Subgroups — C₂³×D₁₃₉₆ #36374966

Order 22336.

1362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₃₄₉	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₃₉₆	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₄₉	—	2	16	no	—
5	C₂³×D₃₄₉	C₂²	1	28	yes	complemented
6	C₂×D₁₃₉₆	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₃₄₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₃₄₉	C₂²	1	14	yes	complemented
9	C₂²×D₃₄₉	—	2	112	no	—
10	C₂²×D₃₄₉	C₂³	1	56	yes	complemented
11	D₁₃₉₆	C₂³	1	64	yes	direct product factor
12	C₂³×C₃₄₉	D₄	1	8	yes	complemented
13	C₂³×C₃₄₉	C₂³	1	7	yes	—
14	C₂×C₄×C₃₄₉	C₂³	1	28	yes	complemented
15	C₂²×C₃₄₉	C₂×D₄	1	28	yes	complemented
16	C₂²×C₃₄₉	C₂⁴	1	7	yes	—
17	C₂×D₃₄₉	—	2	112	no	—
18	C₂×D₃₄₉	C₂⁴	1	16	yes	complemented
19	C₄×C₃₄₉	C₂⁴	1	8	yes	complemented
20	C₂×C₃₄₉	C₂²×D₄	1	14	yes	complemented
21	C₂×C₃₄₉	C₂⁵	1	1	yes	characteristic
22	D₃₄₉	—	2	16	no	—
23	C₃₄₉	C₂³×D₄	1	1	yes	characteristic, complemented, Sylow
24	C₂³×D₄	—	349	1	no	maximal, Sylow
25	C₂⁵	—	349	2	no	—
26	C₂²×D₄	—	349	28	no	—
27	C₂³×C₄	D₃₄₉	1	1	yes	characteristic, complemented
28	C₂⁴	—	349	28	no	—
29	C₂⁴	—	698	16	no	—
30	C₂⁴	C₂×D₃₄₉	1	1	yes	characteristic
31	C₂×D₄	—	349	112	no	—
32	C₂²×C₄	C₂×D₃₄₉	1	14	yes	complemented
33	C₂³	—	349	56	no	—
34	C₂³	—	698	112	no	—
35	C₂³	D₁₃₉₆	1	8	yes	direct product factor
36	C₂³	C₂²×D₃₄₉	1	7	yes	—
37	D₄	—	349	64	no	—
38	C₂×C₄	C₂²×D₃₄₉	1	28	yes	complemented
39	C₂²	—	349	16	no	—
40	C₂²	—	698	112	no	—
41	C₂²	C₂×D₁₃₉₆	1	28	yes	direct product factor
42	C₂²	C₂³×D₃₄₉	1	7	yes	—
43	C₄	C₂³×D₃₄₉	1	8	yes	complemented
44	C₂	—	698	16	no	—
45	C₂	C₂²×D₁₃₉₆	1	14	yes	direct product factor
46	C₂	C₂⁴×D₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₃₄₉
commutator, cosocle	C₂×C₃₄₉	C₂⁵
Frattini	C₂	C₂⁴×D₃₄₉
Fitting	C₂³×C₄×C₃₄₉	C₂
socle	C₂⁴×C₃₄₉	C₂²
hypercenter	C₂³×C₄	D₃₄₉
hypocenter	C₃₄₉	C₂³×D₄
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₃₄₉	C₂	semidirect
2. C₂²×D₁₃₉₆	C₂	direct (N × H)
3. C₂³×C₄×C₃₄₉	C₂	semidirect
5. C₂³×D₃₄₉	C₂²	semidirect
6. C₂×D₁₃₉₆	C₂²	direct (N × H)
8. C₂²×C₄×C₃₄₉	C₂²	semidirect
10. C₂²×D₃₄₉	C₂³	semidirect
11. D₁₃₉₆	C₂³	direct (N × H)
12. C₂³×C₃₄₉	D₄	semidirect
14. C₂×C₄×C₃₄₉	C₂³	semidirect
15. C₂²×C₃₄₉	C₂×D₄	semidirect
18. C₂×D₃₄₉	C₂⁴	semidirect
19. C₄×C₃₄₉	C₂⁴	semidirect
20. C₂×C₃₄₉	C₂²×D₄	semidirect
23. C₃₄₉	C₂³×D₄	semidirect
27. C₂³×C₄	D₃₄₉	semidirect
32. C₂²×C₄	C₂×D₃₄₉	semidirect
35. C₂³	D₁₃₉₆	direct (N × H)
38. C₂×C₄	C₂²×D₃₄₉	semidirect
41. C₂²	C₂×D₁₃₉₆	direct (N × H)
43. C₄	C₂³×D₃₄₉	semidirect
45. C₂	C₂²×D₁₃₉₆	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×D₃₄₉	2
C₂²×D₁₃₉₆	2
C₂³×C₄×C₃₄₉	2
C₂³×D₄	349

Subgroups — C23×D1396 #36374966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₁₃₉₆ #36374966