Subgroups — #41005106

Order 26973.

512 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊#66524	C₃	1	3	yes	maximal
2	#22582441	C₃	1	9	yes	maximal, complemented
3	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	C₃²	1	9	yes	—
5	C₃²⋊#66524	—	3	9	no	—
6	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
7	C₃²×C₉×C₃₇	—	3	3	no	—
8	C₃²×C₉×C₃₇	—	3	6	no	—
9	C₃²×C₉×C₃₇	C₉	1	9	yes	complemented
10	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
11	C₃³×C₃₇	—	3	9	no	—
12	C₃³×C₃₇	C₃×C₉	1	3	yes	—
13	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
14	C₃³×C₃₇	3₋¹⁺²	1	6	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	27	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	3	6	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	9	yes	—
20	C₃×#66524	—	3	9	no	—
21	C₃×#66524	—	3	27	no	—
22	#10624549	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	9	no	—
24	C₃²×C₃₇	—	3	9	no	—
25	C₃²×C₃₇	—	3	18	no	—
26	C₃²×C₃₇	C₃²⋊C₉	1	9	yes	—
27	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
28	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
30	C₃²×C₃₇	#737	1	9	yes	—
31	C₉×C₃₇	—	3	9	no	—
32	#66524	—	9	9	no	—
33	C₃³×C₉	C₃₇⋊C₃	1	1	yes	characteristic
34	C₃×C₃²⋊C₉	—	37	3	no	—
35	#6855	—	37	9	no	—
36	C₃×C₃₇	—	3	9	no	—
37	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
38	C₃×C₃₇	#6887	1	3	yes	—
39	C₃×C₃₇	#6893	1	3	yes	—
40	C₃×C₃₇	#6895	1	6	yes	—
41	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
42	C₃²×C₉	—	3	6	no	—
43	C₃²×C₉	—	3	3	no	—
44	C₃²×C₉	—	37	9	no	—
45	C₃²×C₉	#66524	1	9	yes	complemented
46	C₃²×C₉	C₃×C₃₇⋊C₃	1	3	yes	—
47	C₃²⋊C₉	—	111	9	no	—
48	C₃₇	#10624549	1	1	yes	characteristic, complemented, Sylow
49	C₃³	—	3	9	no	—
50	C₃³	C₃×#66524	1	3	yes	—
51	C₃³	#11758608	1	6	yes	—
52	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
53	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
54	C₃×C₉	—	3	27	no	—
55	C₃×C₉	—	3	3	no	—
56	C₃×C₉	—	3	6	no	—
57	C₃×C₉	—	111	9	no	—
58	C₃×C₉	—	111	27	no	—
59	C₃×C₉	C₃×#66524	1	9	yes	—
60	C₃²	—	3	9	no	—
61	C₃²	—	3	9	no	—
62	C₃²	—	3	18	no	—
63	C₃²	C₃²⋊#66524	1	9	yes	—
64	C₃²	C₃²×#66524	1	1	yes	characteristic
65	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
66	C₃²	C₃×#11758608	1	2	yes	—
67	C₃²	#15688531	1	9	yes	—
68	C₉	—	3	9	no	—
69	C₉	—	333	9	no	—
70	C₃	—	3	9	no	—
71	C₃	C₃×C₃²⋊#66524	1	1	yes	characteristic
72	C₃	#22582538	1	3	yes	—
73	C₃	#22582564	1	3	yes	—
74	C₃	#22582578	1	6	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃³×C₉×C₃₇	C₃
socle, cosocle	C₃³×C₃₇	C₃³
prime radical	C₃⁴×C₃₇	C₃²
hypercenter	C₃³×C₉	C₃₇⋊C₃
hypocenter	C₃₇	#10624549
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
2. #22582441	C₃	semidirect
9. C₃²×C₉×C₃₇	C₉	semidirect
45. C₃²×C₉	#66524	semidirect
48. C₃₇	#10624549	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊#66524	3
#22582441	3
C₃³×C₉×C₃₇	3
#10624549	37