Subgroups — #456×C₄₁₉ #40999949

Order 26816.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×C₄₁₉	C₂	1	1	yes	maximal, characteristic
2	#137×C₄₁₉	C₂	1	2	yes	maximal, complemented
3	C₄⋊Q₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#104×C₄₁₉	C₂	1	2	yes	maximal, complemented
5	#101×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₄₁₉	—	2	2	no	—
7	#41×C₄₁₉	—	2	2	no	—
8	#41×C₄₁₉	C₂²	1	2	yes	—
9	C₂×Q₈×C₄₁₉	—	2	2	no	—
10	C₂×Q₈×C₄₁₉	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₄₁₉	C₂²	1	1	yes	characteristic
12	Q₁₆×C₄₁₉	—	2	2	no	—
13	SD₁₆×C₄₁₉	—	2	2	no	—
14	M₁₆×C₄₁₉	C₂²	1	2	yes	—
15	C₄⋊C₄×C₄₁₉	—	2	2	no	—
16	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
17	Q₈×C₄₁₉	—	2	2	no	—
18	Q₈×C₄₁₉	—	2	2	no	—
19	Q₈×C₄₁₉	—	2	2	no	—
20	Q₈×C₄₁₉	D₄	1	2	yes	—
21	D₄×C₄₁₉	—	2	2	no	—
22	D₄×C₄₁₉	—	2	2	no	—
23	D₄×C₄₁₉	D₄	1	2	yes	—
24	C₂×C₄×C₄₁₉	—	2	4	no	—
25	C₂×C₄×C₄₁₉	—	2	1	no	—
26	C₂×C₄×C₄₁₉	—	2	2	no	—
27	C₂×C₄×C₄₁₉	—	2	1	no	—
28	C₂×C₄×C₄₁₉	D₄	1	2	yes	—
29	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
30	C₈×C₄₁₉	—	2	2	no	—
31	C₂²×C₄₁₉	—	2	2	no	—
32	C₂²×C₄₁₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₄₁₉	—	2	2	no	—
34	C₄×C₄₁₉	—	2	2	no	—
35	C₄×C₄₁₉	—	2	2	no	—
36	C₄×C₄₁₉	—	4	1	no	—
37	C₄×C₄₁₉	C₂×D₄	1	2	yes	—
38	C₂×C₄₁₉	—	2	1	no	—
39	C₂×C₄₁₉	—	4	2	no	—
40	C₂×C₄₁₉	#120	1	1	yes	characteristic
41	C₄₁₉	#456	1	1	yes	characteristic, direct product factor, Sylow
42	#456	C₄₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	2₋¹⁺⁴	C₂×C₄₁₉	1	1	yes	characteristic
44	#137	C₂×C₄₁₉	1	2	yes	complemented
45	C₄⋊Q₈	C₂×C₄₁₉	1	1	yes	characteristic, complemented
46	#104	C₂×C₄₁₉	1	2	yes	complemented
47	#101	C₂×C₄₁₉	1	1	yes	characteristic, complemented
48	#41	—	2	2	no	—
49	#41	—	2	2	no	—
50	#41	C₂²×C₄₁₉	1	2	yes	—
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	C₂²×C₄₁₉	1	1	yes	characteristic
53	C₂×Q₈	C₂²×C₄₁₉	1	1	yes	characteristic
54	Q₁₆	—	2	2	no	—
55	SD₁₆	—	2	2	no	—
56	M₁₆	C₂²×C₄₁₉	1	2	yes	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄²	C₂²×C₄₁₉	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	2	2	no	—
62	Q₈	D₄×C₄₁₉	1	2	yes	—
63	D₄	—	2	2	no	—
64	D₄	—	2	2	no	—
65	D₄	D₄×C₄₁₉	1	2	yes	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	D₄×C₄₁₉	1	2	yes	—
71	C₂×C₄	C₂³×C₄₁₉	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	C₂×D₄×C₄₁₉	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	4	1	no	—
79	C₄	C₂×D₄×C₄₁₉	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	#120×C₄₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁₉	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₄₁₉
prime radical	2₋¹⁺⁴×C₄₁₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁₉ ◁ C₂×C₄×C₄₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #137×C₄₁₉	C₂	semidirect
3. C₄⋊Q₈×C₄₁₉	C₂	semidirect
4. #104×C₄₁₉	C₂	semidirect
5. #101×C₄₁₉	C₂	semidirect
41. C₄₁₉	#456	direct (N × H)
42. #456	C₄₁₉	direct (N × H)
44. #137	C₂×C₄₁₉	semidirect
45. C₄⋊Q₈	C₂×C₄₁₉	semidirect
46. #104	C₂×C₄₁₉	semidirect
47. #101	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
C₄⋊Q₈×C₄₁₉	2
#104×C₄₁₉	2
#101×C₄₁₉	2
#456	419

Subgroups — #456×C419 #40999949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #456×C₄₁₉ #40999949