Subgroups — C₃×C₇×2₋¹⁺⁴ #10621671

Order 672.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

Subgroups — C3×C7×2−1+4 #10621671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×2₋¹⁺⁴ #10621671