Subgroups — #31801250

Order 17739.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊(C₇₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×#10620321	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊#10620321	C₃	1	1	yes	maximal, characteristic, complemented
4	C₇₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₇₃⋊C₃	—	3	1	no	—
6	C₃²⋊(C₇₃⋊C₃)	—	3	1	no	—
7	C₃²⋊(C₇₃⋊C₃)	—	3	1	no	—
8	C₃²⋊(C₇₃⋊C₃)	C₉	1	3	yes	complemented
9	C₃×#10620321	—	3	1	no	—
10	C₃×#10620321	—	3	3	no	—
11	C₃×#10620321	—	3	1	no	—
12	C₃³×C₇₃	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₇₃	—	3	1	no	—
14	C₃²×C₇₃	—	3	1	no	—
15	C₃²×C₇₃	—	3	1	no	—
16	C₃²×C₇₃	—	3	1	no	—
17	C₃²×C₇₃	C₃×C₉	1	1	yes	characteristic
18	C₃²×C₇₃	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₇₃	3₋¹⁺²	1	1	yes	characteristic
20	C₃²×C₇₃	3₋¹⁺²	1	1	yes	characteristic
21	C₃×C₇₃⋊C₃	—	3	3	no	—
22	C₃×C₇₃⋊C₃	—	9	1	no	—
23	C₃×C₇₃⋊C₃	—	9	1	no	—
24	C₃×C₇₃⋊C₃	—	9	1	no	—
25	C₉×C₇₃	—	9	1	no	—
26	#10620321	—	3	3	no	—
27	#10620321	—	9	1	no	—
28	H(3,𝔽₃)⋊C₉	—	73	1	no	maximal, Sylow
29	C₃×C₇₃	—	3	1	no	—
30	C₃×C₇₃	—	3	1	no	—
31	C₃×C₇₃	—	3	1	no	—
32	C₃×C₇₃	C₃²⋊C₉	1	1	yes	characteristic
33	C₃×C₇₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
34	C₃×C₇₃	#731	1	1	yes	characteristic
35	C₃×C₇₃	#732	1	1	yes	characteristic
36	C₇₃⋊C₃	—	9	3	no	—
37	C₃×H(3,𝔽₃)	—	73	1	no	—
38	C₃²×C₉	—	73	1	no	—
39	C₃²⋊C₉	—	73	1	no	—
40	C₃²⋊C₉	C₇₃⋊C₃	1	1	yes	characteristic, complemented
41	C₇₃	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	219	1	no	—
43	C₃³	C₃×C₇₃⋊C₃	1	1	yes	characteristic
44	H(3,𝔽₃)	—	73	3	no	—
45	H(3,𝔽₃)	—	219	1	no	—
46	H(3,𝔽₃)	—	219	1	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	219	1	no	—
49	C₃×C₉	—	219	1	no	—
50	C₃×C₉	—	219	3	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	3	1	no	—
54	C₃²	—	657	1	no	—
55	C₃²	—	657	1	no	—
56	C₃²	—	657	1	no	—
57	C₃²	—	219	3	no	—
58	C₃²	C₉×C₇₃⋊C₃	1	1	yes	characteristic
59	C₃²	C₃²⋊(C₇₃⋊C₃)	1	1	yes	characteristic
60	C₃²	#14520295	1	1	yes	characteristic
61	C₃²	#14520296	1	1	yes	characteristic
62	C₉	—	9	1	no	—
63	C₉	—	657	1	no	—
64	C₉	—	219	3	no	—
65	C₃	—	3	1	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	1	no	—
68	C₃	—	657	3	no	—
69	C₃	#20238257	1	1	yes	characteristic
70	C₃	#20238269	1	1	yes	characteristic
71	C₃	#20238272	1	1	yes	characteristic
72	C₃	#20238273	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₇₃⋊C₃)
commutator	C₃²×C₇₃	C₃×C₉
Frattini	C₃³	C₃×C₇₃⋊C₃
Fitting	C₇₃×C₃²⋊C₉	C₃
socle	C₃²×C₇₃	H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₇₃⋊C₃)	C₃
cosocle	C₃³×C₇₃	C₃²
hypercenter	C₃²⋊C₉	C₇₃⋊C₃
hypocenter	C₇₃	H(3,𝔽₃)⋊C₉
LCS	G ▷ C₃²×C₇₃ ▷ C₃×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃²×#10620321	C₃	semidirect
3. C₃²⋊#10620321	C₃	semidirect
4. C₇₃×C₃²⋊C₉	C₃	semidirect
8. C₃²⋊(C₇₃⋊C₃)	C₉	semidirect
40. C₃²⋊C₉	C₇₃⋊C₃	semidirect
41. C₇₃	H(3,𝔽₃)⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊(C₇₃⋊C₃)	3
C₃²×#10620321	3
C₃²⋊#10620321	3
C₇₃×C₃²⋊C₉	3
H(3,𝔽₃)⋊C₉	73