Subgroups — C₇×H(3,𝔽₃)² #19100302

Order 5103.

2102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇×H(3,𝔽₃)	C₃	1	8	yes	maximal, complemented
2	C₇×C₃²⋊H(3,𝔽₃)	C₃	1	32	yes	maximal, complemented
3	H(3,𝔽₃)²	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₃⁴×C₇	C₃²	1	16	yes	complemented
5	C₃×C₇×H(3,𝔽₃)	—	3	72	no	—
6	C₃×C₇×H(3,𝔽₃)	C₃²	1	2	yes	—
7	C₃×C₇×H(3,𝔽₃)	C₃²	1	64	yes	complemented
8	C₃×C₇×H(3,𝔽₃)	C₃²	1	48	yes	complemented
9	C₃²×H(3,𝔽₃)	C₃×C₇	1	8	yes	complemented
10	C₃²⋊H(3,𝔽₃)	C₃×C₇	1	32	yes	complemented
11	C₃³×C₇	—	3	96	no	—
12	C₃³×C₇	—	9	32	no	—
13	C₃³×C₇	C₃³	1	32	yes	—
14	C₃³×C₇	C₃³	1	8	yes	—
15	C₇×H(3,𝔽₃)	—	3	192	no	—
16	C₇×H(3,𝔽₃)	—	9	48	no	—
17	C₇×H(3,𝔽₃)	H(3,𝔽₃)	1	18	yes	direct product factor
18	C₃⁴	C₃²×C₇	1	16	yes	complemented
19	C₃×H(3,𝔽₃)	—	3	72	no	—
20	C₃×H(3,𝔽₃)	C₃²×C₇	1	2	yes	—
21	C₃×H(3,𝔽₃)	C₃²×C₇	1	64	yes	complemented
22	C₃×H(3,𝔽₃)	C₃²×C₇	1	48	yes	complemented
23	C₃²×C₇	—	3	8	no	—
24	C₃²×C₇	—	3	16	no	—
25	C₃²×C₇	—	3	64	no	—
26	C₃²×C₇	—	3	64	no	—
27	C₃²×C₇	—	9	144	no	—
28	C₃²×C₇	C₃×H(3,𝔽₃)	1	24	yes	complemented
29	C₃²×C₇	C₃⁴	1	1	yes	characteristic
30	C₃³	—	3	96	no	—
31	C₃³	—	9	32	no	—
32	C₃³	C₃³×C₇	1	32	yes	—
33	C₃³	C₃³×C₇	1	8	yes	—
34	H(3,𝔽₃)	—	3	192	no	—
35	H(3,𝔽₃)	—	9	48	no	—
36	H(3,𝔽₃)	C₇×H(3,𝔽₃)	1	18	yes	direct product factor
37	C₃×C₇	—	3	24	no	—
38	C₃×C₇	—	9	32	no	—
39	C₃×C₇	C₃²×H(3,𝔽₃)	1	2	yes	—
40	C₃×C₇	H(5,𝔽₃)	1	2	yes	—
41	C₃²	—	3	8	no	—
42	C₃²	—	3	16	no	—
43	C₃²	—	3	64	no	—
44	C₃²	—	3	64	no	—
45	C₃²	—	9	144	no	—
46	C₃²	C₃×C₇×H(3,𝔽₃)	1	24	yes	complemented
47	C₃²	C₃⁴×C₇	1	1	yes	characteristic
48	C₇	H(3,𝔽₃)²	1	1	yes	characteristic, direct product factor, Sylow
49	C₃	—	3	24	no	—
50	C₃	—	9	32	no	—
51	C₃	C₃²×C₇×H(3,𝔽₃)	1	2	yes	—
52	C₃	C₇×H(5,𝔽₃)	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₇	C₃⁴
commutator, Frattini	C₃²	C₃⁴×C₇
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×C₇×H(3,𝔽₃)	C₃	semidirect
2. C₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
3. H(3,𝔽₃)²	C₇	direct (N × H)
4. C₃⁴×C₇	C₃²	semidirect
7. C₃×C₇×H(3,𝔽₃)	C₃²	semidirect
8. C₃×C₇×H(3,𝔽₃)	C₃²	semidirect
9. C₃²×H(3,𝔽₃)	C₃×C₇	semidirect
10. C₃²⋊H(3,𝔽₃)	C₃×C₇	semidirect
17. C₇×H(3,𝔽₃)	H(3,𝔽₃)	direct (N × H)
18. C₃⁴	C₃²×C₇	semidirect
21. C₃×H(3,𝔽₃)	C₃²×C₇	semidirect
22. C₃×H(3,𝔽₃)	C₃²×C₇	semidirect
28. C₃²×C₇	C₃×H(3,𝔽₃)	semidirect
36. H(3,𝔽₃)	C₇×H(3,𝔽₃)	direct (N × H)
46. C₃²	C₃×C₇×H(3,𝔽₃)	semidirect
48. C₇	H(3,𝔽₃)²	direct (N × H)

Subgroups — C7×H(3,𝔽3)2 #19100302

Conjugacy classes of subgroups

Complements

Subgroups — C₇×H(3,𝔽₃)² #19100302