Subgroups — #34063344

Order 19197.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14523839	C₃	1	1	yes	maximal, characteristic, complemented
2	H(3,𝔽₃)×C₇₉⋊C₃	C₃	1	3	yes	maximal, complemented
3	#20264251	C₃	1	2	yes	maximal, complemented
4	#20264250	C₃	1	6	yes	maximal, complemented
5	C₇₉×#732	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₇₉⋊C₃	—	3	3	no	—
7	C₃²×C₇₉⋊C₃	C₃²	1	1	yes	characteristic
8	#14523839	—	3	3	no	—
9	C₃²⋊(C₇₉⋊C₃)	—	3	6	no	—
10	C₃²⋊(C₇₉⋊C₃)	C₃²	1	6	yes	complemented
11	C₃×C₇₉⋊C₉	C₃²	1	2	yes	complemented
12	H(3,𝔽₃)×C₇₉	C₃²	1	3	yes	complemented
13	C₃×C₉×C₇₉	C₃²	1	1	yes	characteristic, complemented
14	C₃²×C₇₉	—	3	3	no	—
15	C₃²×C₇₉	C₃³	1	1	yes	characteristic
16	C₃×C₇₉⋊C₃	—	3	6	no	—
17	C₃×C₇₉⋊C₃	—	9	1	no	—
18	C₃×C₇₉⋊C₃	—	9	9	no	—
19	C₃×C₇₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
20	C₉×C₇₉	—	3	1	no	—
21	C₇₉⋊C₉	—	3	2	no	—
22	#6911	—	79	1	no	maximal, Sylow
23	C₃×C₇₉	—	3	1	no	—
24	C₃×C₇₉	—	9	3	no	—
25	C₃×C₇₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
26	C₇₉⋊C₃	—	3	3	no	—
27	C₇₉⋊C₃	—	9	6	no	—
28	C₃×3₋¹⁺²	—	79	1	no	—
29	C₃×H(3,𝔽₃)	—	79	3	no	—
30	#732	—	79	6	no	—
31	#732	—	79	2	no	—
32	#732	C₇₉⋊C₃	1	1	yes	characteristic, complemented
33	C₇₉	#6911	1	1	yes	characteristic, complemented, Sylow
34	C₃³	—	79	1	no	—
35	C₃³	—	237	3	no	—
36	3₋¹⁺²	—	237	3	no	—
37	H(3,𝔽₃)	—	79	6	no	—
38	H(3,𝔽₃)	—	237	6	no	—
39	H(3,𝔽₃)	C₃×C₇₉⋊C₃	1	3	yes	complemented
40	C₃×C₉	—	79	2	no	—
41	C₃×C₉	C₃×C₇₉⋊C₃	1	1	yes	characteristic, complemented
42	C₃²	—	3	3	no	—
43	C₃²	—	79	3	no	—
44	C₃²	—	711	1	no	—
45	C₃²	—	711	9	no	—
46	C₃²	—	237	6	no	—
47	C₃²	C₃²×C₇₉⋊C₃	1	1	yes	characteristic
48	C₉	—	3	1	no	—
49	C₉	—	237	2	no	—
50	C₃	—	3	1	no	—
51	C₃	—	9	3	no	—
52	C₃	—	711	6	no	—
53	C₃	—	237	3	no	—
54	C₃	H(3,𝔽₃)×C₇₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₇₉⋊C₃
commutator, cosocle	C₃²×C₇₉	C₃³
Frattini	C₃²	C₃²×C₇₉⋊C₃
Fitting	C₇₉×#732	C₃
socle	C₃×C₇₉	C₃×H(3,𝔽₃)
hypercenter	#732	C₇₉⋊C₃
hypocenter	C₇₉	#6911
LCS	G ▷ C₃²×C₇₉ ▷ C₃×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #732

Complements

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

N	H ≅ G/N	type
1. C₃×#14523839	C₃	semidirect
2. H(3,𝔽₃)×C₇₉⋊C₃	C₃	semidirect
3. #20264251	C₃	semidirect
4. #20264250	C₃	semidirect
5. C₇₉×#732	C₃	semidirect
10. C₃²⋊(C₇₉⋊C₃)	C₃²	semidirect
11. C₃×C₇₉⋊C₉	C₃²	semidirect
12. H(3,𝔽₃)×C₇₉	C₃²	semidirect
13. C₃×C₉×C₇₉	C₃²	semidirect
32. #732	C₇₉⋊C₃	semidirect
33. C₇₉	#6911	semidirect
39. H(3,𝔽₃)	C₃×C₇₉⋊C₃	semidirect
41. C₃×C₉	C₃×C₇₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14523839	3
H(3,𝔽₃)×C₇₉⋊C₃	3
#20264251	3
#20264250	3
C₇₉×#732	3
#6911	79