Subgroups — #41005226

Order 26973.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758606	C₃	1	4	yes	maximal
2	#22582441	C₃	1	8	yes	maximal, complemented
3	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
5	C₃×#11758606	—	3	12	no	—
6	C₃×#11758606	—	3	12	no	—
7	C₃×#11758606	—	3	24	no	—
8	C₃²×#66524	C₃²	1	8	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
10	C₃³×C₃₇	C₃³	1	1	yes	characteristic
11	C₃²×C₃₇⋊C₃	—	3	8	no	—
12	C₃²×C₃₇⋊C₃	—	3	4	no	—
13	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
14	C₃×C₉×C₃₇	—	3	4	no	—
15	C₃×C₉×C₃₇	—	3	8	no	—
16	C₃×C₉×C₃₇	3₋¹⁺²	1	12	yes	complemented
17	C₃×#66524	—	3	8	no	—
18	C₃×#66524	—	3	8	no	—
19	C₃×#66524	—	3	8	no	—
20	C₃×#66524	3₋¹⁺²	1	24	yes	complemented
21	#11758606	—	3	108	no	—
22	#10624571	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
24	C₃²×C₃₇	C₃×3₋¹⁺²	1	8	yes	—
25	C₃²×C₃₇	C₃×3₋¹⁺²	1	4	yes	—
26	C₃×C₃₇⋊C₃	—	3	12	no	—
27	C₃×C₃₇⋊C₃	—	9	1	no	—
28	C₃×C₃₇⋊C₃	—	9	8	no	—
29	C₉×C₃₇	—	3	12	no	—
30	#66524	—	3	24	no	—
31	C₃²×3₋¹⁺²	—	37	4	no	—
32	#6855	—	37	8	no	—
33	#6855	C₃₇⋊C₃	1	1	yes	characteristic, complemented
34	C₃×C₃₇	#6891	1	4	yes	—
35	C₃×C₃₇	#6900	1	1	yes	characteristic
36	C₃×C₃₇	#6900	1	8	yes	—
37	C₃₇⋊C₃	—	9	3	no	—
38	C₃⁴	—	37	1	no	—
39	C₃×3₋¹⁺²	—	111	24	no	—
40	C₃×3₋¹⁺²	—	111	12	no	—
41	C₃×3₋¹⁺²	—	111	12	no	—
42	C₃²×C₉	—	37	8	no	—
43	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
44	C₃₇	#10624571	1	1	yes	characteristic, complemented, Sylow
45	C₃³	—	37	3	no	—
46	C₃³	—	111	8	no	—
47	C₃³	—	111	4	no	—
48	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
49	3₋¹⁺²	—	111	108	no	—
50	C₃×C₉	—	3	4	no	—
51	C₃×C₉	—	3	8	no	—
52	C₃×C₉	—	37	24	no	—
53	C₃×C₉	—	111	8	no	—
54	C₃×C₉	—	111	8	no	—
55	C₃×C₉	—	111	8	no	—
56	C₃×C₉	#11758606	1	12	yes	complemented
57	C₃²	—	333	1	no	—
58	C₃²	—	333	8	no	—
59	C₃²	—	111	12	no	—
60	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
61	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	8	yes	—
62	C₃²	C₃×#11758606	1	4	yes	—
63	C₉	—	3	12	no	—
64	C₉	—	111	24	no	—
65	C₃	—	333	3	no	—
66	C₃	#22582552	1	4	yes	—
67	C₃	#22582603	1	1	yes	characteristic
68	C₃	#22582604	1	8	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃₇×#6855	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624571
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

Complements

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

N	H ≅ G/N	type
2. #22582441	C₃	semidirect
3. C₃₇×#6855	C₃	semidirect
16. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
20. C₃×#66524	3₋¹⁺²	semidirect
33. #6855	C₃₇⋊C₃	semidirect
44. C₃₇	#10624571	semidirect
56. C₃×C₉	#11758606	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11758606	3
#22582441	3
C₃₇×#6855	3
#10624571	37