Subgroups — #34062739

Order 19168.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23751077	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₅₉₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#23751067	C₂	1	1	yes	maximal, characteristic, complemented
4	#23751066	C₂	1	1	yes	maximal, characteristic, complemented
5	#23751063	C₂	1	1	yes	maximal, characteristic, complemented
6	#23751061	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₅₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅₉₉	—	2	1	no	—
9	Dic₁₁₉₈	C₂²	1	1	yes	characteristic, complemented
10	D₂₃₉₆	C₂²	1	1	yes	characteristic
11	C₂²⋊D₅₉₉	—	2	1	no	—
12	C₂²⋊D₅₉₉	—	2	1	no	—
13	C₂×Dic₅₉₉	—	2	1	no	—
14	C₄×D₅₉₉	C₂²	1	1	yes	characteristic
15	C₅₉₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
16	D₄×C₅₉₉	C₂²	1	1	yes	characteristic
17	D₄×C₅₉₉	C₂²	1	1	yes	characteristic, complemented
18	C₈×C₅₉₉	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₅₉₉	—	2	1	no	—
20	C₂²×C₅₉₉	—	2	1	no	—
21	C₂×D₅₉₉	—	2	1	no	—
22	C₂×D₅₉₉	—	4	2	no	—
23	C₂×D₅₉₉	D₄	1	1	yes	characteristic
24	C₄×C₅₉₉	C₂³	1	1	yes	characteristic
25	Dic₅₉₉	—	2	1	no	—
26	Dic₅₉₉	D₄	1	1	yes	characteristic
27	C₂×C₅₉₉	—	4	1	no	—
28	C₂×C₅₉₉	—	4	1	no	—
29	C₂×C₅₉₉	C₂×D₄	1	1	yes	characteristic
30	D₅₉₉	—	2	1	no	—
31	D₅₉₉	—	4	1	no	—
32	C₅₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	599	1	no	maximal, Sylow
34	#41	—	599	1	no	—
35	C₂×D₄	—	599	1	no	—
36	SD₁₆	—	599	1	no	—
37	SD₁₆	—	599	1	no	—
38	D₈	—	599	1	no	—
39	D₈	D₅₉₉	1	1	yes	characteristic, complemented
40	M₁₆	—	599	1	no	—
41	C₂³	—	1198	1	no	—
42	Q₈	—	599	1	no	—
43	D₄	—	599	1	no	—
44	D₄	—	1198	1	no	—
45	D₄	—	1198	1	no	—
46	D₄	C₂×D₅₉₉	1	1	yes	characteristic
47	D₄	C₂×D₅₉₉	1	1	yes	characteristic, complemented
48	C₂×C₄	—	599	1	no	—
49	C₂×C₄	—	1198	1	no	—
50	C₈	—	599	1	no	—
51	C₈	C₂×D₅₉₉	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	599	1	no	—
55	C₂²	—	2396	2	no	—
56	C₂²	—	1198	1	no	—
57	C₄	—	599	1	no	—
58	C₄	—	1198	1	no	—
59	C₄	C₂²×D₅₉₉	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	2396	1	no	—
63	C₂	—	1198	1	no	—
64	C₂	D₄×D₅₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₅₉₉
commutator, cosocle	C₄×C₅₉₉	C₂³
Frattini	C₄	C₂²×D₅₉₉
Fitting	D₈×C₅₉₉	C₂
socle	C₂×C₅₉₉	C₂×D₄
hypercenter	D₈	D₅₉₉
hypocenter	C₅₉₉	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅₉₉ ▷ C₂×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #23751077	C₂	semidirect
2. D₄×D₅₉₉	C₂	semidirect
3. #23751067	C₂	semidirect
4. #23751066	C₂	semidirect
5. #23751063	C₂	semidirect
6. #23751061	C₂	semidirect
7. D₈×C₅₉₉	C₂	semidirect
9. Dic₁₁₉₈	C₂²	semidirect
15. C₅₉₉⋊C₈	C₂²	semidirect
17. D₄×C₅₉₉	C₂²	semidirect
18. C₈×C₅₉₉	C₂²	semidirect
32. C₅₉₉	AGL(1,ℤ/8)	semidirect
39. D₈	D₅₉₉	semidirect
47. D₄	C₂×D₅₉₉	semidirect
51. C₈	C₂×D₅₉₉	semidirect

Maximal subgroups

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

subgroup	index
#23751077	2
D₄×D₅₉₉	2
#23751067	2
#23751066	2
#23751063	2
#23751061	2
D₈×C₅₉₉	2
AGL(1,ℤ/8)	599