Subgroups — #41004800

Order 26973.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688511	C₃	1	2	yes	maximal
2	#22582538	C₃	1	1	yes	maximal, characteristic
3	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic
4	#15688511	—	3	2	no	—
5	#15688511	—	3	2	no	—
6	#15688511	C₉	1	6	yes	complemented
7	C₃×#11758601	—	3	2	no	—
8	#15688506	—	3	1	no	—
9	#15688505	—	3	1	no	—
10	C₃²⋊#66524	C₉	1	3	yes	—
11	C₉×#66524	—	3	1	no	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃×C₂₇×C₃₇	—	3	1	no	—
14	C₃×C₂₇×C₃₇	—	3	2	no	—
15	C₃×C₂₇×C₃₇	C₉	1	3	yes	complemented
16	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
17	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
18	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
21	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
22	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
23	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
24	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
25	C₃×#66524	—	3	3	no	—
26	C₂₇×C₃₇	—	3	3	no	—
27	#11758601	—	3	6	no	—
28	#10624468	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	C₉²	1	1	yes	characteristic
31	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
32	C₃²×C₃₇	C₉⋊C₉	1	2	yes	—
33	C₉×C₃₇	C₃²⋊C₉	1	2	yes	—
34	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
35	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
36	C₉×C₃₇	C₉⋊C₉	1	2	yes	—
37	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
38	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
39	#66524	—	9	3	no	—
40	C₃×C₂₇⋊C₃	—	37	2	no	—
41	C₃²×C₂₇	C₃₇⋊C₃	1	1	yes	characteristic
42	#6887	—	37	1	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	2	no	—
45	C₃×C₃₇	#6855	1	1	yes	characteristic
46	C₃×C₃₇	#6873	1	3	yes	—
47	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
48	C₂₇⋊C₃	—	37	6	no	—
49	C₂₇⋊C₃	—	111	2	no	—
50	C₂₇⋊C₃	—	111	2	no	—
51	C₃×C₂₇	—	3	2	no	—
52	C₃×C₂₇	—	3	1	no	—
53	C₃×C₂₇	—	111	2	no	—
54	C₃×C₂₇	#66524	1	3	yes	complemented
55	C₉⋊C₉	—	111	1	no	—
56	C₉⋊C₉	—	111	1	no	—
57	C₃²⋊C₉	—	37	3	no	—
58	C₉²	—	111	1	no	—
59	C₃₇	#10624468	1	1	yes	characteristic, complemented, Sylow
60	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
61	C₃×C₉	—	111	3	no	—
62	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
63	C₃×C₉	#11758605	1	2	yes	—
64	C₃×C₉	#11758605	1	1	yes	characteristic
65	C₃×C₉	#11758606	1	2	yes	—
66	C₃×C₉	#11758606	1	1	yes	characteristic
67	C₃×C₉	C₃×#66524	1	1	yes	characteristic
68	C₃×C₉	#11758608	1	2	yes	—
69	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
70	C₂₇	—	3	3	no	—
71	C₂₇	—	111	6	no	—
72	C₃²	—	3	3	no	—
73	C₃²	C₉×#66524	1	1	yes	characteristic
74	C₃²	#15688502	1	1	yes	characteristic
75	C₃²	#15688504	1	2	yes	—
76	C₉	—	333	3	no	—
77	C₉	#15688502	1	2	yes	—
78	C₉	C₃²⋊#66524	1	1	yes	characteristic
79	C₉	#15688504	1	2	yes	—
80	C₉	#15688504	1	2	yes	—
81	C₉	#15688505	1	1	yes	characteristic
82	C₉	#15688506	1	1	yes	characteristic
83	C₃	—	3	1	no	—
84	C₃	—	3	2	no	—
85	C₃	#22582441	1	1	yes	characteristic
86	C₃	#22582487	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₉²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃²×C₂₇×C₃₇	C₃
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃³×C₃₇	C₃×C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃²×C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624468
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃²×C₂₇

Complements

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

N	H ≅ G/N	type
6. #15688511	C₉	semidirect
15. C₃×C₂₇×C₃₇	C₉	semidirect
54. C₃×C₂₇	#66524	semidirect
59. C₃₇	#10624468	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688511	3
#22582538	3
C₃²×C₂₇×C₃₇	3
#10624468	37