Subgroups — #41004784

Order 26973.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582538	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊#11758601	C₃	1	2	yes	maximal
3	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758601	—	3	2	no	—
5	#15688506	—	3	1	no	—
6	#15688505	—	3	1	no	—
7	C₃²⋊#66524	C₉	1	2	yes	—
8	C₃²⋊#66524	C₉	1	1	yes	characteristic
9	C₉×#66524	—	3	1	no	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃×C₂₇×C₃₇	—	3	1	no	—
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃×C₉×C₃₇	—	3	1	no	—
14	C₃×C₉×C₃₇	—	3	2	no	—
15	C₃×C₉×C₃₇	H(3,𝔽₃)	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	2	no	—
19	C₃×#66524	—	3	1	no	—
20	C₂₇×C₃₇	—	9	1	no	—
21	#11758601	—	9	2	no	—
22	#10624461	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	1	no	—
24	C₃²×C₃₇	—	3	2	no	—
25	C₃²×C₃₇	—	3	1	no	—
26	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
27	C₉×C₃₇	—	3	1	no	—
28	C₉×C₃₇	—	3	1	no	—
29	C₉×C₃₇	#732	1	1	yes	characteristic
30	C₉×C₃₇	#733	1	2	yes	—
31	#66524	—	9	2	no	—
32	#66524	—	9	1	no	—
33	#6887	—	37	1	no	—
34	C₃²⋊C₂₇	—	37	2	no	—
35	C₃²⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic
36	C₃×C₃₇	—	9	1	no	—
37	C₃×C₃₇	#6868	1	1	yes	characteristic
38	C₃×C₃₇	#6873	1	2	yes	—
39	C₃×C₃₇	#6873	1	1	yes	characteristic
40	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
41	C₃×C₂₇	—	3	1	no	—
42	C₃×C₂₇	—	111	2	no	—
43	C₉⋊C₉	—	111	1	no	—
44	C₉⋊C₉	—	111	1	no	—
45	C₃²⋊C₉	—	37	2	no	—
46	C₃²⋊C₉	—	37	1	no	—
47	C₉²	—	111	1	no	—
48	C₃₇	#10624461	1	1	yes	characteristic, complemented, Sylow
49	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	3	2	no	—
52	C₃×C₉	—	111	2	no	—
53	C₃×C₉	—	111	1	no	—
54	C₃×C₉	#11758605	1	1	yes	characteristic
55	C₃×C₉	#11758606	1	1	yes	characteristic
56	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
57	C₂₇	—	9	1	no	—
58	C₂₇	—	333	2	no	—
59	C₃²	—	3	1	no	—
60	C₃²	—	3	2	no	—
61	C₃²	—	3	1	no	—
62	C₃²	#15688502	1	1	yes	characteristic
63	C₉	—	3	1	no	—
64	C₉	—	3	1	no	—
65	C₉	—	333	2	no	—
66	C₉	—	333	1	no	—
67	C₉	#15688518	1	1	yes	characteristic
68	C₉	#15688520	1	2	yes	—
69	C₃	—	9	1	no	—
70	C₃	#22582471	1	1	yes	characteristic
71	C₃	#22582485	1	2	yes	—
72	C₃	#22582487	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator, prime radical	C₃³×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×C₃²⋊C₂₇	C₃
socle	C₃²×C₃₇	C₃²⋊C₉
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃²⋊C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624461
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×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
48. C₃₇	#10624461	semidirect

Maximal subgroups

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

subgroup	index
#22582538	3
C₃²⋊#11758601	3
C₃₇×C₃²⋊C₂₇	3
#10624461	37