Subgroups — C₂²×D₄×C₇₅₁ #37562338

Order 24032.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇₅₁	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×C₇₅₁	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₇₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₇₅₁	—	2	8	no	—
5	C₂³×C₇₅₁	C₂²	1	1	yes	characteristic
6	C₂³×C₇₅₁	C₂²	1	12	yes	complemented
7	D₄×C₇₅₁	C₂²	1	16	yes	direct product factor
8	C₂×C₄×C₇₅₁	C₂²	1	6	yes	complemented
9	C₂²×C₇₅₁	—	2	24	no	—
10	C₂²×C₇₅₁	D₄	1	4	yes	direct product factor
11	C₂²×C₇₅₁	C₂³	1	3	yes	—
12	C₂²×C₇₅₁	C₂³	1	8	yes	complemented
13	C₄×C₇₅₁	C₂³	1	4	yes	complemented
14	C₂×C₇₅₁	—	2	8	no	—
15	C₂×C₇₅₁	C₂×D₄	1	6	yes	direct product factor
16	C₂×C₇₅₁	C₂⁴	1	1	yes	characteristic
17	C₇₅₁	C₂²×D₄	1	1	yes	characteristic, direct product factor, Sylow
18	C₂²×D₄	C₇₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂⁴	C₂×C₇₅₁	1	2	yes	complemented
20	C₂×D₄	C₂×C₇₅₁	1	12	yes	direct product factor
21	C₂²×C₄	C₂×C₇₅₁	1	1	yes	characteristic, complemented
22	C₂³	—	2	8	no	—
23	C₂³	C₂²×C₇₅₁	1	1	yes	characteristic
24	C₂³	C₂²×C₇₅₁	1	12	yes	complemented
25	D₄	C₂²×C₇₅₁	1	16	yes	direct product factor
26	C₂×C₄	C₂²×C₇₅₁	1	6	yes	complemented
27	C₂²	—	2	24	no	—
28	C₂²	D₄×C₇₅₁	1	4	yes	direct product factor
29	C₂²	C₂³×C₇₅₁	1	3	yes	—
30	C₂²	C₂³×C₇₅₁	1	8	yes	complemented
31	C₄	C₂³×C₇₅₁	1	4	yes	complemented
32	C₂	—	2	8	no	—
33	C₂	C₂×D₄×C₇₅₁	1	6	yes	direct product factor
34	C₂	C₂⁴×C₇₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₅₁	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₇₅₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₇₅₁	C₂	semidirect
2. C₂×D₄×C₇₅₁	C₂	direct (N × H)
3. C₂²×C₄×C₇₅₁	C₂	semidirect
6. C₂³×C₇₅₁	C₂²	semidirect
7. D₄×C₇₅₁	C₂²	direct (N × H)
8. C₂×C₄×C₇₅₁	C₂²	semidirect
10. C₂²×C₇₅₁	D₄	direct (N × H)
12. C₂²×C₇₅₁	C₂³	semidirect
13. C₄×C₇₅₁	C₂³	semidirect
15. C₂×C₇₅₁	C₂×D₄	direct (N × H)
17. C₇₅₁	C₂²×D₄	direct (N × H)
18. C₂²×D₄	C₇₅₁	direct (N × H)
19. C₂⁴	C₂×C₇₅₁	semidirect
20. C₂×D₄	C₂×C₇₅₁	direct (N × H)
21. C₂²×C₄	C₂×C₇₅₁	semidirect
24. C₂³	C₂²×C₇₅₁	semidirect
25. D₄	C₂²×C₇₅₁	direct (N × H)
26. C₂×C₄	C₂²×C₇₅₁	semidirect
28. C₂²	D₄×C₇₅₁	direct (N × H)
30. C₂²	C₂³×C₇₅₁	semidirect
31. C₄	C₂³×C₇₅₁	semidirect
33. C₂	C₂×D₄×C₇₅₁	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×C₇₅₁	2
C₂×D₄×C₇₅₁	2
C₂²×C₄×C₇₅₁	2
C₂²×D₄	751

Subgroups — C22×D4×C751 #37562338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×C₇₅₁ #37562338