Subgroups — #41006021

Order 26973.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582552	C₃	1	2	yes	maximal
2	#22582551	C₃	1	2	yes	maximal
3	#22582455	C₃	1	4	yes	maximal, complemented
4	#22582454	C₃	1	4	yes	maximal, complemented
5	C₃₇×#6860	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×#11758608	C₃²	1	4	yes	—
8	C₃×#11758606	—	3	2	no	—
9	C₃²⋊#66524	C₃²	1	4	yes	—
10	#15688502	—	3	4	no	—
11	#15688502	—	3	2	no	—
12	#15688502	—	3	4	no	—
13	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
14	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	—	9	2	no	—
17	C₃²×C₃₇⋊C₃	—	9	2	no	—
18	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
19	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
20	3₋¹⁺²×C₃₇	—	9	2	no	—
21	C₃×C₉×C₃₇	—	3	2	no	—
22	C₃×C₉×C₃₇	—	3	2	no	—
23	#11758608	—	9	4	no	—
24	C₃×#66524	—	3	4	no	—
25	C₃×#66524	—	3	4	no	—
26	#11758606	—	9	6	no	—
27	#10624662	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	—	3	2	no	—
30	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	—	3	4	no	—
32	C₃×C₃₇⋊C₃	—	3	2	no	—
33	C₃×C₃₇⋊C₃	—	3	2	no	—
34	C₃×C₃₇⋊C₃	—	3	4	no	—
35	C₃×C₃₇⋊C₃	—	9	9	no	—
36	C₉×C₃₇	—	9	2	no	—
37	C₉×C₃₇	—	9	2	no	—
38	#66524	—	9	4	no	—
39	#66524	—	9	4	no	—
40	#6891	—	37	2	no	—
41	#6891	—	37	2	no	—
42	#6860	—	37	4	no	—
43	#6860	—	37	4	no	—
44	#6860	C₃₇⋊C₃	1	1	yes	characteristic, complemented
45	C₃×C₃₇	—	9	1	no	—
46	C₃×C₃₇	#6909	1	2	yes	—
47	C₃×C₃₇	#6912	1	2	yes	—
48	C₃₇⋊C₃	—	9	1	no	—
49	C₃₇⋊C₃	—	9	2	no	—
50	C₃⁴	—	37	1	no	—
51	C₃×3₋¹⁺²	—	37	4	no	—
52	C₃×3₋¹⁺²	—	111	2	no	—
53	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
54	C₃²⋊C₉	—	37	4	no	—
55	C₃²⋊C₉	—	111	4	no	—
56	C₃²⋊C₉	—	111	2	no	—
57	C₃²⋊C₉	—	111	4	no	—
58	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
59	C₃₇	#10624662	1	1	yes	characteristic, complemented, Sylow
60	C₃³	—	37	2	no	—
61	C₃³	—	37	1	no	—
62	C₃³	—	333	2	no	—
63	C₃³	—	333	2	no	—
64	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
65	3₋¹⁺²	—	9	2	no	—
66	3₋¹⁺²	—	333	6	no	—
67	3₋¹⁺²	—	333	4	no	—
68	C₃×C₉	—	3	2	no	—
69	C₃×C₉	—	3	2	no	—
70	C₃×C₉	—	111	4	no	—
71	C₃×C₉	—	111	4	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	333	9	no	—
75	C₃²	—	111	2	no	—
76	C₃²	—	111	4	no	—
77	C₃²	—	111	4	no	—
78	C₃²	—	111	2	no	—
79	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
80	C₉	—	9	2	no	—
81	C₉	—	9	2	no	—
82	C₉	—	333	4	no	—
83	C₉	—	333	4	no	—
84	C₃	—	9	1	no	—
85	C₃	—	333	1	no	—
86	C₃	—	333	2	no	—
87	C₃	#22582635	1	2	yes	—
88	C₃	#22582650	1	2	yes	—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6860	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6860	C₃₇⋊C₃
hypocenter	C₃₇	#10624662
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6860

Complements

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

N	H ≅ G/N	type
3. #22582455	C₃	semidirect
4. #22582454	C₃	semidirect
5. C₃₇×#6860	C₃	semidirect
44. #6860	C₃₇⋊C₃	semidirect
59. C₃₇	#10624662	semidirect

Maximal subgroups

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

subgroup	index
#22582552	3
#22582551	3
#22582455	3
#22582454	3
C₃₇×#6860	3
#10624662	37