Subgroups — #509×C₄₁₉ #41000002

Order 26816.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₃₂×C₄₁₉	C₂	1	2	yes	maximal, complemented
4	D₁₆×C₄₁₉	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₁₉	—	2	1	no	—
7	C₂×D₄×C₄₁₉	—	2	1	no	—
8	Q₁₆×C₄₁₉	C₂²	1	1	yes	characteristic, complemented
9	SD₁₆×C₄₁₉	—	2	1	no	—
10	D₈×C₄₁₉	—	2	1	no	—
11	D₈×C₄₁₉	C₂²	1	2	yes	—
12	D₈×C₄₁₉	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
14	C₁₆×C₄₁₉	C₂²	1	2	yes	complemented
15	C₂³×C₄₁₉	—	4	1	no	—
16	Q₈×C₄₁₉	—	2	1	no	—
17	D₄×C₄₁₉	—	2	2	no	—
18	D₄×C₄₁₉	—	2	1	no	—
19	D₄×C₄₁₉	—	4	1	no	—
20	D₄×C₄₁₉	—	4	1	no	—
21	C₂×C₄×C₄₁₉	—	4	1	no	—
22	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
23	C₈×C₄₁₉	D₄	1	1	yes	characteristic
24	C₈×C₄₁₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄₁₉	—	4	1	no	—
26	C₂²×C₄₁₉	—	4	2	no	—
27	C₂²×C₄₁₉	—	8	2	no	—
28	C₂²×C₄₁₉	D₈	1	1	yes	characteristic
29	C₄×C₄₁₉	—	4	1	no	—
30	C₄×C₄₁₉	D₈	1	1	yes	characteristic
31	C₄×C₄₁₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄₁₉	—	2	1	no	—
33	C₂×C₄₁₉	—	8	1	no	—
34	C₂×C₄₁₉	—	8	2	no	—
35	C₂×C₄₁₉	C₂×D₈	1	1	yes	characteristic
36	C₄₁₉	#509	1	1	yes	characteristic, direct product factor, Sylow
37	#509	C₄₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	#135	C₂×C₄₁₉	1	1	yes	characteristic, complemented
39	C₂×D₈	C₂×C₄₁₉	1	1	yes	characteristic, complemented
40	SD₃₂	C₂×C₄₁₉	1	2	yes	complemented
41	D₁₆	C₂×C₄₁₉	1	2	yes	complemented
42	M₃₂	C₂×C₄₁₉	1	1	yes	characteristic, complemented
43	#41	—	2	1	no	—
44	C₂×D₄	—	2	1	no	—
45	Q₁₆	C₂²×C₄₁₉	1	1	yes	characteristic, complemented
46	SD₁₆	—	2	1	no	—
47	D₈	—	2	1	no	—
48	D₈	C₂²×C₄₁₉	1	2	yes	—
49	D₈	C₂²×C₄₁₉	1	1	yes	characteristic, complemented
50	C₂×C₈	C₂²×C₄₁₉	1	1	yes	characteristic
51	C₁₆	C₂²×C₄₁₉	1	2	yes	complemented
52	C₂³	—	4	1	no	—
53	Q₈	—	2	1	no	—
54	D₄	—	2	2	no	—
55	D₄	—	2	1	no	—
56	D₄	—	4	1	no	—
57	D₄	—	4	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	D₄×C₄₁₉	1	1	yes	characteristic
60	C₈	D₄×C₄₁₉	1	1	yes	characteristic
61	C₈	C₂³×C₄₁₉	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	—	4	2	no	—
64	C₂²	—	8	2	no	—
65	C₂²	D₈×C₄₁₉	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	D₈×C₄₁₉	1	1	yes	characteristic
68	C₄	C₂×D₄×C₄₁₉	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	2	no	—
71	C₂	—	8	1	no	—
72	C₂	C₂×D₈×C₄₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁₉	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂³×C₄₁₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁₉ ◁ C₂×C₄×C₄₁₉ ◁ C₂×C₈×C₄₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #135×C₄₁₉	C₂	semidirect
2. C₂×D₈×C₄₁₉	C₂	semidirect
3. SD₃₂×C₄₁₉	C₂	semidirect
4. D₁₆×C₄₁₉	C₂	semidirect
5. M₃₂×C₄₁₉	C₂	semidirect
8. Q₁₆×C₄₁₉	C₂²	semidirect
12. D₈×C₄₁₉	C₂²	semidirect
14. C₁₆×C₄₁₉	C₂²	semidirect
36. C₄₁₉	#509	direct (N × H)
37. #509	C₄₁₉	direct (N × H)
38. #135	C₂×C₄₁₉	semidirect
39. C₂×D₈	C₂×C₄₁₉	semidirect
40. SD₃₂	C₂×C₄₁₉	semidirect
41. D₁₆	C₂×C₄₁₉	semidirect
42. M₃₂	C₂×C₄₁₉	semidirect
45. Q₁₆	C₂²×C₄₁₉	semidirect
49. D₈	C₂²×C₄₁₉	semidirect
51. C₁₆	C₂²×C₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
#135×C₄₁₉	2
C₂×D₈×C₄₁₉	2
SD₃₂×C₄₁₉	2
D₁₆×C₄₁₉	2
M₃₂×C₄₁₉	2
#509	419

Subgroups — #509×C419 #41000002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #509×C₄₁₉ #41000002