Subgroups — #41004927

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	2	no	—
12	C₃×C₉×C₃₇	—	9	1	no	—
13	C₃×#66524	—	3	3	no	—
14	C₃×#66524	—	9	1	no	—
15	C₃×#66524	—	9	1	no	—
16	C₃×#66524	—	9	1	no	—
17	C₃×#66524	—	9	2	no	—
18	#10624495	—	37	1	no	maximal, Sylow
19	C₃²×C₃₇	—	3	1	no	—
20	C₃²×C₃₇	—	3	1	no	—
21	C₃²×C₃₇	—	3	1	no	—
22	C₃²×C₃₇	#731	1	1	yes	characteristic
23	C₃²×C₃₇	#732	1	1	yes	characteristic
24	C₃²×C₃₇	#733	1	2	yes	—
25	C₉×C₃₇	—	9	1	no	—
26	C₉×C₃₇	—	27	1	no	—
27	#66524	—	9	3	no	—
28	#66524	—	27	2	no	—
29	#6868	—	37	2	no	—
30	#6868	C₃₇⋊C₃	1	1	yes	characteristic
31	#6855	—	37	1	no	—
32	C₃×C₃₇	—	3	1	no	—
33	C₃×C₃₇	—	3	2	no	—
34	C₃×C₃₇	#6861	1	1	yes	characteristic
35	C₃×C₃₇	#6880	1	1	yes	characteristic
36	C₃×C₃₇	#6881	1	1	yes	characteristic
37	C₃×C₃₇	#6883	1	1	yes	characteristic
38	C₃²×C₉	—	111	1	no	—
39	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
40	C₃²⋊C₉	—	3	1	no	—
41	C₃²⋊C₉	—	111	2	no	—
42	C₃₇	#10624495	1	1	yes	characteristic, complemented, Sylow
43	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	3	1	no	—
46	C₃×C₉	—	3	2	no	—
47	C₃×C₉	—	9	1	no	—
48	C₃×C₉	—	333	1	no	—
49	C₃×C₉	—	333	1	no	—
50	C₃×C₉	—	333	1	no	—
51	C₃×C₉	—	333	2	no	—
52	C₃×C₉	—	111	3	no	—
53	C₃²	—	3	1	no	—
54	C₃²	—	3	1	no	—
55	C₃²	—	3	1	no	—
56	C₃²	#15688516	1	1	yes	characteristic
57	C₃²	#15688518	1	1	yes	characteristic
58	C₃²	#15688520	1	2	yes	—
59	C₉	—	9	1	no	—
60	C₉	—	27	1	no	—
61	C₉	—	333	3	no	—
62	C₉	—	999	2	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	2	no	—
65	C₃	#22582457	1	1	yes	characteristic
66	C₃	#22582508	1	1	yes	characteristic
67	C₃	#22582515	1	1	yes	characteristic
68	C₃	#22582524	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	#10624495
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
42. C₃₇	#10624495	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
#10624495	37