Subgroups — #41004871

Order 26973.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688519	C₃	1	1	yes	maximal, characteristic
2	#22582578	C₃	1	2	yes	maximal, complemented
3	C₃₇×#6867	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
5	#15688519	—	3	3	no	—
6	#15688504	—	3	6	no	—
7	C₃²⋊#66524	—	3	2	no	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₃₇×C₃²⋊C₉	—	3	1	no	—
10	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
11	C₃²×C₃₇⋊C₃	—	9	1	no	—
12	C₃×C₉×C₃₇	—	3	2	no	—
13	C₃×C₉×C₃₇	—	3	1	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	9	1	no	—
16	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
17	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
18	C₃×#66524	—	3	6	no	—
19	#10624484	—	37	1	no	maximal, Sylow
20	C₃²×C₃₇	—	9	1	no	—
21	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
22	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	2	yes	—
23	C₃²×C₃₇	#731	1	1	yes	characteristic
24	C₃×C₃₇⋊C₃	—	9	3	no	—
25	C₃×C₃₇⋊C₃	—	27	3	no	—
26	C₉×C₃₇	—	9	1	no	—
27	C₉×C₃₇	—	27	1	no	—
28	#66524	—	9	6	no	—
29	C₃×#732	—	37	1	no	—
30	#6895	—	37	2	no	—
31	#6867	C₃₇⋊C₃	1	1	yes	characteristic, complemented
32	C₃×C₃₇	—	3	2	no	—
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	—	3	1	no	—
35	C₃×C₃₇	#6857	1	1	yes	characteristic
36	C₃₇⋊C₃	—	27	3	no	—
37	C₃×H(3,𝔽₃)	—	111	1	no	—
38	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
39	#732	—	111	3	no	—
40	C₉⋊C₉	—	111	6	no	—
41	C₃²⋊C₉	—	3	1	no	—
42	C₃²⋊C₉	—	111	2	no	—
43	C₃₇	#10624484	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	333	1	no	—
45	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
46	H(3,𝔽₃)	—	333	2	no	—
47	H(3,𝔽₃)	—	111	3	no	—
48	C₃×C₉	—	3	1	no	—
49	C₃×C₉	—	3	2	no	—
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	9	1	no	—
52	C₃×C₉	—	111	6	no	—
53	C₃²	—	9	1	no	—
54	C₃²	—	333	3	no	—
55	C₃²	—	999	3	no	—
56	C₃²	#15688512	1	1	yes	characteristic
57	C₃²	#15688514	1	2	yes	—
58	C₃²	#15688516	1	1	yes	characteristic
59	C₉	—	9	1	no	—
60	C₉	—	27	1	no	—
61	C₉	—	333	6	no	—
62	C₃	—	3	1	no	—
63	C₃	—	3	2	no	—
64	C₃	—	3	1	no	—
65	C₃	—	999	3	no	—
66	C₃	#22582446	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582446
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6867	C₃
socle	C₃×C₃₇	#6857
prime radical	C₃×#15688519	C₃
hypercenter	#6867	C₃₇⋊C₃
hypocenter	C₃₇	#10624484
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6867

Complements

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

N	H ≅ G/N	type
2. #22582578	C₃	semidirect
3. C₃₇×#6867	C₃	semidirect
31. #6867	C₃₇⋊C₃	semidirect
43. C₃₇	#10624484	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688519	3
#22582578	3
C₃₇×#6867	3
#10624484	37