Subgroups — #41000420

Order 26816.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211675	C₂	1	1	yes	maximal, characteristic
2	Dic₄₁₉⋊C₈	C₂	1	2	yes	maximal, complemented
3	(C₂×D₄₁₉)⋊C₈	C₂	1	2	yes	maximal
4	#27211629	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₄₁₉⋊C₈	C₂²	1	2	yes	—
8	Dic₄₁₉⋊C₄	C₄	1	2	yes	—
9	(C₂×D₄₁₉)⋊C₄	C₄	1	2	yes	—
10	C₄×Dic₄₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₄₁₉	C₂²	1	2	yes	—
12	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄₁₉	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₄₁₉	C₂×C₄	1	2	yes	—
15	C₂×Dic₄₁₉	C₂×C₄	1	1	yes	characteristic
16	C₄×D₄₁₉	—	2	2	no	—
17	C₄₁₉⋊C₈	—	2	2	no	—
18	C₂×C₄×C₄₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
20	C₈×C₄₁₉	—	2	2	no	—
21	C₂²×C₄₁₉	C₂²×C₄	1	1	yes	characteristic
22	C₂×D₄₁₉	—	2	1	no	—
23	C₂×D₄₁₉	—	2	2	no	—
24	C₄×C₄₁₉	—	2	2	no	—
25	C₄×C₄₁₉	#41	1	2	yes	—
26	Dic₄₁₉	—	2	1	no	—
27	Dic₄₁₉	—	2	2	no	—
28	C₂×C₄₁₉	#117	1	1	yes	characteristic
29	C₂×C₄₁₉	#131	1	2	yes	—
30	D₄₁₉	—	4	1	no	—
31	C₄₁₉	#433	1	1	yes	characteristic, complemented, Sylow
32	#433	—	419	1	no	maximal, Sylow
33	#117	—	419	1	no	—
34	C₄⋊C₈	—	419	2	no	—
35	C₂²⋊C₈	—	419	2	no	—
36	#97	—	419	1	no	—
37	C₄×C₈	D₄₁₉	1	1	yes	characteristic, complemented
38	C₂²×C₄	—	419	1	no	—
39	C₂×C₈	—	419	2	no	—
40	C₂×C₈	C₂×D₄₁₉	1	2	yes	—
41	C₄⋊C₄	—	419	2	no	—
42	C₂²⋊C₄	—	419	2	no	—
43	C₄²	—	419	1	no	—
44	C₄²	C₂×D₄₁₉	1	1	yes	characteristic
45	C₂³	—	419	1	no	—
46	C₂×C₄	—	838	2	no	—
47	C₂×C₄	—	419	2	no	—
48	C₂×C₄	—	419	1	no	—
49	C₂×C₄	C₄×D₄₁₉	1	2	yes	—
50	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	838	2	no	—
53	C₂²	—	838	1	no	—
54	C₂²	—	838	2	no	—
55	C₂²	C₂×C₄×D₄₁₉	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	838	1	no	—
58	C₄	—	838	2	no	—
59	C₄	#20269411	1	2	yes	—
60	C₂	—	1676	1	no	—
61	C₂	#27211675	1	1	yes	characteristic
62	C₂	#27211724	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₁₉
commutator, socle	C₂²×C₄₁₉	C₂²×C₄
Fitting	C₄×C₈×C₄₁₉	C₂
prime radical	C₂²×D₄₁₉	C₂×C₄
cosocle	C₂×C₄×C₄₁₉	C₂³
hypercenter	C₄×C₈	D₄₁₉
hypocenter	C₄₁₉	#433
LCS	G ▷ C₂²×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. Dic₄₁₉⋊C₈	C₂	semidirect
4. #27211629	C₂	semidirect
5. C₄×C₈×C₄₁₉	C₂	semidirect
31. C₄₁₉	#433	semidirect
37. C₄×C₈	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
#27211675	2
Dic₄₁₉⋊C₈	2
(C₂×D₄₁₉)⋊C₈	2
#27211629	2
C₄×C₈×C₄₁₉	2
#433	419