Subgroups — #578×C₄₁₉ #41000071

Order 26816.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×C₄₁₉	C₂	1	2	yes	maximal, complemented
2	#137×C₄₁₉	C₂	1	6	yes	maximal, complemented
3	#135×C₄₁₉	C₂	1	3	yes	maximal, complemented
4	C₂×Q₁₆×C₄₁₉	C₂	1	3	yes	maximal, complemented
5	#131×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₁₉	—	2	6	no	—
7	#41×C₄₁₉	C₂²	1	6	yes	—
8	#41×C₄₁₉	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₄₁₉	—	2	2	no	—
10	C₂×Q₈×C₄₁₉	C₂²	1	6	yes	—
11	Q₁₆×C₄₁₉	C₂²	1	3	yes	—
12	Q₁₆×C₄₁₉	C₂²	1	6	yes	—
13	SD₁₆×C₄₁₉	C₂²	1	6	yes	—
14	D₈×C₄₁₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₄₁₉	C₂²	1	3	yes	—
16	C₂×C₈×C₄₁₉	C₂²	1	3	yes	—
17	Q₈×C₄₁₉	—	2	6	no	—
18	Q₈×C₄₁₉	D₄	1	1	yes	characteristic
19	Q₈×C₄₁₉	C₂³	1	6	yes	—
20	D₄×C₄₁₉	—	2	6	no	—
21	D₄×C₄₁₉	D₄	1	3	yes	—
22	D₄×C₄₁₉	C₂³	1	2	yes	—
23	C₂×C₄×C₄₁₉	—	2	6	no	—
24	C₂×C₄×C₄₁₉	—	2	6	no	—
25	C₂×C₄×C₄₁₉	C₂³	1	3	yes	—
26	C₈×C₄₁₉	C₂³	1	1	yes	characteristic
27	C₈×C₄₁₉	C₂³	1	3	yes	—
28	C₂²×C₄₁₉	—	2	2	no	—
29	C₂²×C₄₁₉	C₂×D₄	1	3	yes	—
30	C₄×C₄₁₉	—	2	6	no	—
31	C₄×C₄₁₉	C₂×D₄	1	3	yes	—
32	C₄×C₄₁₉	C₂⁴	1	1	yes	characteristic
33	C₂×C₄₁₉	—	2	3	no	—
34	C₂×C₄₁₉	—	4	2	no	—
35	C₂×C₄₁₉	C₂²×D₄	1	1	yes	characteristic
36	C₄₁₉	#578	1	1	yes	characteristic, direct product factor, Sylow
37	#578	C₄₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	2₋¹⁺⁴	C₂×C₄₁₉	1	2	yes	complemented
39	#137	C₂×C₄₁₉	1	6	yes	complemented
40	#135	C₂×C₄₁₉	1	3	yes	complemented
41	C₂×Q₁₆	C₂×C₄₁₉	1	3	yes	complemented
42	#131	C₂×C₄₁₉	1	1	yes	characteristic, complemented
43	#41	—	2	6	no	—
44	#41	C₂²×C₄₁₉	1	6	yes	—
45	#41	C₂²×C₄₁₉	1	1	yes	characteristic
46	C₂×Q₈	—	2	2	no	—
47	C₂×Q₈	C₂²×C₄₁₉	1	6	yes	—
48	Q₁₆	C₂²×C₄₁₉	1	3	yes	—
49	Q₁₆	C₂²×C₄₁₉	1	6	yes	—
50	SD₁₆	C₂²×C₄₁₉	1	6	yes	—
51	D₈	C₂²×C₄₁₉	1	1	yes	characteristic
52	M₁₆	C₂²×C₄₁₉	1	3	yes	—
53	C₂×C₈	C₂²×C₄₁₉	1	3	yes	—
54	Q₈	—	2	6	no	—
55	Q₈	D₄×C₄₁₉	1	1	yes	characteristic
56	Q₈	C₂³×C₄₁₉	1	6	yes	—
57	D₄	—	2	6	no	—
58	D₄	D₄×C₄₁₉	1	3	yes	—
59	D₄	C₂³×C₄₁₉	1	2	yes	—
60	C₂×C₄	—	2	6	no	—
61	C₂×C₄	—	2	6	no	—
62	C₂×C₄	C₂³×C₄₁₉	1	3	yes	—
63	C₈	C₂³×C₄₁₉	1	1	yes	characteristic
64	C₈	C₂³×C₄₁₉	1	3	yes	—
65	C₂²	—	2	2	no	—
66	C₂²	C₂×D₄×C₄₁₉	1	3	yes	—
67	C₄	—	2	6	no	—
68	C₄	C₂×D₄×C₄₁₉	1	3	yes	—
69	C₄	C₂⁴×C₄₁₉	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	4	2	no	—
72	C₂	C₂²×D₄×C₄₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₄₁₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁₉ ◁ #41×C₄₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×C₄₁₉	C₂	semidirect
2. #137×C₄₁₉	C₂	semidirect
3. #135×C₄₁₉	C₂	semidirect
4. C₂×Q₁₆×C₄₁₉	C₂	semidirect
5. #131×C₄₁₉	C₂	semidirect
36. C₄₁₉	#578	direct (N × H)
37. #578	C₄₁₉	direct (N × H)
38. 2₋¹⁺⁴	C₂×C₄₁₉	semidirect
39. #137	C₂×C₄₁₉	semidirect
40. #135	C₂×C₄₁₉	semidirect
41. C₂×Q₁₆	C₂×C₄₁₉	semidirect
42. #131	C₂×C₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×C₄₁₉	2
#137×C₄₁₉	2
#135×C₄₁₉	2
C₂×Q₁₆×C₄₁₉	2
#131×C₄₁₉	2
#578	419

Subgroups — #578×C419 #41000071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #578×C₄₁₉ #41000071