Subgroups — #34063166

Order 19197.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₇₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊(C₇₉⋊C₉)	C₃	1	2	yes	maximal, complemented
3	C₇₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₇₉⋊C₃	—	3	1	no	—
5	C₃²×C₇₉⋊C₃	—	3	1	no	—
6	C₃²×C₇₉⋊C₃	—	3	1	no	—
7	C₃²×C₇₉⋊C₃	—	3	3	no	—
8	C₃²×C₇₉⋊C₃	—	3	6	no	—
9	C₃²×C₇₉⋊C₃	C₉	1	3	yes	complemented
10	C₃×C₇₉⋊C₉	—	3	2	no	—
11	C₃³×C₇₉	C₃²	1	1	yes	characteristic
12	C₃×C₉×C₇₉	—	3	1	no	—
13	C₃²×C₇₉	—	3	1	no	—
14	C₃²×C₇₉	—	3	2	no	—
15	C₃²×C₇₉	C₃×C₉	1	1	yes	characteristic
16	C₃²×C₇₉	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃²×C₇₉	3₋¹⁺²	1	1	yes	characteristic
18	C₃²×C₇₉	3₋¹⁺²	1	1	yes	characteristic
19	C₃×C₇₉⋊C₃	—	3	9	no	—
20	C₃×C₇₉⋊C₃	—	3	9	no	—
21	C₃×C₇₉⋊C₃	—	3	3	no	—
22	C₃×C₇₉⋊C₃	—	3	3	no	—
23	C₃×C₇₉⋊C₃	—	3	6	no	—
24	C₃×C₇₉⋊C₃	—	3	9	no	—
25	C₉×C₇₉	—	9	1	no	—
26	C₇₉⋊C₉	—	9	2	no	—
27	#6866	—	79	1	no	maximal, Sylow
28	C₃×C₇₉	—	3	1	no	—
29	C₃×C₇₉	—	3	1	no	—
30	C₃×C₇₉	—	3	1	no	—
31	C₃×C₇₉	C₃²⋊C₉	1	1	yes	characteristic
32	C₃×C₇₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
33	C₃×C₇₉	AΣL(1,𝔽₂₇)	1	2	yes	—
34	C₇₉⋊C₃	—	3	9	no	—
35	C₃⁴	—	79	1	no	—
36	C₃²⋊C₉	—	79	2	no	—
37	C₃²⋊C₉	C₇₉⋊C₃	1	1	yes	characteristic, complemented
38	C₇₉	#6866	1	1	yes	characteristic, complemented, Sylow
39	C₃³	—	79	3	no	—
40	C₃³	—	237	3	no	—
41	C₃³	—	237	6	no	—
42	C₃³	—	237	1	no	—
43	C₃³	—	237	1	no	—
44	C₃³	—	237	1	no	—
45	C₃³	C₃×C₇₉⋊C₃	1	1	yes	characteristic
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	237	2	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	3	2	no	—
50	C₃²	—	237	9	no	—
51	C₃²	—	237	9	no	—
52	C₃²	—	237	9	no	—
53	C₃²	—	237	3	no	—
54	C₃²	—	237	3	no	—
55	C₃²	—	237	6	no	—
56	C₃²	C₉×C₇₉⋊C₃	1	1	yes	characteristic
57	C₃²	C₃²⋊(C₇₉⋊C₃)	1	1	yes	characteristic
58	C₃²	#14523838	1	1	yes	characteristic
59	C₃²	#14523839	1	1	yes	characteristic
60	C₉	—	9	1	no	—
61	C₉	—	711	2	no	—
62	C₃	—	3	1	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	1	no	—
65	C₃	—	237	9	no	—
66	C₃	(C₃×C₇₉⋊C₃)⋊C₉	1	1	yes	characteristic
67	C₃	#20264244	1	1	yes	characteristic
68	C₃	#20264246	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃²⋊(C₇₉⋊C₃)
commutator	C₃²×C₇₉	C₃×C₉
Frattini	C₃³	C₃×C₇₉⋊C₃
Fitting	C₇₉×C₃²⋊C₉	C₃
socle	C₃²×C₇₉	H(3,𝔽₃)
prime radical	C₃³×C₇₉⋊C₃	C₃
cosocle	C₃³×C₇₉	C₃²
hypercenter	C₃²⋊C₉	C₇₉⋊C₃
hypocenter	C₇₉	#6866
LCS	G ▷ C₃²×C₇₉ ▷ C₃×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃²⋊(C₇₉⋊C₉)	C₃	semidirect
3. C₇₉×C₃²⋊C₉	C₃	semidirect
9. C₃²×C₇₉⋊C₃	C₉	semidirect
37. C₃²⋊C₉	C₇₉⋊C₃	semidirect
38. C₇₉	#6866	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₇₉⋊C₃	3
C₃²⋊(C₇₉⋊C₉)	3
C₇₉×C₃²⋊C₉	3
#6866	79