—
50	C₃×C₉	—	333	36	no	—
51	C₃²	—	3	12	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	3	2	no	—
54	C₃²	—	3	24	no	—
55	C₃²	—	9	36	no	—
56	C₃²	C₃²×#66524	1	36	yes	—
57	C₃²	C₃²×#66524	1	3	yes	—
58	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
59	C₉	—	111	27	no	—
60	C₃	—	3	3	no	—
61	C₃	—	3	36	no	—
62	C₃	C₃³×#66524	1	1	yes	characteristic
63	C₃	C₃₇⋊H(5,𝔽₃)	1	1	yes	characteristic
64	C₃	#22582680	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle, cosocle	C₃²×C₃₇	C₃⁴
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624862
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)×#66524	C₃	semidirect
2. C₃×C₃²⋊#66524	C₃	semidirect
4. C₃²×#66524	C₃²	semidirect
6. C₃²⋊#66524	C₃²	semidirect
8. C₃×H(3,𝔽₃)×C₃₇	C₉	semidirect
12. C₃³×C₃₇	C₃×C₉	semidirect
15. H(3,𝔽₃)×C₃₇	C₃×C₉	semidirect
37. C₃×H(3,𝔽₃)	#66524	semidirect
42. C₃₇	#10624862	semidirect
45. C₃³	C₃×#66524	semidirect
48. H(3,𝔽₃)	C₃×#66524	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₃)×#66524	3
C₃×C₃²⋊#66524	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624862	37