Subgroups — #31801248

Order 17739.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₇₃⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#20238266	C₃	1	2	yes	maximal, complemented
3	C₇₃×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₇₃⋊C₃	C₉	1	1	yes	characteristic
5	C₃×#10620321	C₉	1	2	yes	—
6	C₉×C₇₃⋊C₃	—	3	1	no	—
7	C₉×C₇₃⋊C₃	—	3	1	no	—
8	C₉×C₇₃⋊C₃	—	3	1	no	—
9	#14520291	—	3	2	no	—
10	C₃×C₉×C₇₃	C₃²	1	1	yes	characteristic
11	C₂₇×C₇₃	—	3	1	no	—
12	C₃²×C₇₃	C₃×C₉	1	1	yes	characteristic
13	C₃×C₇₃⋊C₃	—	3	1	no	—
14	C₃×C₇₃⋊C₃	—	3	3	no	—
15	C₉×C₇₃	H(3,𝔽₃)	1	1	yes	characteristic
16	C₉×C₇₃	3₋¹⁺²	1	1	yes	characteristic
17	C₉×C₇₃	3₋¹⁺²	1	1	yes	characteristic
18	#10620321	—	3	2	no	—
19	#6869	—	73	1	no	maximal, Sylow
20	C₃×C₇₃	—	3	1	no	—
21	C₃×C₇₃	C₃²⋊C₉	1	1	yes	characteristic
22	C₇₃⋊C₃	—	3	3	no	—
23	C₃²×C₉	—	73	1	no	—
24	C₂₇⋊C₃	—	73	2	no	—
25	C₂₇⋊C₃	C₇₃⋊C₃	1	1	yes	characteristic, complemented
26	C₇₃	#6869	1	1	yes	characteristic, complemented, Sylow
27	C₃³	—	73	1	no	—
28	C₃×C₉	—	73	2	no	—
29	C₃×C₉	—	219	1	no	—
30	C₃×C₉	—	219	1	no	—
31	C₃×C₉	—	219	1	no	—
32	C₃×C₉	C₃×C₇₃⋊C₃	1	1	yes	characteristic
33	C₂₇	—	3	1	no	—
34	C₂₇	—	219	2	no	—
35	C₃²	—	219	3	no	—
36	C₃²	—	219	1	no	—
37	C₃²	C₉×C₇₃⋊C₃	1	1	yes	characteristic
38	C₉	—	219	2	no	—
39	C₉	C₃²⋊(C₇₃⋊C₃)	1	1	yes	characteristic
40	C₉	#14520295	1	1	yes	characteristic
41	C₉	#14520296	1	1	yes	characteristic
42	C₃	—	3	1	no	—
43	C₃	—	219	3	no	—
44	C₃	#20238257	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₃²⋊(C₇₃⋊C₃)
commutator	C₃²×C₇₃	C₃×C₉
Frattini	C₃×C₉	C₃×C₇₃⋊C₃
Fitting	C₇₃×C₂₇⋊C₃	C₃
socle	C₃×C₇₃	C₃²⋊C₉
prime radical	C₃²×C₇₃⋊C₃	C₉
cosocle	C₃×C₉×C₇₃	C₃²
hypercenter	C₂₇⋊C₃	C₇₃⋊C₃
hypocenter	C₇₃	#6869
LCS	G ▷ C₃²×C₇₃ ▷ C₃×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
2. #20238266	C₃	semidirect
3. C₇₃×C₂₇⋊C₃	C₃	semidirect
25. C₂₇⋊C₃	C₇₃⋊C₃	semidirect
26. C₇₃	#6869	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×C₇₃⋊C₃	3
#20238266	3
C₇₃×C₂₇⋊C₃	3
#6869	73