Subgroups — #41000389

Order 26816.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#27211677	C₂	1	1	yes	maximal, characteristic
3	Dic₄₁₉⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₄₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₁₉⋊C₈	—	2	2	no	—
6	C₂×C₄₁₉⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₄₁₉	C₄	1	2	yes	—
8	C₂²⋊Dic₄₁₉	C₄	1	2	yes	—
9	C₄×Dic₄₁₉	C₂²	1	2	yes	—
10	C₂²×C₄×C₄₁₉	C₂²	1	1	yes	characteristic
11	M₁₆×C₄₁₉	—	2	2	no	—
12	C₂×C₈×C₄₁₉	C₂²	1	2	yes	—
13	C₂×Dic₄₁₉	C₂×C₄	1	4	yes	—
14	C₄₁₉⋊C₈	—	2	2	no	—
15	C₂³×C₄₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₄₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₄₁₉	D₄	1	2	yes	—
18	C₂×C₄×C₄₁₉	Q₈	1	2	yes	—
19	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
20	C₈×C₄₁₉	—	2	2	no	—
21	C₂²×C₄₁₉	—	2	2	no	—
22	C₂²×C₄₁₉	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄₁₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₄₁₉	C₄⋊C₄	1	2	yes	—
25	C₄×C₄₁₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₄₁₉	C₂×Q₈	1	1	yes	characteristic
27	Dic₄₁₉	—	2	4	no	—
28	C₂×C₄₁₉	—	2	2	no	—
29	C₂×C₄₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄₁₉	#131	1	2	yes	—
31	C₄₁₉	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	419	1	no	maximal, Sylow
33	C₂×M₁₆	D₄₁₉	1	1	yes	characteristic
34	C₂²×C₈	—	419	1	no	—
35	#117	—	419	1	no	—
36	C₄⋊C₈	—	419	4	no	—
37	C₂²×C₄	C₂×D₄₁₉	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	C₂×C₈	—	838	2	no	—
40	C₂×C₈	—	419	2	no	—
41	C₂×C₈	C₂×D₄₁₉	1	2	yes	—
42	C₄⋊C₄	—	419	2	no	—
43	C₂²⋊C₄	—	419	2	no	—
44	C₄²	—	419	2	no	—
45	C₂³	C₄×D₄₁₉	1	1	yes	characteristic
46	C₂×C₄	—	419	4	no	—
47	C₂×C₄	C₄×D₄₁₉	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₄₁₉	1	2	yes	—
49	C₂×C₄	Dic₈₃₈	1	2	yes	—
50	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	838	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	Dic₄₁₉⋊C₄	1	2	yes	—
55	C₂²	C₂×C₄×D₄₁₉	1	1	yes	characteristic
56	C₄	—	838	4	no	—
57	C₄	Dic₄₁₉⋊C₄	1	2	yes	—
58	C₄	C₂×C₂²⋊D₄₁₉	1	1	yes	characteristic
59	C₄	C₂×Dic₈₃₈	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×Dic₄₁₉⋊C₄	1	1	yes	characteristic
62	C₂	#27211726	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₁₉
commutator, socle	C₂²×C₄₁₉	C₂²×C₄
Fitting	C₂×M₁₆×C₄₁₉	C₂
prime radical	C₂³×C₄₁₉	C₂×C₄
cosocle	C₂×C₄×C₄₁₉	C₂³
hypercenter	C₂×M₁₆	D₄₁₉
hypocenter	C₄₁₉	#424
LCS	G ▷ C₂²×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. Dic₄₁₉⋊C₈	C₂	semidirect
31. C₄₁₉	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄₁₉⋊C₈	2
#27211677	2
Dic₄₁₉⋊C₈	2
C₂×M₁₆×C₄₁₉	2
#424	419