Subgroups — #34063377

Order 19197.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20264264	C₃	1	16	yes	maximal, complemented
2	#20264263	C₃	1	8	yes	maximal, complemented
3	#20264262	C₃	1	2	yes	maximal, complemented
4	C₃×#14523839	C₃	1	4	yes	maximal, complemented
5	3₋¹⁺²×C₇₉⋊C₃	C₃	1	8	yes	maximal, complemented
6	H(3,𝔽₃)×C₇₉⋊C₃	C₃	1	1	yes	maximal, characteristic
7	C₇₉×#737	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₇₉⋊C₃	C₃²	1	4	yes	—
9	#14523840	C₃²	1	16	yes	complemented
10	#14523839	C₃²	1	8	yes	—
11	#14523839	C₃²	1	1	yes	characteristic
12	#14523839	C₃²	1	8	yes	complemented
13	#14523839	C₃²	1	16	yes	complemented
14	#14523838	C₃²	1	24	yes	complemented
15	C₃²⋊(C₇₉⋊C₃)	C₃²	1	8	yes	—
16	C₃×C₇₉⋊C₉	C₃²	1	8	yes	complemented
17	C₉×C₇₉⋊C₃	C₃²	1	24	yes	complemented
18	3₋¹⁺²×C₇₉	C₃²	1	8	yes	complemented
19	H(3,𝔽₃)×C₇₉	C₃²	1	1	yes	characteristic
20	C₃×C₉×C₇₉	C₃²	1	4	yes	complemented
21	C₃²×C₇₉	C₃³	1	4	yes	—
22	C₃×C₇₉⋊C₃	—	9	4	no	—
23	C₃×C₇₉⋊C₃	C₃³	1	8	yes	—
24	C₃×C₇₉⋊C₃	C₃³	1	1	yes	characteristic
25	C₉×C₇₉	C₃³	1	1	yes	characteristic
26	C₉×C₇₉	C₃³	1	8	yes	—
27	C₇₉⋊C₉	C₃³	1	16	yes	—
28	C₇₉⋊C₉	C₃³	1	2	yes	—
29	3₋¹⁺⁴	—	79	1	no	maximal, Sylow
30	C₃×C₇₉	—	3	4	no	—
31	C₃×C₇₉	C₃⁴	1	1	yes	characteristic
32	C₇₉⋊C₃	—	3	1	no	—
33	C₇₉⋊C₃	—	3	8	no	—
34	#737	—	79	16	no	—
35	#737	—	79	2	no	—
36	#737	—	79	8	no	—
37	#737	C₇₉⋊C₃	1	1	yes	characteristic, complemented
38	C₃×3₋¹⁺²	—	79	8	no	—
39	C₃×3₋¹⁺²	—	79	4	no	—
40	C₃×H(3,𝔽₃)	—	79	1	no	—
41	C₇₉	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	79	4	no	—
43	3₋¹⁺²	—	79	16	no	—
44	3₋¹⁺²	—	79	24	no	—
45	3₋¹⁺²	—	79	8	no	—
46	3₋¹⁺²	—	79	1	no	—
47	3₋¹⁺²	—	79	16	no	—
48	3₋¹⁺²	—	79	8	no	—
49	3₋¹⁺²	C₃×C₇₉⋊C₃	1	8	yes	complemented
50	H(3,𝔽₃)	—	79	8	no	—
51	H(3,𝔽₃)	C₃×C₇₉⋊C₃	1	1	yes	characteristic
52	C₃×C₉	—	79	8	no	—
53	C₃×C₉	—	79	24	no	—
54	C₃×C₉	C₃×C₇₉⋊C₃	1	4	yes	complemented
55	C₃²	—	79	8	no	—
56	C₃²	—	79	1	no	—
57	C₃²	—	711	4	no	—
58	C₃²	C₃²×C₇₉⋊C₃	1	4	yes	—
59	C₉	—	79	16	no	—
60	C₉	—	79	2	no	—
61	C₉	C₃²×C₇₉⋊C₃	1	8	yes	—
62	C₉	C₃²×C₇₉⋊C₃	1	1	yes	characteristic
63	C₃	—	3	4	no	—
64	C₃	—	237	1	no	—
65	C₃	—	237	8	no	—
66	C₃	C₃³×C₇₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×C₇₉⋊C₃
commutator, socle, cosocle	C₃×C₇₉	C₃⁴
Fitting	C₇₉×#737	C₃
prime radical	H(3,𝔽₃)×C₇₉⋊C₃	C₃
hypercenter	#737	C₇₉⋊C₃
hypocenter	C₇₉	3₋¹⁺⁴
LCS	G ▷ C₃×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃ ◁ #737

Complements

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

N	H ≅ G/N	type
1. #20264264	C₃	semidirect
2. #20264263	C₃	semidirect
3. #20264262	C₃	semidirect
4. C₃×#14523839	C₃	semidirect
5. 3₋¹⁺²×C₇₉⋊C₃	C₃	semidirect
7. C₇₉×#737	C₃	semidirect
9. #14523840	C₃²	semidirect
12. #14523839	C₃²	semidirect
13. #14523839	C₃²	semidirect
14. #14523838	C₃²	semidirect
16. C₃×C₇₉⋊C₉	C₃²	semidirect
17. C₉×C₇₉⋊C₃	C₃²	semidirect
18. 3₋¹⁺²×C₇₉	C₃²	semidirect
20. C₃×C₉×C₇₉	C₃²	semidirect
37. #737	C₇₉⋊C₃	semidirect
41. C₇₉	3₋¹⁺⁴	semidirect
49. 3₋¹⁺²	C₃×C₇₉⋊C₃	semidirect
54. 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
#20264264	3
#20264263	3
#20264262	3
C₃×#14523839	3
3₋¹⁺²×C₇₉⋊C₃	3
H(3,𝔽₃)×C₇₉⋊C₃	3
C₇₉×#737	3
3₋¹⁺⁴	79