Subgroups — #41004997

Order 26973.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
2	#22582472	C₃	1	2	yes	maximal
3	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	1	no	—
5	C₃²⋊#66524	—	3	2	no	—
6	C₉×#66524	—	3	1	no	—
7	C₉×#66524	—	3	1	no	—
8	C₉×#66524	C₉	1	3	yes	complemented
9	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
10	C₃₇×C₃²⋊C₉	—	3	1	no	—
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃×C₉×C₃₇	—	3	1	no	—
13	C₃×C₉×C₃₇	—	3	2	no	—
14	C₃×C₉×C₃₇	—	9	1	no	—
15	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
17	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
18	C₃×#66524	—	3	3	no	—
19	C₃×#66524	—	3	3	no	—
20	C₃×#66524	—	3	3	no	—
21	C₃×#66524	—	3	3	no	—
22	C₃×#66524	—	9	2	no	—
23	#10624519	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	—	3	1	no	—
25	C₃²×C₃₇	—	3	1	no	—
26	C₃²×C₃₇	—	3	1	no	—
27	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
28	C₃²×C₃₇	#732	1	1	yes	characteristic
29	C₃²×C₃₇	#733	1	2	yes	—
30	C₉×C₃₇	—	3	1	no	—
31	C₉×C₃₇	—	3	1	no	—
32	C₉×C₃₇	—	3	1	no	—
33	C₉×C₃₇	—	27	1	no	—
34	#66524	—	3	9	no	—
35	#66524	—	27	2	no	—
36	C₃×C₉²	—	37	1	no	—
37	#6868	—	37	2	no	—
38	#6868	C₃₇⋊C₃	1	1	yes	characteristic
39	C₃×C₃₇	—	3	1	no	—
40	C₃×C₃₇	—	3	2	no	—
41	C₃×C₃₇	#6868	1	1	yes	characteristic
42	C₃×C₃₇	#6879	1	1	yes	characteristic
43	C₃×C₃₇	#6880	1	1	yes	characteristic
44	C₃×C₃₇	#6880	1	1	yes	characteristic
45	C₃²×C₉	—	111	1	no	—
46	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
47	C₃²⋊C₉	—	3	1	no	—
48	C₃²⋊C₉	—	111	2	no	—
49	C₉²	—	37	3	no	—
50	C₉²	—	111	1	no	—
51	C₉²	—	111	1	no	—
52	C₃₇	#10624519	1	1	yes	characteristic, complemented, Sylow
53	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
54	C₃×C₉	—	3	1	no	—
55	C₃×C₉	—	3	2	no	—
56	C₃×C₉	—	9	1	no	—
57	C₃×C₉	—	333	2	no	—
58	C₃×C₉	—	111	3	no	—
59	C₃×C₉	—	111	3	no	—
60	C₃×C₉	—	111	3	no	—
61	C₃×C₉	—	111	3	no	—
62	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
63	C₃×C₉	#11758605	1	1	yes	characteristic
64	C₃×C₉	#11758606	1	1	yes	characteristic
65	C₃²	—	3	1	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	3	1	no	—
68	C₃²	#15688502	1	1	yes	characteristic
69	C₃²	#15688518	1	1	yes	characteristic
70	C₃²	#15688520	1	2	yes	—
71	C₉	—	3	1	no	—
72	C₉	—	3	1	no	—
73	C₉	—	3	1	no	—
74	C₉	—	27	1	no	—
75	C₉	—	111	9	no	—
76	C₉	—	999	2	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	2	no	—
79	C₃	#22582471	1	1	yes	characteristic
80	C₃	#22582506	1	1	yes	characteristic
81	C₃	#22582508	1	1	yes	characteristic
82	C₃	#22582509	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624519
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ #6868

Complements

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

N	H ≅ G/N	type
8. C₉×#66524	C₉	semidirect
52. C₃₇	#10624519	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#66524	3
#22582472	3
C₃₇×#6868	3
#10624519	37