Subgroups — #41005214

Order 26973.

524 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758608	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃²⋊#66524	C₃	1	2	yes	maximal
3	#22582441	C₃	1	9	yes	maximal, complemented
4	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	3	no	—
6	C₃×#11758608	—	3	9	no	—
7	C₃²×#66524	C₃²	1	6	yes	—
8	C₃²×#66524	C₃²	1	3	yes	—
9	C₃²⋊#66524	—	3	6	no	—
10	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
11	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
12	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
13	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
14	C₃³×C₃₇	—	3	3	no	—
15	C₃³×C₃₇	—	3	3	no	—
16	C₃³×C₃₇	—	3	6	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	3₋¹⁺²×C₃₇	—	3	27	no	—
20	C₃×C₉×C₃₇	—	3	3	no	—
21	C₃×C₉×C₃₇	—	3	6	no	—
22	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
23	#11758608	—	3	27	no	—
24	C₃×#66524	—	3	3	no	—
25	C₃×#66524	—	3	6	no	—
26	C₃×#66524	—	3	6	no	—
27	C₃×#66524	—	3	18	no	—
28	C₃×#66524	3₋¹⁺²	1	9	yes	complemented
29	#10624567	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	6	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
36	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
37	C₃²×C₃₇	#737	1	6	yes	—
38	C₉×C₃₇	—	3	9	no	—
39	#66524	—	3	9	no	—
40	#66524	—	9	6	no	—
41	C₃²×3₋¹⁺²	—	37	1	no	—
42	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
43	C₃×C₃²⋊C₉	—	37	2	no	—
44	#6855	—	37	9	no	—
45	C₃×C₃₇	—	9	3	no	—
46	C₃×C₃₇	#6891	1	1	yes	characteristic
47	C₃×C₃₇	#6891	1	1	yes	characteristic
48	C₃×C₃₇	#6893	1	2	yes	—
49	C₃×C₃₇	#6896	1	9	yes	—
50	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	C₃×3₋¹⁺²	—	3	3	no	—
52	C₃×3₋¹⁺²	—	3	9	no	—
53	C₃×3₋¹⁺²	—	111	9	no	—
54	C₃×3₋¹⁺²	—	111	3	no	—
55	C₃²×C₉	—	37	3	no	—
56	C₃²×C₉	—	37	6	no	—
57	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
58	C₃²⋊C₉	—	111	6	no	—
59	C₃₇	#10624567	1	1	yes	characteristic, complemented, Sylow
60	C₃³	—	3	3	no	—
61	C₃³	—	3	3	no	—
62	C₃³	—	3	6	no	—
63	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
64	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
65	3₋¹⁺²	—	3	27	no	—
66	3₋¹⁺²	—	111	27	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	3	6	no	—
69	C₃×C₉	—	37	9	no	—
70	C₃×C₉	—	111	6	no	—
71	C₃×C₉	—	111	18	no	—
72	C₃×C₉	—	111	3	no	—
73	C₃×C₉	—	111	6	no	—
74	C₃×C₉	#11758608	1	9	yes	complemented
75	C₃²	—	3	3	no	—
76	C₃²	—	3	3	no	—
77	C₃²	—	3	6	no	—
78	C₃²	—	9	9	no	—
79	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
80	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
81	C₃²	C₃×#11758608	1	3	yes	—
82	C₃²	#15688532	1	6	yes	—
83	C₉	—	3	9	no	—
84	C₉	—	333	6	no	—
85	C₉	—	111	9	no	—
86	C₃	—	9	3	no	—
87	C₃	#22582551	1	1	yes	characteristic
88	C₃	#22582554	1	1	yes	characteristic
89	C₃	#22582565	1	2	yes	—
90	C₃	#22582580	1	9	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624567
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
3. #22582441	C₃	semidirect
22. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
28. C₃×#66524	3₋¹⁺²	semidirect
59. C₃₇	#10624567	semidirect
74. C₃×C₉	#11758608	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11758608	3
C₃×C₃²⋊#66524	3
#22582441	3
C₃²×3₋¹⁺²×C₃₇	3
#10624567	37