Subgroups — C₃×H(7,𝔽₂) #87584

Order 384.

5992 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#585	C₂	1	28	yes	maximal, complemented
2	C₂×C₃×H(5,𝔽₂)	C₂	1	35	yes	maximal, complemented
3	H(7,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₃×2₋¹⁺⁴	C₂²	1	56	yes	complemented
5	C₃×H(5,𝔽₂)	C₂²	1	280	yes	complemented
6	C₂×C₃×#41	C₂²	1	210	yes	complemented
7	C₂²×C₃×D₄	C₂²	1	105	yes	complemented
8	#585	C₂×C₃	1	28	yes	complemented
9	C₂×H(5,𝔽₂)	C₂×C₃	1	35	yes	complemented
10	C₂⁴×C₃	C₂³	1	30	yes	complemented
11	C₃×#41	C₂³	1	560	yes	complemented
12	C₂×C₃×Q₈	C₂³	1	70	yes	complemented
13	C₂×C₃×D₄	C₂³	1	630	yes	complemented
14	C₂²×C₃×C₄	C₂³	1	105	yes	complemented
15	2₋¹⁺⁴	C₂²×C₃	1	56	yes	complemented
16	H(5,𝔽₂)	C₂²×C₃	1	280	yes	complemented
17	C₂×#41	C₂²×C₃	1	210	yes	complemented
18	C₂²×D₄	C₂²×C₃	1	105	yes	complemented
19	C₂³×C₃	—	8	30	no	—
20	C₂³×C₃	C₂⁴	1	105	yes	—
21	C₃×Q₈	C₂⁴	1	56	yes	—
22	C₃×D₄	C₂⁴	1	280	yes	—
23	C₂×C₃×C₄	C₂⁴	1	210	yes	—
24	C₂⁴	C₂³×C₃	1	30	yes	complemented
25	#41	C₂³×C₃	1	560	yes	complemented
26	C₂×Q₈	C₂³×C₃	1	70	yes	complemented
27	C₂×D₄	C₂³×C₃	1	630	yes	complemented
28	C₂²×C₄	C₂³×C₃	1	105	yes	complemented
29	C₂²×C₃	—	4	105	no	—
30	C₂²×C₃	C₂⁵	1	35	yes	—
31	C₃×C₄	C₂⁵	1	28	yes	—
32	C₂³	—	8	30	no	—
33	C₂³	C₂⁴×C₃	1	105	yes	—
34	Q₈	C₂⁴×C₃	1	56	yes	—
35	D₄	C₂⁴×C₃	1	280	yes	—
36	C₂×C₄	C₂⁴×C₃	1	210	yes	—
37	C₂×C₃	—	2	35	no	—
38	C₂×C₃	C₂⁶	1	1	yes	characteristic
39	C₂²	—	4	105	no	—
40	C₂²	C₂⁵×C₃	1	35	yes	—
41	C₄	C₂⁵×C₃	1	28	yes	—
42	C₃	H(7,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
43	C₂	—	2	35	no	—
44	C₂	C₂⁶×C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂⁶
commutator, Frattini, cosocle	C₂	C₂⁶×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#585	C₂	semidirect
2. C₂×C₃×H(5,𝔽₂)	C₂	semidirect
3. H(7,𝔽₂)	C₃	direct (N × H)
4. C₃×2₋¹⁺⁴	C₂²	semidirect
5. C₃×H(5,𝔽₂)	C₂²	semidirect
6. C₂×C₃×#41	C₂²	semidirect
7. C₂²×C₃×D₄	C₂²	semidirect
8. #585	C₂×C₃	semidirect
9. C₂×H(5,𝔽₂)	C₂×C₃	semidirect
10. C₂⁴×C₃	C₂³	semidirect
11. C₃×#41	C₂³	semidirect
12. C₂×C₃×Q₈	C₂³	semidirect
13. C₂×C₃×D₄	C₂³	semidirect
14. C₂²×C₃×C₄	C₂³	semidirect
15. 2₋¹⁺⁴	C₂²×C₃	semidirect
16. H(5,𝔽₂)	C₂²×C₃	semidirect
17. C₂×#41	C₂²×C₃	semidirect
18. C₂²×D₄	C₂²×C₃	semidirect
24. C₂⁴	C₂³×C₃	semidirect
25. #41	C₂³×C₃	semidirect
26. C₂×Q₈	C₂³×C₃	semidirect
27. C₂×D₄	C₂³×C₃	semidirect
28. C₂²×C₄	C₂³×C₃	semidirect
42. C₃	H(7,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#585	2
C₂×C₃×H(5,𝔽₂)	2
H(7,𝔽₂)	3

Subgroups — C3×H(7,𝔽2) #87584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×H(7,𝔽₂) #87584