Subgroups — #41000145

Order 26816.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211658	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×C₄₁₉	C₂	1	1	yes	maximal, characteristic
4	C₄₁₉⋊C₁₆	—	2	1	no	—
5	C₄₁₉⋊C₁₆	—	2	1	no	—
6	#41×C₄₁₉	C₄	1	1	yes	characteristic
7	M₁₆×C₄₁₉	—	2	1	no	—
8	M₁₆×C₄₁₉	C₄	1	1	yes	characteristic
9	C₂×C₈×C₄₁₉	—	2	1	no	—
10	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
11	Q₈×C₄₁₉	C₈	1	1	yes	characteristic
12	D₄×C₄₁₉	—	2	1	no	—
13	D₄×C₄₁₉	C₈	1	1	yes	characteristic
14	C₂×C₄×C₄₁₉	—	2	1	no	—
15	C₂×C₄×C₄₁₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₄₁₉	—	2	1	no	—
17	C₈×C₄₁₉	D₄	1	1	yes	characteristic
18	C₈×C₄₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₄₁₉	—	2	1	no	—
20	C₂²×C₄₁₉	M₁₆	1	1	yes	characteristic
21	C₄×C₄₁₉	—	2	1	no	—
22	C₄×C₄₁₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₄₁₉	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄₁₉	—	2	1	no	—
25	C₂×C₄₁₉	—	4	1	no	—
26	C₂×C₄₁₉	C₂²⋊C₈	1	1	yes	characteristic
27	C₄₁₉	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	419	1	no	maximal, Sylow
29	#131	D₄₁₉	1	1	yes	characteristic
30	M₃₂	—	419	1	no	—
31	C₂×C₁₆	—	419	1	no	—
32	#41	Dic₄₁₉	1	1	yes	characteristic
33	M₁₆	—	2	1	no	—
34	M₁₆	Dic₄₁₉	1	1	yes	characteristic
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	C₂×D₄₁₉	1	1	yes	characteristic
37	C₁₆	—	838	1	no	—
38	C₁₆	—	838	1	no	—
39	Q₈	C₄₁₉⋊C₈	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	C₄₁₉⋊C₈	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×Dic₄₁₉	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	C₂²⋊D₄₁₉	1	1	yes	characteristic
46	C₈	C₂²⋊D₄₁₉	1	1	yes	characteristic
47	C₂²	—	2	1	no	—
48	C₂²	#20269395	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂²⋊Dic₄₁₉	1	1	yes	characteristic
51	C₄	C₂×C₄₁₉⋊C₈	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊(C₄₁₉⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₄₁₉
commutator	C₄×C₄₁₉	C₂×C₈
Frattini	C₂×C₈	C₂×D₄₁₉
Fitting	#131×C₄₁₉	C₂
socle	C₂×C₄₁₉	C₂²⋊C₈
prime radical	#41×C₄₁₉	C₄
cosocle	C₂×C₈×C₄₁₉	C₂²
hypercenter	#131	D₄₁₉
hypocenter	C₄₁₉	#350
LCS	G ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #27211658	C₂	semidirect
2. C₂×C₄₁₉⋊C₁₆	C₂	semidirect
27. C₄₁₉	#350	semidirect

Maximal subgroups

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

subgroup	index
#27211658	2
C₂×C₄₁₉⋊C₁₆	2
#131×C₄₁₉	2
#350	419