Subgroups — #41004955

Order 26973.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582468	C₃	1	3	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66524	—	3	3	no	—
4	C₃⁴×C₃₇	C₉	1	3	yes	complemented
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×H(3,𝔽₃)×C₃₇	—	3	1	no	—
7	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
8	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
9	C₃³×C₃₇	—	3	9	no	—
10	C₃³×C₃₇	—	3	3	no	—
11	C₃³×C₃₇	—	3	3	no	—
12	C₃³×C₃₇	—	9	9	no	—
13	C₃³×C₃₇	—	9	3	no	—
14	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃³×C₃₇	3₋¹⁺²	1	2	yes	—
17	H(3,𝔽₃)×C₃₇	—	3	18	no	—
18	H(3,𝔽₃)×C₃₇	—	3	9	no	—
19	C₃×#66524	—	9	3	no	—
20	#10624505	—	37	1	no	maximal, Sylow
21	C₃²×C₃₇	—	3	3	no	—
22	C₃²×C₃₇	—	3	1	no	—
23	C₃²×C₃₇	—	3	9	no	—
24	C₃²×C₃₇	—	3	2	no	—
25	C₃²×C₃₇	—	9	1	no	—
26	C₃²×C₃₇	—	9	9	no	—
27	C₃²×C₃₇	—	9	2	no	—
28	C₃²×C₃₇	—	9	27	no	—
29	C₃²×C₃₇	—	9	9	no	—
30	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
31	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
32	#66524	—	27	3	no	—
33	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
34	#6866	—	37	3	no	—
35	C₃×C₃₇	—	3	3	no	—
36	C₃×C₃₇	—	3	3	no	—
37	C₃×C₃₇	—	3	1	no	—
38	C₃×C₃₇	—	9	9	no	—
39	C₃×C₃₇	—	9	2	no	—
40	C₃×C₃₇	#6866	1	1	yes	characteristic
41	C₃⁴	#66524	1	3	yes	complemented
42	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
43	C₃×H(3,𝔽₃)	—	3	2	no	—
44	C₃×H(3,𝔽₃)	—	3	1	no	—
45	C₃×H(3,𝔽₃)	—	3	9	no	—
46	C₃²⋊C₉	—	111	3	no	—
47	C₃₇	#10624505	1	1	yes	characteristic, complemented, Sylow
48	C₃³	—	3	9	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	9	9	no	—
52	C₃³	—	9	3	no	—
53	C₃³	C₃×#66524	1	1	yes	characteristic
54	C₃³	#11758608	1	2	yes	—
55	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
56	H(3,𝔽₃)	—	3	18	no	—
57	H(3,𝔽₃)	—	3	9	no	—
58	C₃×C₉	—	333	3	no	—
59	C₃²	—	3	3	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	2	no	—
62	C₃²	—	3	9	no	—
63	C₃²	—	9	1	no	—
64	C₃²	—	9	9	no	—
65	C₃²	—	9	2	no	—
66	C₃²	—	9	27	no	—
67	C₃²	—	9	9	no	—
68	C₃²	C₃²⋊#66524	1	1	yes	characteristic
69	C₃²	#15688513	1	3	yes	—
70	C₉	—	999	3	no	—
71	C₃	—	3	3	no	—
72	C₃	—	3	1	no	—
73	C₃	—	3	3	no	—
74	C₃	—	9	2	no	—
75	C₃	—	9	9	no	—
76	C₃	#22582468	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582468
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃⁴	C₃×C₃₇⋊C₃
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃×C₃₇	#6866
cosocle	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624505
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #22582468	C₃	semidirect
4. C₃⁴×C₃₇	C₉	semidirect
41. C₃⁴	#66524	semidirect
47. C₃₇	#10624505	semidirect

Maximal subgroups

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

subgroup	index
#22582468	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624505	37