Subgroups — C₂⁴×#168 #10619113

Order 640.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×#168	C₂	1	30	yes	maximal, direct product factor
3	C₂⁵×C₅	C₄	1	1	yes	characteristic
4	C₂³×C₄×C₅	C₄	1	15	yes	—
5	C₂³×C₄×C₅	C₂²	1	15	yes	—
6	C₂²×#168	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₈	—	5	1	no	maximal, Sylow
8	C₂⁴×C₅	C₈	1	16	yes	complemented
9	C₂⁴×C₅	C₂×C₄	1	15	yes	—
10	C₂²×C₄×C₅	C₂×C₄	1	105	yes	—
11	C₂²×C₄×C₅	C₂³	1	35	yes	—
12	C₂×#168	C₂³	1	120	yes	direct product factor
13	C₂⁴×C₄	D₅	1	1	yes	characteristic
14	C₂³×C₈	—	5	30	no	—
15	C₂³×C₅	C₂×C₈	1	120	yes	complemented
16	C₂³×C₅	C₂²×C₄	1	35	yes	—
17	C₂×C₄×C₅	C₂²×C₄	1	105	yes	—
18	C₂×C₄×C₅	C₂⁴	1	15	yes	—
19	#168	C₂⁴	1	16	yes	direct product factor
20	C₂⁵	Dic₅	1	1	yes	characteristic
21	C₂³×C₄	Dic₅	1	15	yes	—
22	C₂³×C₄	C₂×D₅	1	15	yes	—
23	C₂²×C₈	—	5	140	no	—
24	C₂²×C₅	C₂²×C₈	1	140	yes	complemented
25	C₂²×C₅	C₂³×C₄	1	15	yes	—
26	C₄×C₅	C₂³×C₄	1	15	yes	—
27	C₄×C₅	C₂⁵	1	1	yes	characteristic
28	C₂⁴	#168	1	16	yes	direct product factor
29	C₂⁴	C₂×Dic₅	1	15	yes	—
30	C₂²×C₄	C₂×Dic₅	1	105	yes	—
31	C₂²×C₄	C₂²×D₅	1	35	yes	—
32	C₂×C₈	—	5	120	no	—
33	C₂×C₅	C₂³×C₈	1	30	yes	complemented
34	C₂×C₅	C₂⁴×C₄	1	1	yes	characteristic
35	C₂³	C₂×#168	1	120	yes	direct product factor
36	C₂³	C₂²×Dic₅	1	35	yes	—
37	C₂×C₄	C₂²×Dic₅	1	105	yes	—
38	C₂×C₄	C₂³×D₅	1	15	yes	—
39	C₈	—	5	16	no	—
40	C₅	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
41	C₂²	C₂²×#168	1	140	yes	direct product factor
42	C₂²	C₂³×Dic₅	1	15	yes	—
43	C₄	C₂³×Dic₅	1	15	yes	—
44	C₄	C₂⁴×D₅	1	1	yes	characteristic
45	C₂	C₂³×#168	1	30	yes	direct product factor
46	C₂	C₂⁴×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	D₅
commutator, hypocenter	C₅	C₂⁴×C₈
Frattini	C₄	C₂⁴×D₅
Fitting	C₂⁴×C₄×C₅	C₂
socle, prime radical	C₂⁵×C₅	C₄
cosocle	C₄×C₅	C₂⁵
LCS	G ▷ C₅
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×#168	C₂	direct (N × H)
6. C₂²×#168	C₂²	direct (N × H)
8. C₂⁴×C₅	C₈	semidirect
12. C₂×#168	C₂³	direct (N × H)
15. C₂³×C₅	C₂×C₈	semidirect
19. #168	C₂⁴	direct (N × H)
24. C₂²×C₅	C₂²×C₈	semidirect
28. C₂⁴	#168	direct (N × H)
33. C₂×C₅	C₂³×C₈	semidirect
35. C₂³	C₂×#168	direct (N × H)
40. C₅	C₂⁴×C₈	semidirect
41. C₂²	C₂²×#168	direct (N × H)
45. C₂	C₂³×#168	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₅	2
C₂³×#168	2
C₂⁴×C₈	5

Subgroups — C24×#168 #10619113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#168 #10619113