Subgroups — #41000746

Order 26816.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269398	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₃₅₂	C₂	1	1	yes	maximal, characteristic
3	#27211707	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₇₆	C₂²	1	2	yes	—
7	#20269398	—	2	4	no	—
8	D₃₃₅₂	—	2	2	no	—
9	C₂×C₄₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	(C₂×D₄₁₉)⋊C₄	—	2	4	no	—
11	C₄×Dic₄₁₉	C₂²	1	1	yes	characteristic
12	C₂×Q₈×C₄₁₉	C₂²	1	2	yes	—
13	Q₁₆×C₄₁₉	—	2	2	no	—
14	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
15	C₂²×D₄₁₉	—	2	2	no	—
16	D₁₆₇₆	—	2	2	no	—
17	D₁₆₇₆	—	2	2	no	—
18	C₂×Dic₄₁₉	D₄	1	2	yes	—
19	C₄₁₉⋊C₈	D₄	1	2	yes	—
20	Q₈×C₄₁₉	—	2	2	no	—
21	Q₈×C₄₁₉	—	2	2	no	—
22	C₂×C₄×C₄₁₉	—	2	2	no	—
23	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
24	C₈×C₄₁₉	D₄	1	2	yes	—
25	C₂²×C₄₁₉	C₂×D₄	1	1	yes	characteristic
26	C₂×D₄₁₉	—	4	2	no	—
27	C₂×D₄₁₉	—	4	4	no	—
28	C₄×C₄₁₉	—	4	2	no	—
29	C₄×C₄₁₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₁₉	C₂×D₄	1	1	yes	characteristic
31	Dic₄₁₉	—	2	2	no	—
32	C₂×C₄₁₉	#127	1	1	yes	characteristic
33	C₂×C₄₁₉	#135	1	2	yes	—
34	D₄₁₉	—	8	2	no	—
35	C₄₁₉	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	419	1	no	maximal, Sylow
37	C₂×Q₁₆	D₄₁₉	1	1	yes	characteristic, complemented
38	C₂×SD₁₆	—	419	2	no	—
39	C₂×D₈	—	419	1	no	—
40	#124	—	419	2	no	—
41	C₄×C₈	—	419	1	no	—
42	C₂×Q₈	C₂×D₄₁₉	1	2	yes	—
43	C₂×D₄	—	419	2	no	—
44	Q₁₆	—	2	2	no	—
45	SD₁₆	—	838	4	no	—
46	D₈	—	838	2	no	—
47	C₂×C₈	—	419	1	no	—
48	C₂×C₈	C₂×D₄₁₉	1	1	yes	characteristic
49	C₂²⋊C₄	—	838	4	no	—
50	C₄²	—	419	1	no	—
51	C₂³	—	838	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	D₄	—	838	2	no	—
55	D₄	—	838	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	419	2	no	—
58	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
59	C₈	—	419	2	no	—
60	C₈	C₂²⋊D₄₁₉	1	2	yes	—
61	C₂²	—	1676	2	no	—
62	C₂²	—	1676	4	no	—
63	C₂²	D₄×D₄₁₉	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	838	2	no	—
66	C₄	D₄×D₄₁₉	1	1	yes	characteristic
67	C₄	C₂×C₂²⋊D₄₁₉	1	1	yes	characteristic
68	C₂	—	3352	2	no	—
69	C₂	#27211714	1	1	yes	characteristic
70	C₂	#27211741	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₄₁₉
commutator, cosocle	C₂×C₄×C₄₁₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₁₉
Fitting	C₂×Q₁₆×C₄₁₉	C₂
socle	C₂²×C₄₁₉	C₂×D₄
prime radical	C₂×D₃₃₅₂	C₂
hypercenter	C₂×Q₁₆	D₄₁₉
hypocenter	C₄₁₉	#495
LCS	G ▷ C₂×C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#20269398	C₂	semidirect
3. #27211707	C₂	semidirect
4. C₈×Dic₄₁₉	C₂	semidirect
5. C₂×Q₁₆×C₄₁₉	C₂	semidirect
35. C₄₁₉	#495	semidirect
37. C₂×Q₁₆	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269398	2
C₂×D₃₃₅₂	2
#27211707	2
C₈×Dic₄₁₉	2
C₂×Q₁₆×C₄₁₉	2
#495	419