Subgroups — C₂²×#41×C₁₉₉ #27163732

Order 12736.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₁₉₉	C₂	1	24	yes	maximal, direct product factor
2	C₂²×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₄×C₁₉₉	C₂	1	3	yes	maximal, complemented
4	C₂³×C₄×C₁₉₉	C₂	1	3	yes	maximal, complemented
5	C₂⁴×C₁₉₉	C₂²	1	3	yes	—
6	#41×C₁₉₉	C₂²	1	64	yes	direct product factor
7	C₂×Q₈×C₁₉₉	C₂²	1	12	yes	complemented
8	C₂×D₄×C₁₉₉	C₂²	1	36	yes	complemented
9	C₂²×C₄×C₁₉₉	C₂²	1	3	yes	—
10	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₁₉₉	C₂²	1	36	yes	complemented
12	C₂³×C₁₉₉	—	2	12	no	—
13	C₂³×C₁₉₉	C₂³	1	18	yes	—
14	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
15	Q₈×C₁₉₉	C₂³	1	16	yes	complemented
16	D₄×C₁₉₉	C₂³	1	48	yes	complemented
17	C₂×C₄×C₁₉₉	C₂³	1	18	yes	—
18	C₂×C₄×C₁₉₉	C₂³	1	6	yes	—
19	C₂×C₄×C₁₉₉	C₂³	1	48	yes	complemented
20	C₂²×C₁₉₉	—	2	36	no	—
21	C₂²×C₁₉₉	#41	1	4	yes	direct product factor
22	C₂²×C₁₉₉	C₂⁴	1	12	yes	—
23	C₂²×C₁₉₉	C₂⁴	1	3	yes	—
24	C₄×C₁₉₉	C₂⁴	1	4	yes	—
25	C₄×C₁₉₉	C₂⁴	1	12	yes	—
26	C₂×C₁₉₉	—	2	12	no	—
27	C₂×C₁₉₉	C₂×#41	1	6	yes	direct product factor
28	C₂×C₁₉₉	C₂⁵	1	1	yes	characteristic
29	C₁₉₉	C₂²×#41	1	1	yes	characteristic, direct product factor, Sylow
30	C₂²×#41	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×#41	C₂×C₁₉₉	1	24	yes	direct product factor
32	C₂²×Q₈	C₂×C₁₉₉	1	1	yes	characteristic, complemented
33	C₂²×D₄	C₂×C₁₉₉	1	3	yes	complemented
34	C₂³×C₄	C₂×C₁₉₉	1	3	yes	complemented
35	C₂⁴	C₂²×C₁₉₉	1	3	yes	—
36	#41	C₂²×C₁₉₉	1	64	yes	direct product factor
37	C₂×Q₈	C₂²×C₁₉₉	1	12	yes	complemented
38	C₂×D₄	C₂²×C₁₉₉	1	36	yes	complemented
39	C₂²×C₄	C₂²×C₁₉₉	1	3	yes	—
40	C₂²×C₄	C₂²×C₁₉₉	1	1	yes	characteristic
41	C₂²×C₄	C₂²×C₁₉₉	1	36	yes	complemented
42	C₂³	—	2	12	no	—
43	C₂³	C₂³×C₁₉₉	1	18	yes	—
44	C₂³	C₂³×C₁₉₉	1	1	yes	characteristic
45	Q₈	C₂³×C₁₉₉	1	16	yes	complemented
46	D₄	C₂³×C₁₉₉	1	48	yes	complemented
47	C₂×C₄	C₂³×C₁₉₉	1	6	yes	—
48	C₂×C₄	C₂³×C₁₉₉	1	18	yes	—
49	C₂×C₄	C₂³×C₁₉₉	1	48	yes	complemented
50	C₂²	—	2	36	no	—
51	C₂²	#41×C₁₉₉	1	4	yes	direct product factor
52	C₂²	C₂⁴×C₁₉₉	1	12	yes	—
53	C₂²	C₂⁴×C₁₉₉	1	3	yes	—
54	C₄	C₂⁴×C₁₉₉	1	4	yes	—
55	C₄	C₂⁴×C₁₉₉	1	12	yes	—
56	C₂	—	2	12	no	—
57	C₂	C₂×#41×C₁₉₉	1	6	yes	direct product factor
58	C₂	C₂⁵×C₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₉₉	C₂²
commutator, Frattini, cosocle	C₂	C₂⁵×C₁₉₉
socle	C₂³×C₁₉₉	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#41×C₁₉₉	C₂	direct (N × H)
2. C₂²×Q₈×C₁₉₉	C₂	semidirect
3. C₂²×D₄×C₁₉₉	C₂	semidirect
4. C₂³×C₄×C₁₉₉	C₂	semidirect
6. #41×C₁₉₉	C₂²	direct (N × H)
7. C₂×Q₈×C₁₉₉	C₂²	semidirect
8. C₂×D₄×C₁₉₉	C₂²	semidirect
11. C₂²×C₄×C₁₉₉	C₂²	semidirect
15. Q₈×C₁₉₉	C₂³	semidirect
16. D₄×C₁₉₉	C₂³	semidirect
19. C₂×C₄×C₁₉₉	C₂³	semidirect
21. C₂²×C₁₉₉	#41	direct (N × H)
27. C₂×C₁₉₉	C₂×#41	direct (N × H)
29. C₁₉₉	C₂²×#41	direct (N × H)
30. C₂²×#41	C₁₉₉	direct (N × H)
31. C₂×#41	C₂×C₁₉₉	direct (N × H)
32. C₂²×Q₈	C₂×C₁₉₉	semidirect
33. C₂²×D₄	C₂×C₁₉₉	semidirect
34. C₂³×C₄	C₂×C₁₉₉	semidirect
36. #41	C₂²×C₁₉₉	direct (N × H)
37. C₂×Q₈	C₂²×C₁₉₉	semidirect
38. C₂×D₄	C₂²×C₁₉₉	semidirect
41. C₂²×C₄	C₂²×C₁₉₉	semidirect
45. Q₈	C₂³×C₁₉₉	semidirect
46. D₄	C₂³×C₁₉₉	semidirect
49. C₂×C₄	C₂³×C₁₉₉	semidirect
51. C₂²	#41×C₁₉₉	direct (N × H)
57. C₂	C₂×#41×C₁₉₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#41×C₁₉₉	2
C₂²×Q₈×C₁₉₉	2
C₂²×D₄×C₁₉₉	2
C₂³×C₄×C₁₉₉	2
C₂²×#41	199

Subgroups — C22×#41×C199 #27163732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#41×C₁₉₉ #27163732