Subgroups — #41000313

Order 26816.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269394	C₂	1	2	yes	maximal
2	C₄²×D₄₁₉	C₂	1	1	yes	maximal, characteristic
3	#27211629	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₄₁₉	C₂	1	2	yes	maximal, complemented
5	#97×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₁₉	C₄	1	2	yes	—
7	C₂×C₄×D₄₁₉	C₂²	1	1	yes	characteristic
8	#20269394	C₄	1	8	yes	complemented
9	C₂×C₄₁₉⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₄₁₉	C₄	1	2	yes	—
11	C₄×Dic₄₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₄₁₉	C₂²	1	2	yes	—
13	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₄₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₄₁₉	C₂×C₄	1	2	yes	—
17	C₄×D₄₁₉	—	2	4	no	—
18	C₄×D₄₁₉	C₂×C₄	1	4	yes	—
19	C₄₁₉⋊C₈	C₂×C₄	1	4	yes	complemented
20	C₂×C₄×C₄₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
22	C₈×C₄₁₉	C₂×C₄	1	4	yes	complemented
23	C₂²×C₄₁₉	C₂²×C₄	1	1	yes	characteristic
24	C₂×D₄₁₉	—	2	2	no	—
25	C₂×D₄₁₉	C₄²	1	2	yes	—
26	C₄×C₄₁₉	—	2	2	no	—
27	C₄×C₄₁₉	C₂²×C₄	1	2	yes	—
28	Dic₄₁₉	C₄²	1	2	yes	—
29	Dic₄₁₉	M₁₆	1	4	yes	complemented
30	C₂×C₄₁₉	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄₁₉	C₂×M₁₆	1	2	yes	—
32	D₄₁₉	—	2	2	no	—
33	C₄₁₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×M₁₆	—	419	1	no	maximal, Sylow
35	C₂×M₁₆	—	419	2	no	—
36	C₂×C₄²	—	419	1	no	—
37	#97	—	419	1	no	—
38	#97	D₄₁₉	1	1	yes	characteristic, complemented
39	C₄×C₈	—	419	2	no	—
40	C₂²×C₄	—	419	2	no	—
41	C₂²×C₄	—	419	1	no	—
42	M₁₆	—	419	8	no	—
43	C₂×C₈	—	419	2	no	—
44	C₂×C₈	C₂×D₄₁₉	1	2	yes	—
45	C₄²	—	419	2	no	—
46	C₄²	—	419	1	no	—
47	C₄²	C₂×D₄₁₉	1	1	yes	characteristic
48	C₂³	—	419	1	no	—
49	C₂×C₄	—	838	4	no	—
50	C₂×C₄	—	419	1	no	—
51	C₂×C₄	—	419	2	no	—
52	C₂×C₄	—	419	4	no	—
53	C₂×C₄	C₄×D₄₁₉	1	2	yes	—
54	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
55	C₈	—	419	4	no	—
56	C₈	C₄×D₄₁₉	1	4	yes	complemented
57	C₂²	—	838	2	no	—
58	C₂²	—	419	2	no	—
59	C₂²	C₂×C₄×D₄₁₉	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	419	2	no	—
62	C₄	—	419	4	no	—
63	C₄	C₂×C₄×D₄₁₉	1	2	yes	—
64	C₂	—	838	2	no	—
65	C₂	C₄²×D₄₁₉	1	1	yes	characteristic
66	C₂	M₁₆×D₄₁₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₁₉
commutator	C₂×C₄₁₉	C₂×C₄²
Fitting	#97×C₄₁₉	C₂
socle	C₂²×C₄₁₉	C₂²×C₄
prime radical	C₂²×D₄₁₉	C₂×C₄
cosocle	C₂×C₄×C₄₁₉	C₂³
hypercenter	#97	D₄₁₉
hypocenter	C₄₁₉	C₄×M₁₆
LCS	G ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. #27211629	C₂	semidirect
4. C₈×Dic₄₁₉	C₂	semidirect
5. #97×C₄₁₉	C₂	semidirect
8. #20269394	C₄	semidirect
19. C₄₁₉⋊C₈	C₂×C₄	semidirect
22. C₈×C₄₁₉	C₂×C₄	semidirect
29. Dic₄₁₉	M₁₆	semidirect
33. C₄₁₉	C₄×M₁₆	semidirect
38. #97	D₄₁₉	semidirect
56. C₈	C₄×D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269394	2
C₄²×D₄₁₉	2
#27211629	2
C₈×Dic₄₁₉	2
#97×C₄₁₉	2
C₄×M₁₆	419