Subgroups — (C₃×C₇₉⋊C₃)⋊H(3,𝔽₃) #34063248

Order 19197.

378 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, complemented
2	C₃×C₃²⋊(C₇₉⋊C₃)	C₃	1	2	yes	maximal, complemented
3	C₃×C₃²⋊(C₇₉⋊C₃)	C₃	1	9	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₇₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₇₉⋊C₃	—	3	3	no	—
6	C₃²×C₇₉⋊C₃	—	3	9	no	—
7	C₃²×C₇₉⋊C₃	C₃²	1	3	yes	—
8	C₃²×C₇₉⋊C₃	C₃²	1	6	yes	complemented
9	C₃²⋊(C₇₉⋊C₃)	—	3	6	no	—
10	C₃²⋊(C₇₉⋊C₃)	—	3	27	no	—
11	C₃³×C₇₉	C₃²	1	1	yes	characteristic
12	C₃³×C₇₉	C₃²	1	3	yes	complemented
13	H(3,𝔽₃)×C₇₉	—	3	3	no	—
14	C₃²×C₇₉	—	3	3	no	—
15	C₃²×C₇₉	—	3	3	no	—
16	C₃²×C₇₉	—	3	9	no	—
17	C₃²×C₇₉	H(3,𝔽₃)	1	3	yes	complemented
18	C₃²×C₇₉	C₃³	1	1	yes	characteristic
19	C₃×C₇₉⋊C₃	—	3	27	no	—
20	C₃×C₇₉⋊C₃	—	3	6	no	—
21	C₃×C₇₉⋊C₃	—	3	3	no	—
22	C₃×C₇₉⋊C₃	—	3	18	no	—
23	C₃×C₇₉⋊C₃	—	3	6	no	—
24	C₃×C₇₉⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
25	C₃²⋊H(3,𝔽₃)	—	79	1	no	maximal, Sylow
26	C₃×C₇₉	—	3	3	no	—
27	C₃×C₇₉	—	9	3	no	—
28	C₃×C₇₉	C₃×H(3,𝔽₃)	1	3	yes	—
29	C₃×C₇₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₇₉⋊C₃	—	3	9	no	—
31	C₇₉⋊C₃	—	9	6	no	—
32	C₃⁴	—	79	1	no	—
33	C₃×H(3,𝔽₃)	—	79	2	no	—
34	C₃×H(3,𝔽₃)	—	79	9	no	—
35	C₃×H(3,𝔽₃)	C₇₉⋊C₃	1	1	yes	characteristic, complemented
36	C₇₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
37	C₃³	—	79	3	no	—
38	C₃³	—	79	6	no	—
39	C₃³	—	237	9	no	—
40	C₃³	—	237	3	no	—
41	C₃³	C₃×C₇₉⋊C₃	1	1	yes	characteristic
42	C₃³	C₃×C₇₉⋊C₃	1	3	yes	complemented
43	H(3,𝔽₃)	—	3	3	no	—
44	H(3,𝔽₃)	—	237	27	no	—
45	H(3,𝔽₃)	—	237	6	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	3	9	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	79	9	no	—
50	C₃²	—	237	27	no	—
51	C₃²	—	237	18	no	—
52	C₃²	—	237	6	no	—
53	C₃²	—	237	6	no	—
54	C₃²	—	237	3	no	—
55	C₃²	C₃²⋊(C₇₉⋊C₃)	1	3	yes	complemented
56	C₃²	C₃²×C₇₉⋊C₃	1	1	yes	characteristic
57	C₃	—	3	3	no	—
58	C₃	—	9	3	no	—
59	C₃	—	711	6	no	—
60	C₃	—	237	9	no	—
61	C₃	H(3,𝔽₃)×C₇₉⋊C₃	1	3	yes	—
62	C₃	C₃×C₃²⋊(C₇₉⋊C₃)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃³×C₇₉⋊C₃	C₃	semidirect
2. C₃×C₃²⋊(C₇₉⋊C₃)	C₃	semidirect
3. C₃×C₃²⋊(C₇₉⋊C₃)	C₃	semidirect
4. C₃×H(3,𝔽₃)×C₇₉	C₃	semidirect
8. C₃²×C₇₉⋊C₃	C₃²	semidirect
12. C₃³×C₇₉	C₃²	semidirect
17. C₃²×C₇₉	H(3,𝔽₃)	semidirect
24. C₃×C₇₉⋊C₃	H(3,𝔽₃)	semidirect
35. C₃×H(3,𝔽₃)	C₇₉⋊C₃	semidirect
36. C₇₉	C₃²⋊H(3,𝔽₃)	semidirect
42. C₃³	C₃×C₇₉⋊C₃	semidirect
55. C₃²	C₃²⋊(C₇₉⋊C₃)	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
C₃×C₃²⋊(C₇₉⋊C₃)	3
C₃×H(3,𝔽₃)×C₇₉	3
C₃²⋊H(3,𝔽₃)	79

Subgroups — (C3×C79⋊C3)⋊H(3,𝔽3) #34063248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₃×C₇₉⋊C₃)⋊H(3,𝔽₃) #34063248