Subgroups — #41005116

Order 26973.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688506	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688505	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
4	#22582441	C₃	1	9	yes	maximal
5	C₃×C₃₇×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₃²	1	3	yes	—
7	C₃²×#66524	C₃²	1	3	yes	—
8	C₃²×#66524	C₃²	1	3	yes	—
9	#15688506	—	3	3	no	—
10	#15688505	—	3	3	no	—
11	C₉×#66524	C₉	1	9	yes	complemented
12	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×C₉⋊C₉	—	3	3	no	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	3	no	—
17	C₃×C₉×C₃₇	—	3	9	no	—
18	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
19	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
20	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
21	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
22	C₃×#66524	—	3	9	no	—
23	C₃×#66524	—	3	3	no	—
24	C₃×#66524	—	3	3	no	—
25	C₃×#66524	—	3	9	no	—
26	C₃×#66524	—	3	9	no	—
27	C₃×#66524	C₃×C₉	1	9	yes	—
28	#10624553	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
30	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
31	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
32	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
33	C₃²×C₃₇	#737	1	9	yes	—
34	C₉×C₃₇	—	9	3	no	—
35	C₉×C₃₇	C₉⋊C₉	1	9	yes	—
36	#66524	—	3	9	no	—
37	#66524	—	9	3	no	—
38	#66524	—	9	3	no	—
39	C₃×C₉⋊C₉	—	37	1	no	—
40	C₃×C₉⋊C₉	—	37	1	no	—
41	C₃×C₉⋊C₉	C₃₇⋊C₃	1	1	yes	characteristic
42	C₃×C₉²	—	37	1	no	—
43	#6855	—	37	9	no	—
44	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
45	C₃×C₃₇	#6887	1	3	yes	—
46	C₃×C₃₇	#6899	1	3	yes	—
47	C₃×C₃₇	#6899	1	3	yes	—
48	C₃×C₃₇	#6899	1	3	yes	—
49	C₃²×C₉	—	37	3	no	—
50	C₃²×C₉	—	37	3	no	—
51	C₃²×C₉	—	37	3	no	—
52	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
53	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
54	C₉⋊C₉	—	3	3	no	—
55	C₉⋊C₉	—	111	3	no	—
56	C₉⋊C₉	—	111	3	no	—
57	C₉²	—	37	9	no	—
58	C₃₇	#10624553	1	1	yes	characteristic, complemented, Sylow
59	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	9	no	—
62	C₃×C₉	—	37	9	no	—
63	C₃×C₉	—	111	3	no	—
64	C₃×C₉	—	111	3	no	—
65	C₃×C₉	—	111	9	no	—
66	C₃×C₉	—	111	9	no	—
67	C₃×C₉	—	111	9	no	—
68	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
69	C₃×C₉	#11758605	1	3	yes	—
70	C₃×C₉	#11758606	1	3	yes	—
71	C₃×C₉	#11758608	1	3	yes	—
72	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
73	C₃²	C₃×#11758606	1	1	yes	characteristic
74	C₃²	C₃×#11758605	1	1	yes	characteristic
75	C₃²	C₃×#11758608	1	1	yes	characteristic
76	C₃²	#15688532	1	9	yes	—
77	C₉	—	9	3	no	—
78	C₉	—	333	3	no	—
79	C₉	—	333	3	no	—
80	C₉	—	111	9	no	—
81	C₉	#15688504	1	9	yes	—
82	C₃	C₃×#15688504	1	1	yes	characteristic
83	C₃	#22582537	1	3	yes	—
84	C₃	#22582594	1	3	yes	—
85	C₃	#22582599	1	3	yes	—
86	C₃	#22582600	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃×C₃₇×C₉⋊C₉	C₃
socle, prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×C₉⋊C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624553
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃×C₉⋊C₉

Complements

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

N	H ≅ G/N	type
11. C₉×#66524	C₉	semidirect
58. C₃₇	#10624553	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688506	3
C₃×#15688505	3
C₃×C₉×#66524	3
#22582441	3
C₃×C₃₇×C₉⋊C₉	3
#10624553	37