Subgroups — C₃×C₇×H(5,𝔽₂) #10621670

Order 672.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#41	C₂	1	6	yes	maximal, complemented
2	C₂×C₃×C₇×D₄	C₂	1	9	yes	maximal, complemented
3	C₇×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₇	C₂²	1	6	yes	complemented
5	C₃×C₇×Q₈	C₂²	1	2	yes	complemented
6	C₃×C₇×D₄	C₂²	1	18	yes	complemented
7	C₂×C₃×C₄×C₇	C₂²	1	9	yes	complemented
8	C₇×#41	C₂×C₃	1	6	yes	complemented
9	C₂×C₇×D₄	C₂×C₃	1	9	yes	complemented
10	C₃×H(5,𝔽₂)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₃×C₇	—	4	6	no	—
12	C₂²×C₃×C₇	C₂³	1	9	yes	—
13	C₃×C₄×C₇	C₂³	1	6	yes	—
14	C₂³×C₇	C₂²×C₃	1	6	yes	complemented
15	C₇×Q₈	C₂²×C₃	1	2	yes	complemented
16	C₇×D₄	C₂²×C₃	1	18	yes	complemented
17	C₂×C₄×C₇	C₂²×C₃	1	9	yes	complemented
18	C₃×#41	C₂×C₇	1	6	yes	complemented
19	C₂×C₃×D₄	C₂×C₇	1	9	yes	complemented
20	C₂×C₃×C₇	—	2	9	no	—
21	C₂×C₃×C₇	C₂⁴	1	1	yes	characteristic
22	H(5,𝔽₂)	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
23	C₂²×C₇	—	4	6	no	—
24	C₂²×C₇	C₂³×C₃	1	9	yes	—
25	C₄×C₇	C₂³×C₃	1	6	yes	—
26	C₂³×C₃	C₂²×C₇	1	6	yes	complemented
27	C₃×Q₈	C₂²×C₇	1	2	yes	complemented
28	C₃×D₄	C₂²×C₇	1	18	yes	complemented
29	C₂×C₃×C₄	C₂²×C₇	1	9	yes	complemented
30	C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Hall
31	#41	C₂×C₃×C₇	1	6	yes	complemented
32	C₂×D₄	C₂×C₃×C₇	1	9	yes	complemented
33	C₂×C₇	—	2	9	no	—
34	C₂×C₇	C₂⁴×C₃	1	1	yes	characteristic
35	C₂²×C₃	—	4	6	no	—
36	C₂²×C₃	C₂³×C₇	1	9	yes	—
37	C₃×C₄	C₂³×C₇	1	6	yes	—
38	C₂³	C₂²×C₃×C₇	1	6	yes	complemented
39	Q₈	C₂²×C₃×C₇	1	2	yes	complemented
40	D₄	C₂²×C₃×C₇	1	18	yes	complemented
41	C₂×C₄	C₂²×C₃×C₇	1	9	yes	complemented
42	C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃	—	2	9	no	—
44	C₂×C₃	C₂⁴×C₇	1	1	yes	characteristic
45	C₂²	—	4	6	no	—
46	C₂²	C₂³×C₃×C₇	1	9	yes	—
47	C₄	C₂³×C₃×C₇	1	6	yes	—
48	C₃	C₇×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
49	C₂	—	2	9	no	—
50	C₂	C₂⁴×C₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₃×C₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#41	C₂	semidirect
2. C₂×C₃×C₇×D₄	C₂	semidirect
3. C₇×H(5,𝔽₂)	C₃	direct (N × H)
4. C₂³×C₃×C₇	C₂²	semidirect
5. C₃×C₇×Q₈	C₂²	semidirect
6. C₃×C₇×D₄	C₂²	semidirect
7. C₂×C₃×C₄×C₇	C₂²	semidirect
8. C₇×#41	C₂×C₃	semidirect
9. C₂×C₇×D₄	C₂×C₃	semidirect
10. C₃×H(5,𝔽₂)	C₇	direct (N × H)
14. C₂³×C₇	C₂²×C₃	semidirect
15. C₇×Q₈	C₂²×C₃	semidirect
16. C₇×D₄	C₂²×C₃	semidirect
17. C₂×C₄×C₇	C₂²×C₃	semidirect
18. C₃×#41	C₂×C₇	semidirect
19. C₂×C₃×D₄	C₂×C₇	semidirect
22. H(5,𝔽₂)	C₃×C₇	direct (N × H)
26. C₂³×C₃	C₂²×C₇	semidirect
27. C₃×Q₈	C₂²×C₇	semidirect
28. C₃×D₄	C₂²×C₇	semidirect
29. C₂×C₃×C₄	C₂²×C₇	semidirect
30. C₃×C₇	H(5,𝔽₂)	direct (N × H)
31. #41	C₂×C₃×C₇	semidirect
32. C₂×D₄	C₂×C₃×C₇	semidirect
38. C₂³	C₂²×C₃×C₇	semidirect
39. Q₈	C₂²×C₃×C₇	semidirect
40. D₄	C₂²×C₃×C₇	semidirect
41. C₂×C₄	C₂²×C₃×C₇	semidirect
42. C₇	C₃×H(5,𝔽₂)	direct (N × H)
48. C₃	C₇×H(5,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×C₇×#41	2
C₂×C₃×C₇×D₄	2
C₇×H(5,𝔽₂)	3
C₃×H(5,𝔽₂)	7

Subgroups — C3×C7×H(5,𝔽2) #10621670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×H(5,𝔽₂) #10621670