Subgroups — #41004924

Order 26973.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582472	C₃	1	2	yes	maximal
2	#22582441	C₃	1	1	yes	maximal, characteristic
3	C₃₇×#6868	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	1	no	—
5	C₃²⋊#66524	—	3	2	no	—
6	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
7	C₃₇×C₃²⋊C₉	—	3	1	no	—
8	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
9	C₃×C₉×C₃₇	—	3	1	no	—
10	C₃×C₉×C₃₇	—	3	1	no	—
11	C₃×C₉×C₃₇	—	3	1	no	—
12	C₃×C₉×C₃₇	—	3	1	no	—
13	C₃×C₉×C₃₇	—	9	1	no	—
14	C₃×#66524	—	3	3	no	—
15	C₃×#66524	—	9	1	no	—
16	C₃×#66524	—	9	1	no	—
17	C₃×#66524	—	9	1	no	—
18	C₃×#66524	—	9	2	no	—
19	#10624494	—	37	1	no	maximal, Sylow
20	C₃²×C₃₇	—	3	1	no	—
21	C₃²×C₃₇	—	3	1	no	—
22	C₃²×C₃₇	—	3	1	no	—
23	C₃²×C₃₇	#731	1	1	yes	characteristic
24	C₃²×C₃₇	#732	1	1	yes	characteristic
25	C₃²×C₃₇	#733	1	1	yes	characteristic
26	C₃²×C₃₇	#733	1	1	yes	characteristic
27	C₉×C₃₇	—	9	1	no	—
28	C₉×C₃₇	—	27	1	no	—
29	#66524	—	9	3	no	—
30	#66524	—	27	2	no	—
31	#6868	—	37	2	no	—
32	#6868	C₃₇⋊C₃	1	1	yes	characteristic
33	#6855	—	37	1	no	—
34	C₃×C₃₇	—	3	1	no	—
35	C₃×C₃₇	—	3	1	no	—
36	C₃×C₃₇	—	3	1	no	—
37	C₃×C₃₇	#6861	1	1	yes	characteristic
38	C₃×C₃₇	#6879	1	1	yes	characteristic
39	C₃×C₃₇	#6882	1	1	yes	characteristic
40	C₃×C₃₇	#6882	1	1	yes	characteristic
41	C₃²×C₉	—	111	1	no	—
42	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
43	C₃²⋊C₉	—	3	1	no	—
44	C₃²⋊C₉	—	111	2	no	—
45	C₃₇	#10624494	1	1	yes	characteristic, complemented, Sylow
46	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	3	1	no	—
49	C₃×C₉	—	3	1	no	—
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	9	1	no	—
52	C₃×C₉	—	333	1	no	—
53	C₃×C₉	—	333	1	no	—
54	C₃×C₉	—	333	1	no	—
55	C₃×C₉	—	333	2	no	—
56	C₃×C₉	—	111	3	no	—
57	C₃²	—	3	1	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	1	no	—
60	C₃²	#15688517	1	1	yes	characteristic
61	C₃²	#15688518	1	1	yes	characteristic
62	C₃²	#15688520	1	1	yes	characteristic
63	C₃²	#15688521	1	1	yes	characteristic
64	C₉	—	9	1	no	—
65	C₉	—	27	1	no	—
66	C₉	—	333	3	no	—
67	C₉	—	999	2	no	—
68	C₃	—	3	1	no	—
69	C₃	—	3	1	no	—
70	C₃	—	3	1	no	—
71	C₃	#22582458	1	1	yes	characteristic
72	C₃	#22582506	1	1	yes	characteristic
73	C₃	#22582516	1	1	yes	characteristic
74	C₃	#22582517	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6868	C₃
socle	C₃²×C₃₇	#732
prime radical	C₃³×C₃₇	H(3,𝔽₃)
hypercenter	#6868	C₃₇⋊C₃
hypocenter	C₃₇	#10624494
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×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
45. C₃₇	#10624494	semidirect

Maximal subgroups

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

subgroup	index
#22582472	3
#22582441	3
C₃₇×#6868	3
#10624494	37