Subgroups — #41005217

Order 26973.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688502	C₃	1	4	yes	maximal
2	#22582441	C₃	1	8	yes	maximal, complemented
3	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
5	C₃²×#66524	C₃²	1	8	yes	—
6	#15688502	—	3	12	no	—
7	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
8	C₃³×C₃₇	C₃³	1	1	yes	characteristic
9	C₃²×C₃₇⋊C₃	—	3	4	no	—
10	C₃²×C₃₇⋊C₃	—	3	8	no	—
11	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
12	C₃×C₉×C₃₇	—	3	4	no	—
13	C₃×C₉×C₃₇	—	3	4	no	—
14	C₃×C₉×C₃₇	—	3	8	no	—
15	C₃×#66524	—	3	8	no	—
16	C₃×#66524	—	3	8	no	—
17	C₃×#66524	—	3	16	no	—
18	#10624568	—	37	1	no	maximal, Sylow
19	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₃₇	#737	1	4	yes	—
21	C₃²×C₃₇	#737	1	8	yes	—
22	C₃×C₃₇⋊C₃	—	3	12	no	—
23	C₃×C₃₇⋊C₃	—	9	1	no	—
24	C₃×C₃₇⋊C₃	—	9	8	no	—
25	C₉×C₃₇	—	9	4	no	—
26	#66524	—	9	8	no	—
27	C₃×C₃²⋊C₉	—	37	4	no	—
28	#6855	—	37	8	no	—
29	#6855	C₃₇⋊C₃	1	1	yes	characteristic, complemented
30	C₃×C₃₇	#6893	1	4	yes	—
31	C₃×C₃₇	#6897	1	1	yes	characteristic
32	C₃×C₃₇	#6897	1	8	yes	—
33	C₃₇⋊C₃	—	9	3	no	—
34	C₃⁴	—	37	1	no	—
35	C₃²×C₉	—	37	8	no	—
36	C₃²×C₉	C₃×C₃₇⋊C₃	1	4	yes	—
37	C₃²⋊C₉	—	111	12	no	—
38	C₃₇	#10624568	1	1	yes	characteristic, complemented, Sylow
39	C₃³	—	37	3	no	—
40	C₃³	—	111	4	no	—
41	C₃³	—	111	8	no	—
42	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
43	C₃×C₉	—	3	4	no	—
44	C₃×C₉	—	3	4	no	—
45	C₃×C₉	—	3	8	no	—
46	C₃×C₉	—	111	8	no	—
47	C₃×C₉	—	111	8	no	—
48	C₃×C₉	—	111	16	no	—
49	C₃²	—	333	1	no	—
50	C₃²	—	333	8	no	—
51	C₃²	—	111	12	no	—
52	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃²	#15688531	1	4	yes	—
54	C₃²	#15688532	1	8	yes	—
55	C₉	—	9	4	no	—
56	C₉	—	333	8	no	—
57	C₃	—	333	3	no	—
58	C₃	#22582563	1	4	yes	—
59	C₃	#22582587	1	1	yes	characteristic
60	C₃	#22582588	1	8	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃₇×#6855	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624568
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

Complements

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

N	H ≅ G/N	type
2. #22582441	C₃	semidirect
3. C₃₇×#6855	C₃	semidirect
29. #6855	C₃₇⋊C₃	semidirect
38. C₃₇	#10624568	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688502	3
#22582441	3
C₃₇×#6855	3
#10624568	37