Subgroups — #41007346

Order 26973.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688519	C₃	1	3	yes	maximal, complemented
2	#22582507	C₃	1	9	yes	maximal, complemented
3	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
5	#15688519	—	3	6	no	—
6	#15688519	C₃²	1	9	yes	complemented
7	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
8	C₃₇×C₉⋊C₉	—	3	2	no	—
9	C₃₇×C₃²⋊C₉	—	3	1	no	—
10	C₉²×C₃₇	C₃²	1	3	yes	complemented
11	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃²×C₃₇⋊C₃	—	9	3	no	—
13	C₃×C₉×C₃₇	—	3	3	no	—
14	C₃×C₉×C₃₇	—	3	3	no	—
15	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
16	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
17	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
18	C₃²⋊(C₃₇⋊C₃)	—	9	6	no	—
19	#10624839	—	37	1	no	maximal, Sylow
20	C₃²×C₃₇	—	9	1	no	—
21	C₃²×C₃₇	#732	1	3	yes	—
22	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×C₃₇⋊C₃	—	9	9	no	—
24	C₃×C₃₇⋊C₃	—	27	9	no	—
25	C₉×C₃₇	—	3	2	no	—
26	C₉×C₃₇	—	3	1	no	—
27	C₉×C₃₇	—	9	3	no	—
28	C₃×#732	—	37	3	no	—
29	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
30	#6879	—	37	9	no	—
31	C₃×C₃₇	—	3	1	no	—
32	C₃×C₃₇	—	3	3	no	—
33	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
34	C₃₇⋊C₃	—	27	9	no	—
35	C₃×H(3,𝔽₃)	—	111	3	no	—
36	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
37	#732	—	37	9	no	—
38	#732	—	111	6	no	—
39	C₉⋊C₉	—	3	2	no	—
40	C₃²⋊C₉	—	3	1	no	—
41	C₉²	C₃×C₃₇⋊C₃	1	3	yes	complemented
42	C₃₇	#10624839	1	1	yes	characteristic, complemented, Sylow
43	C₃³	—	333	3	no	—
44	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
45	H(3,𝔽₃)	—	333	6	no	—
46	H(3,𝔽₃)	—	111	9	no	—
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
50	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
51	C₃²	—	9	1	no	—
52	C₃²	—	333	9	no	—
53	C₃²	—	999	9	no	—
54	C₃²	#15688519	1	3	yes	—
55	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
56	C₉	—	3	2	no	—
57	C₉	—	3	1	no	—
58	C₉	—	9	3	no	—
59	C₃	—	3	1	no	—
60	C₃	—	3	3	no	—
61	C₃	—	999	9	no	—
62	C₃	C₃×#15688519	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×#15688519
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624839
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. C₃×#15688519	C₃	semidirect
2. #22582507	C₃	semidirect
3. C₃₇×#6887	C₃	semidirect
6. #15688519	C₃²	semidirect
10. C₉²×C₃₇	C₃²	semidirect
29. #6887	C₃₇⋊C₃	semidirect
41. C₉²	C₃×C₃₇⋊C₃	semidirect
42. C₃₇	#10624839	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
#22582507	3
C₃₇×#6887	3
#10624839	37