Subgroups — #41000113

Order 26816.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₄₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁₉⋊C₈	—	2	1	no	—
5	C₂×C₄₁₉⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄₁₉⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×C₄₁₉	—	2	1	no	—
8	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
9	C₄₁₉⋊C₈	—	2	1	no	—
10	C₄₁₉⋊C₈	—	4	1	no	—
11	C₄₁₉⋊C₈	C₈	1	2	yes	complemented
12	C₂×C₄×C₄₁₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₄₁₉	D₄	1	1	yes	characteristic
14	C₂×C₄×C₄₁₉	Q₈	1	1	yes	characteristic
15	C₈×C₄₁₉	—	4	1	no	—
16	C₂²×C₄₁₉	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₄₁₉	—	2	1	no	—
18	C₄×C₄₁₉	C₂×C₈	1	1	yes	characteristic
19	C₄×C₄₁₉	M₁₆	1	1	yes	characteristic
20	C₄×C₄₁₉	D₈	1	1	yes	characteristic
21	C₄×C₄₁₉	Q₁₆	1	1	yes	characteristic
22	C₂×C₄₁₉	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₄₁₉	#107	1	1	yes	characteristic
24	C₂×C₄₁₉	#108	1	1	yes	characteristic
25	C₄₁₉	#335	1	1	yes	characteristic, complemented, Sylow
26	#335	—	419	1	no	maximal, Sylow
27	C₄⋊C₈	—	419	1	no	—
28	C₄⋊C₈	D₄₁₉	1	1	yes	characteristic
29	C₄×C₈	—	419	1	no	—
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	—	838	1	no	—
32	C₂×C₈	—	419	1	no	—
33	C₂×C₈	—	419	1	no	—
34	C₄²	C₂×D₄₁₉	1	1	yes	characteristic
35	C₂×C₄	C₄×D₄₁₉	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊D₄₁₉	1	1	yes	characteristic
37	C₂×C₄	Dic₈₃₈	1	1	yes	characteristic
38	C₈	—	4	1	no	—
39	C₈	—	838	1	no	—
40	C₈	—	1676	1	no	—
41	C₈	—	419	2	no	—
42	C₂²	Dic₄₁₉⋊C₄	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	C₈×D₄₁₉	1	1	yes	characteristic
45	C₄	#20269394	1	1	yes	characteristic
46	C₄	#20269396	1	1	yes	characteristic
47	C₄	#20269401	1	1	yes	characteristic
48	C₂	Dic₄₁₉⋊C₈	1	1	yes	characteristic
49	C₂	#27211651	1	1	yes	characteristic
50	C₂	#27211653	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₄₁₉
commutator	C₄×C₄₁₉	C₂×C₈
Frattini	C₄²	C₂×D₄₁₉
Fitting	C₄⋊C₈×C₄₁₉	C₂
socle, prime radical	C₂²×C₄₁₉	C₄⋊C₄
cosocle	C₄²×C₄₁₉	C₂²
hypercenter	C₄⋊C₈	D₄₁₉
hypocenter	C₄₁₉	#335
LCS	G ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
11. C₄₁₉⋊C₈	C₈	semidirect
25. C₄₁₉	#335	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₄₁₉⋊C₈)	2
C₄×C₄₁₉⋊C₈	2
C₄⋊C₈×C₄₁₉	2
#335	419