Subgroups — #34063340

Order 19197.

136 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	3₋¹⁺²×C₇₉⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₇₉⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#20264249	C₃	1	4	yes	maximal, complemented
5	#20264248	C₃	1	2	yes	maximal, complemented
6	#20264247	C₃	1	2	yes	maximal, complemented
7	C₇₉×#731	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₇₉⋊C₃	—	3	1	no	—
9	C₃²×C₇₉⋊C₃	C₃²	1	1	yes	characteristic
10	#14523840	C₃²	1	4	yes	complemented
11	#14523839	—	3	2	no	—
12	#14523839	—	3	2	no	—
13	#14523839	—	3	1	no	—
14	#14523838	—	3	2	no	—
15	C₃²⋊(C₇₉⋊C₃)	—	3	2	no	—
16	C₃²⋊(C₇₉⋊C₃)	C₃²	1	2	yes	—
17	C₃×C₇₉⋊C₉	C₃²	1	2	yes	complemented
18	C₉×C₇₉⋊C₃	—	3	2	no	—
19	3₋¹⁺²×C₇₉	C₃²	1	2	yes	complemented
20	H(3,𝔽₃)×C₇₉	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₇₉	C₃²	1	1	yes	characteristic, complemented
22	C₃²×C₇₉	—	3	1	no	—
23	C₃²×C₇₉	C₃³	1	1	yes	characteristic
24	C₃×C₇₉⋊C₃	—	3	2	no	—
25	C₃×C₇₉⋊C₃	—	9	1	no	—
26	C₃×C₇₉⋊C₃	—	9	3	no	—
27	C₃×C₇₉⋊C₃	H(3,𝔽₃)	1	2	yes	—
28	C₃×C₇₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₉×C₇₉	—	3	1	no	—
30	C₉×C₇₉	—	3	2	no	—
31	C₇₉⋊C₉	—	3	4	no	—
32	C₇₉⋊C₉	—	3	2	no	—
33	#6910	—	79	1	no	maximal, Sylow
34	C₃×C₇₉	—	3	1	no	—
35	C₃×C₇₉	—	9	1	no	—
36	C₃×C₇₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₇₉⋊C₃	—	3	1	no	—
38	C₇₉⋊C₃	—	3	2	no	—
39	C₇₉⋊C₃	—	9	2	no	—
40	C₃×3₋¹⁺²	—	79	1	no	—
41	C₃×3₋¹⁺²	—	79	2	no	—
42	C₃×H(3,𝔽₃)	—	79	1	no	—
43	#731	—	79	4	no	—
44	#731	—	79	2	no	—
45	#731	—	79	2	no	—
46	#731	C₇₉⋊C₃	1	1	yes	characteristic, complemented
47	C₇₉	#6910	1	1	yes	characteristic, complemented, Sylow
48	C₃³	—	79	1	no	—
49	C₃³	—	237	1	no	—
50	3₋¹⁺²	—	79	4	no	—
51	3₋¹⁺²	—	237	2	no	—
52	3₋¹⁺²	—	237	2	no	—
53	3₋¹⁺²	—	237	2	no	—
54	3₋¹⁺²	—	237	1	no	—
55	3₋¹⁺²	C₃×C₇₉⋊C₃	1	2	yes	complemented
56	H(3,𝔽₃)	—	79	2	no	—
57	H(3,𝔽₃)	—	237	2	no	—
58	H(3,𝔽₃)	C₃×C₇₉⋊C₃	1	1	yes	characteristic
59	C₃×C₉	—	79	2	no	—
60	C₃×C₉	—	237	2	no	—
61	C₃×C₉	C₃×C₇₉⋊C₃	1	1	yes	characteristic, complemented
62	C₃²	—	3	1	no	—
63	C₃²	—	79	2	no	—
64	C₃²	—	79	1	no	—
65	C₃²	—	711	1	no	—
66	C₃²	—	711	3	no	—
67	C₃²	—	237	2	no	—
68	C₃²	C₃²×C₇₉⋊C₃	1	1	yes	characteristic
69	C₉	—	3	2	no	—
70	C₉	—	3	1	no	—
71	C₉	—	237	2	no	—
72	C₉	—	237	4	no	—
73	C₃	—	3	1	no	—
74	C₃	—	9	1	no	—
75	C₃	—	711	2	no	—
76	C₃	—	237	1	no	—
77	C₃	—	237	2	no	—
78	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×#731	C₃
socle	C₃×C₇₉	C₃×H(3,𝔽₃)
prime radical	H(3,𝔽₃)×C₇₉⋊C₃	C₃
hypercenter	#731	C₇₉⋊C₃
hypocenter	C₇₉	#6910
LCS	G ▷ C₃²×C₇₉ ▷ C₃×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

Complements

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

N	H ≅ G/N	type
1. C₃×#14523839	C₃	semidirect
2. 3₋¹⁺²×C₇₉⋊C₃	C₃	semidirect
4. #20264249	C₃	semidirect
5. #20264248	C₃	semidirect
6. #20264247	C₃	semidirect
7. C₇₉×#731	C₃	semidirect
10. #14523840	C₃²	semidirect
17. C₃×C₇₉⋊C₉	C₃²	semidirect
19. 3₋¹⁺²×C₇₉	C₃²	semidirect
21. C₃×C₉×C₇₉	C₃²	semidirect
46. #731	C₇₉⋊C₃	semidirect
47. C₇₉	#6910	semidirect
55. 3₋¹⁺²	C₃×C₇₉⋊C₃	semidirect
61. 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
3₋¹⁺²×C₇₉⋊C₃	3
H(3,𝔽₃)×C₇₉⋊C₃	3
#20264249	3
#20264248	3
#20264247	3
C₇₉×#731	3
#6910	79