Subgroups — #41000144

Order 26816.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211724	C₂	1	1	yes	maximal, characteristic
2	#27211658	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269411	C₄	1	1	yes	characteristic
5	#20269395	C₄	1	1	yes	characteristic
6	#20269394	—	2	1	no	—
7	C₈×D₄₁₉	—	2	1	no	—
8	C₄₁₉⋊C₁₆	—	2	1	no	—
9	C₂×C₈×C₄₁₉	C₂²	1	1	yes	characteristic
10	C₁₆×C₄₁₉	—	2	1	no	—
11	Dic₈₃₈	C₈	1	1	yes	characteristic
12	D₁₆₇₆	C₈	1	1	yes	characteristic
13	C₂²⋊D₄₁₉	—	2	1	no	—
14	C₄×D₄₁₉	—	2	1	no	—
15	C₄₁₉⋊C₈	—	2	1	no	—
16	C₂×C₄×C₄₁₉	C₂×C₄	1	1	yes	characteristic
17	C₈×C₄₁₉	D₄	1	1	yes	characteristic
18	C₈×C₄₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₄₁₉	M₁₆	1	1	yes	characteristic
20	C₂×D₄₁₉	—	2	1	no	—
21	C₄×C₄₁₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₄₁₉	C₂×C₈	1	1	yes	characteristic
23	Dic₄₁₉	—	2	1	no	—
24	C₂×C₄₁₉	—	2	1	no	—
25	C₂×C₄₁₉	C₂²⋊C₈	1	1	yes	characteristic
26	D₄₁₉	—	4	1	no	—
27	C₄₁₉	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	419	1	no	maximal, Sylow
29	#131	—	419	1	no	—
30	M₃₂	—	419	1	no	—
31	C₂×C₁₆	D₄₁₉	1	1	yes	characteristic, complemented
32	#41	—	419	1	no	—
33	M₁₆	—	838	1	no	—
34	M₁₆	—	419	1	no	—
35	C₂×C₈	—	838	1	no	—
36	C₂×C₈	C₂×D₄₁₉	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	838	1	no	—
39	Q₈	—	419	1	no	—
40	D₄	—	838	1	no	—
41	D₄	—	419	1	no	—
42	C₂×C₄	—	838	1	no	—
43	C₂×C₄	C₄×D₄₁₉	1	1	yes	characteristic
44	C₈	—	838	1	no	—
45	C₈	C₂²⋊D₄₁₉	1	1	yes	characteristic
46	C₈	D₁₆₇₆	1	1	yes	characteristic
47	C₂²	—	838	1	no	—
48	C₂²	#20269394	1	1	yes	characteristic
49	C₄	—	838	1	no	—
50	C₄	(C₂×D₄₁₉)⋊C₄	1	1	yes	characteristic
51	C₄	C₈×D₄₁₉	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1676	1	no	—
54	C₂	(C₂×D₄₁₉)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₆₇₆
commutator	C₄×C₄₁₉	C₂×C₈
Frattini	C₂×C₈	C₂×D₄₁₉
Fitting	C₂×C₁₆×C₄₁₉	C₂
socle	C₂×C₄₁₉	C₂²⋊C₈
prime radical	#20269411	C₄
cosocle	C₂×C₈×C₄₁₉	C₂²
hypercenter	C₂×C₁₆	D₄₁₉
hypocenter	C₄₁₉	#350
LCS	G ▷ C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #27211658	C₂	semidirect
3. C₂×C₁₆×C₄₁₉	C₂	semidirect
27. C₄₁₉	#350	semidirect
31. C₂×C₁₆	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
#27211724	2
#27211658	2
C₂×C₁₆×C₄₁₉	2
#350	419